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Authors Basich Whitney • Brown • Dawson • Gonsalves • Silbey • Vielhaber
Photo Credits Cover Jupiter Images; iv (tl bl br) File Photo, (tc tr) The McGraw-Hill Companies, (cl c) Doug Martin, (cr) Aaron Haupt; v (1 2 3 4 6 7 8 9 11 12) The McGraw-Hill Companies, (5 10 13 14) File Photo; viii CORBIS; viii Mitchell Funk/Getty Images; ix S. Alden/PhotoLink/Getty Images; xi Peter Barritt/Alamy; 2–3 Mike Brinson/Getty Images; 9 (Frame) Getty Images; 9 (inset) Ryan McVay/Getty Images; 15 Comstock/CORBIS; 16 CORBIS; 25 Jerry Irwin/Photo Researchers, Inc.; 34 42 CORBIS; 52 Getty Images; 53 CORBIS; 061 Ingram Publishing/ SuperStock; 068 (b) CORBIS, (t) Rudi Von Briel/PhotoEdit; 70 71 CORBIS; 77 Getty Images; 78 CORBIS; 79 Comstock/Imagestate; 80 CORBIS; 95 Punchstock
Copyright © 2008 by The McGraw-Hill Companies, Inc. All rights reserved. Except as permitted under the United States Copyright Act, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without prior permission of the publisher. Send all inquiries to: Glencoe/McGraw-Hill 8787 Orion Place Columbus, OH 43240-4027 ISBN: 978-0-07-878214 MHID: 0-07-878214-7 Printed in the United States of America. 1 2 3 4 5 6 7 8 9 10 055/027 16 15 14 13 12 11 10 09 08 07
California Math Triumphs Volume 6B
California Math Triumphs Volume 1 Place Value and Basic Number Skills 1A Chapter 1 Counting 1A Chapter 2 Place Value 1A Chapter 3 Addition and Subtraction 1B Chapter 4 Multiplication 1B Chapter 5 Division 1B Chapter 6 Integers Volume 2 Fractions and Decimals 2A Chapter 1 Parts of a Whole 2A Chapter 2 Equivalence of Fractions 2B Chapter 3 Operations with Fractions 2B Chapter 4 Positive and Negative Fractions and Decimals
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Volume 3 Ratios, Rates, and Percents 3A Chapter 1 Ratios and Rates 3A Chapter 2 Percents, Fractions, and Decimals 3B Chapter 3 Using Percents 3B Chapter 4 Rates and Proportional Reasoning Volume 4 The Core Processes of Mathematics 4A Chapter 1 Operations and Equality 4A Chapter 2 Math Fundamentals 4B Chapter 3 Math Expressions 4B Chapter 4 Linear Equations 4B Chapter 5 Inequalities Volume 5 Functions and Equations 5A Chapter 1 Patterns and Relationships 5A Chapter 2 Graphing 5B Chapter 3 Proportional Relationships 5B Chapter 4 The Relationship Between Graphs and Functions Volume 6 Measurement 6A Chapter 1 How Measurements Are Made 6A Chapter 2 Length and Area in the Real World 6B Chapter 3 Exact Measures in Geometry 6B Chapter 4 Angles and Circles iii
Authors and Consultants AUTHORS
Frances Basich Whitney
Kathleen M. Brown
Dixie Dawson
Project Director, Mathematics K–12 Santa Cruz County Office of Education Capitola, California
Math Curriculum Staff Developer Washington Middle School Long Beach, California
Math Curriculum Leader Long Beach Unified Long Beach, California
Philip Gonsalves
Robyn Silbey
Kathy Vielhaber
Mathematics Coordinator Alameda County Office of Education Hayward, California
Math Specialist Montgomery County Public Schools Gaithersburg, Maryland
Mathematics Consultant St. Louis, Missouri
Viken Hovsepian Professor of Mathematics Rio Hondo College Whittier, California
Dinah Zike Educational Consultant, Dinah-Might Activities, Inc. San Antonio, Texas
CONSULTANTS Assessment Donna M. Kopenski, Ed.D. Math Coordinator K–5 City Heights Educational Collaborative San Diego, California
iv
Instructional Planning and Support
ELL Support and Vocabulary
Beatrice Luchin
ReLeah Cossett Lent
Mathematics Consultant League City, Texas
Author/Educational Consultant Alford, Florida
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
CONTRIBUTING AUTHORS
California Advisory Board CALIFORNIA ADVISORY BOARD
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Glencoe wishes to thank the following professionals for their invaluable feedback during the development of the program. They reviewed the table of contents, the prototype of the Student Study Guide, the prototype of the Teacher Wraparound Edition, and the professional development plan.
Linda Anderson
Cheryl L. Avalos
Bonnie Awes
Kathleen M. Brown
4th/5th Grade Teacher Oliveira Elementary School, Fremont, California
Mathematics Consultant Retired Teacher Hacienda Heights, California
Teacher, 6th Grade Math Monroe Clark Middle School San Diego, California
Math Curriculum Staff Developer Washington Middle School Long Beach, California
Carol Cronk
Audrey M. Day
Jill Fetters
Grant A. Fraser, Ph.D.
Mathematics Program Specialist San Bernardino City Unified School District San Bernardino, California
Classroom Teacher Rosa Parks Elementary School San Diego, California
Math Teacher Tevis Jr. High School Bakersfield, California
Professor of Mathematics California State University, Los Angeles Los Angeles, California
Eric Kimmel
Donna M. Kopenski, Ed.D.
Michael A. Pease
Chuck Podhorsky, Ph.D.
Mathematics Department Chair Frontier High School Bakersfield, California
Math Coordinator K–5 City Heights Educational Collaborative San Diego, California
Instructional Math Coach Aspire Public Schools Oakland, California
Math Director City Heights Educational Collaborative San Diego, California
Arthur K. Wayman, Ph.D.
Frances Basich Whitney
Mario Borrayo
Melissa Bray
Professor Emeritus California State University, Long Beach Long Beach, California
Project Director, Mathematics K–12 Santa Cruz County Office of Education Capitola, CA
Teacher Rosa Parks Elementary San Diego, California
K–8 Math Resource Teacher Modesto City Schools Modesto, California
v
California Reviewers CALIFORNIA REVIEWERS Each California Reviewer reviewed at least two chapters of the Student Study Guides, providing feedback and suggestions for improving the effectiveness of the mathematics instruction. Melody McGuire
Math Teacher California College Preparatory Academy Oakland, California
6th and 7th Grade Math Teacher McKinleyville Middle School McKinleyville, California
Eppie Leamy Chung
Monica S. Patterson
Teacher Modesto City Schools Modesto, California
Educator Aspire Public Schools Modesto, California
Judy Descoteaux
Rechelle Pearlman
Mathematics Teacher Thornton Junior High School Fremont, California
4th Grade Teacher Wanda Hirsch Elementary School Tracy, California
Paul J. Fogarty
Armida Picon
Mathematics Lead Aspire Public Schools Modesto, California
5th Grade Teacher Mineral King School Visalia, California
Lisa Majarian
Anthony J. Solina
Classroom Teacher Cottonwood Creek Elementary Visalia, California
Lead Educator Aspire Public Schools Stockton, California
vi
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Bobbi Anne Barnowsky
Volume 6A
Measurement
Chapter
How Measurements Are Made
1
1-1 Unit Conversions: Metric Length ...................................4. 3AF1.4, 3MG1.4, 6AF2.1
1-2 Unit Conversions: Customary Length .........................11 3AF1.4, 3MG1.4, 6AF2.1
Progress Check 1 .............................................................18 1-3 Unit Conversions: Metric Capacity and Mass ............19 3AF1.4, 3MG1.4, 6AF2.1, 7MG1.1
1-4 Unit Conversions: Customary Capacity and Weight… ...................................................................25 3AF1.4, 3MG1.4, 6AF2.1
Progress Check 2 .............................................................32 1-5 Time and Temperature ...................................................33 3AF1.4, 3MG1.4, 6AF2.1, 7MG1.1
1-6 Analyze Units of Measure .............................................39
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
6AF2.1, 7MG1.1, 7MG1.3
Progress Check 3 .............................................................46 Assessment Study Guide .....................................................................47
Chapters 1 and 2 are contained in Volume 6A. Chapters 3 and 4 are contained in Volume 6B.
Standards Addressed in This Chapter 3AF1.4 Express simple unit conversions in symbolic form (e.g., ___ inches = ___ feet × 12). 3MG1.4 Carry out simple unit conversions within a system of measurement (e.g., centimeters and meters, hours and minutes). 6AF2.1 Convert one unit of measurement to another (e.g., from feet to miles, from centimeters to inches). 7MG1.1 Compare weights, capacities, geometric measures, times, and temperatures within and between measurement systems. (e.g., miles per hour and feet per second, cubic inches to cubic centimeters) 7MG1.3 Use measures expressed as rates (e.g., speed, density) and measures expressed as products (e.g., person-days) to solve problems; check the units of the solutions; and use dimensional analysis to check the reasonableness of the answer.
Chapter Test .....................................................................50 Standards Practice ...................................................52
Lake Tahoe
vii
Contents Chapter
Length and Area in the Real World
2
Standards Addressed in This Chapter 2-1 Length ..............................................................................56 2MG1.3, 4MG2.2, 4MG2.3
2-2 Perimeter ..........................................................................63 3MG1.3
Progress Check 1 .............................................................70 2-3 Introduction to Area .......................................................71 3MG1.2
2-4 Introduction to Volume ................................................. 77 3MG1.2
Progress Check 2 .............................................................83 Assessment
2MG1.3 Measure the length of an object to the nearest inch and/or centimeter. 3MG1.2 Estimate or determine the area and volume of solid figures by covering them with squares or by counting the number of cubes that would fill them. 3MG1.3 Find the perimeter of a polygon with integer sides. 4MG2.2 Understand that the length of a horizontal line segment equals the difference of the x-coordinates. 4MG2.3 Understand that the length of a vertical line segment equals the difference of the y-coordinates.
Study Guide .....................................................................84 Chapter Test .....................................................................88 Standards Practice ...................................................90
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
viii
Alamo Square, San Francisco
Contents Chapter
Exact Measures in Geometry
3
3-1 Area of a Rectangle ..........................................................4 3MG1.2, 4MG1.1
3-2 Area of a Parallelogram..................................................11 4MG1.1, 5MG1.1
Progress Check 1 .............................................................18 3-3 Area of a Triangle ............................................................19 3MG1.2, 5MG1.1
3-4 Surface Area of Rectangular Solids ............................. 27 3MG1.2, 4MG1.1, 5MG1.2
Progress Check 2 .............................................................36 3-5 Volume of Rectangular Solids .......................................37 3MG1.2, 5MG1.3
Assessment
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Study Guide .....................................................................43 Chapter Test .....................................................................48 Standards Practice ...................................................50 Santa Cruz
Chapters 1 and 2 are contained in Volume 6A. Chapters 3 and 4 are contained in Volume 6B.
Standards Addressed in This Chapter 3MG1.2 Estimate or determine the area and volume of solid figures by covering them with squares or by counting the number of cubes that would fill them. 4MG1.1 Measure the area of rectangular shapes by using appropriate units, such as square centimeter (cm2), square meter (m2), square kilometer (km2), square inch (in.2), square yard (yd.2), or square mile (mi.2). 5MG1.1 Derive and use the formula for the area of a triangle and of a parallelogram by comparing each with the formula for the area of a rectangle (i.e., two of the same triangles make a parallelogram with twice the area; a parallelogram is compared with a rectangle of the same area by pasting and cutting a right triangle on the parallelogram). 5MG1.2 Construct a cube and rectangular box from two-dimensional patterns and use these patterns to complete the surface area for these objects. 5MG1.3 Understand the concept of volume and use the appropriate units in common measuring systems (i.e., cubic centimeter [cm3], cubic meter [m3], cubic inch [in.3], cubic yard [yd.3]) to compute the volume of rectangular solids.
ix
Contents Chapter
Angles and Circles
4
Standards Addressed in This Chapter 4-1 Lines 5MG2.1 ....................................................................54 4-2 Angles 5MG2.1 .................................................................63 Progress Check 1.............................................................72 4-3 Triangles and Quadrilaterals 5MG2.1 ...........................73 4-4 Add Angles 5MG2.1, 5MG2.2, 6MG2.2 ............................. 81 Progress Check 2.............................................................90 4-5 Congruent Figures 7MG3.4 ............................................91 4-6 Pythagorean Theorem 5MG2.1, 7MG3.3 ........................ 99 Progress Check 3...........................................................108 4-7 Circles 6MG1.2 ...............................................................109 4-8 Volume of Triangular Prisms and Cylinders ........... 117 6MG1.3
Progress Check 4...........................................................127
Study Guide ..................................................................128 Chapter Test ..................................................................134 Standards Practice.................................................136 Mono Lake Tufa State Reserve
5MG2.2 Know that the sum of the angles of any triangle is 180° and the sum of the angles of any quadrilateral is 360° and use this information to solve problems. 6MG1.2 Know common estimates 22 of π (3.14, ___) and use these values to 7 estimate and calculate the circumference and the area of circles; compare with actual measurements. 6MG1.3 Know and use the formulas for the volume of triangular prisms and cylinders (area of base × height); compare these formulas and explain the similarity between them and the formula for the volume of a rectangular solid. 6MG2.2 Use the properties of complementary and supplementary angles and the sum of the angles of a triangle to solve problems involving an unknown angle. 7MG3.3 Know and understand the Pythagorean theorem and its converse and use it to find the length of the missing side of a right triangle and the lengths of other line segments and, in some situations, empirically verify the Pythagorean theorem by direct measurement. 7MG3.4 Demonstrate an understanding of conditions that indicate two geometrical figures are congruent and what congruence means about the relationship between the sides and angles of the two figures.
x
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Assessment
5MG2.1 Measure, identify, and draw angles, perpendicular and parallel lines, rectangles, and triangles by using appropriate tools (e.g., straight edge, ruler, compass, protractor, drawing software).
R E G N E V A SC HUNT Let’s Get Started Use the Scavenger Hunt below to learn where things are located in each chapter. 1 What is the title of Chapter 4? 2
Angles and Circles
What is the Key Concept of Lesson 3-3?
Area of a Triangle 3
What is the definition of surface area on page 27?
the area of the surface of a three-dimensional figure 4
What are the vocabulary words for Lesson 4-4?
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
complementary angles, supplementary angles 5
How many Examples are presented in the Chapter 4 Study Guide? 9
6
What are the California Standards covered in Lesson 4-3?
5MG2.1 7
According to Lesson 3-1, what is the formula for the area of a rectangle? A = × w
8
What do you think is the purpose of the Progress Check on page 36? It allows you to practice problems from the last
two lessons. 9
On what pages will you find the Study Guide for Chapter 4?
pages 126–131 10
In Chapter 3, find the logo and Internet address that tells you where you can take the Online Readiness Quiz. It is
found on page 3. The URL is ca.mathtriumphs.com. 1
Chapter
3
Exact Measures in Geometry Why is area important? What if you compare your statistics in volleyball with your cousin’s, but the two of you play on different-sized courts? It would be an unfair comparison. Most school teams play on volleyball courts that are of regulation size.
Copyright © by The McGraw-Hill Companies, Inc.
2
Chapter 3 Exact Measures in Geometry
Mike Brinson/Getty Images
STEP
STEP
1 Quiz
Are you ready for Chapter 3? Take the Online Readiness www.mathtriumphs.com to to find find out. out. Quiz at ca.mathtriumphs.com
2 Preview
Get ready for Chapter 3. Review these skills and compare them with what you’ll learn in this chapter.
What You Know
What You Will Learn
You know how to measure the lengths of items.
5IFMFOHUIPGUIJT SFDUBOHMFJTJO
JO JO
You know that you can make a parallelogram into a rectangle. Step 1: Step 2: Copyright © by The McGraw-Hill Companies, Inc.
To find the area of a rectangle, you multiply the length by the width. Area = length × width Area = 2 inches × 1 inch Area = 2 square inches
JO
5IFXJEUI PGUIJT SFDUBOHMF JO JTJO
Lesson 3-1
Lesson 3-2 To find the area of a parallelogram, you multiply the base by the height. Area = base × height Area = 3 inches × 2 inches Area = 6 square inches JO
Step 3: JO
You know you can separate a rectangle into two triangles.
Lesson 3-3 A triangle is half of a rectangle. So the area of a triangle is one-half the area of a rectangle. "SFB CBTFIFJHIU
"SFB CBTFIFJHIU
3
Lesson
3-1 Area of a Rectangle KEY Concept Find the area of a rectangle using the formula below. is the length of the rectangle. A is the area of the rectangle.
A=×w
AK
ŰVU
w is the width of the rectangle.
AKVAK AK
3MG1.2 Estimate or determine the area and volume of solid figures by covering them with squares or by counting the number of cubes that would fill them. 4MG1.1 Measure the area of rectangular shapes by using appropriate units, such as square centimeter (cm2), square meter (m2), square kilometer (km2), square inch (in2), square yard (yd2), or square mile (mi2).
VOCABULARY area the number of square units needed to cover a region or a plane figure (Lesson 2-3, p. 71)
square unit a unit for measuring area
AK
(Lesson 2-3, p. 71)
rectangle a quadrilateral with four right angles; opposite sides are parallel and equal in length
The area of the rectangle is 12 square centimeters.
The units of area are square units.
JO JO
JO JO
square a rectangle with congruent sides
What is the area of the rectangle?
ZE
1. The length of the rectangle is 5 inches, and the width is 3 inches.
GL GL
2. Substitute these values into the formula. Multiply. A=×w A = 5 in. × 3 in. A = 15 in2 The area of the rectangle is 15 square inches.
4
Chapter 3 Exact Measures in Geometry
ZE
ZE ZE
Copyright © by The McGraw-Hill Companies, Inc.
Example 1
YOUR TURN! What is the area of the rectangle? 1. The length of the rectangle is width is 2 yards.
7
yards, and the
WB WB
2. Substitute these values into the formula. Multiply. A=×w A=
7
yd ×
A=
14
yd2
2
yd
The area of the rectangle is
14
square yards.
Example 2
YOUR TURN! What is the area of the square?
What is the area of the square?
1. The length of the square is 5 kilometers, and the width is 5 kilometers.
1. The length of the square is 4 feet, and the width is 4 feet.
Copyright © by The McGraw-Hill Companies, Inc.
2. Substitute these values into the formula. Multiply.
DR
IK
IK
2. Substitute these values into the formula. Multiply.
DR
A=×w A = 4 ft × 4 ft A = 16 ft2
A=×w A = 5 km × A = 25 km2
The area of the square is 16 square feet.
5
The area of the square is kilometers.
km
25
square
Who is Correct? What is the area of the rectangle at the right?
Bena 2 A = 4 + 8 = 12 mi
Ryan 2
A = 4 × 8 = 32 mi
Arturo
KG
2 A = 8 × 8 = 64 mi
KG
Circle correct answer(s). Cross out incorrect answer(s). Lesson 3-1 Area of a Rectangle
5
Guided Practice Draw a rectangle for each given area. 1
18 cm2
Sample answer:
2
4 cm2
Sample answer: AK
AK AK
AK
Step by Step Practice 3
What is the area of the rectangle?
GU
6
Step 1 The length of the rectangle is 5 feet.
feet, and the width
GU
Step 2 Substitute these values into the formula. Multiply. A=l×w
6 30
A= A=
ft ×
5
ft
2
ft
The area of the rectangle is
30
square feet.
4
The length of the rectangle is 6 inches.
8
JO
inches. The width is JO
A=l×w A= A= 5
in. ×
A=
9 27
6
in.
2
in
The length is A=
6
8 48
9
m×
meters. The width is
3
m
m2
Chapter 3 Exact Measures in Geometry
3
meters.
N N
Copyright © by The McGraw-Hill Companies, Inc.
Find the area of each rectangle.
6
ZE
36 yd2
A=
7
2 A = 100 cm
DN
ZE DN
8
40 mi2
A=
9
A=
36 km2
NJ
LN LN NJ
Step by Step Problem-Solving Practice
Problem-Solving Strategies ✓ Use a formula.
Solve. 10
Understand
Copyright © by The McGraw-Hill Companies, Inc.
Look for a pattern. Guess and check. Act it out. Solve a simpler problem.
BASKETBALL A high-school basketball court is 94 feet long and 50 feet wide. What is the area of the court? Read the problem. Write what you know. The length of the basketball court is 94 and the width of the court is 50 feet. Plan
feet,
Pick a strategy. One strategy is to use a formula. Substitute values for length and width into the area formula.
Solve
Use the formula. A=×w A=
94
ft ×
50
ft
A = 4,700 ft2 The area of the high-school basketball court is 4,700 square feet. Check
Use a calculator to check your answer. GO ON Lesson 3-1 Area of a Rectangle
7
11
CONSTRUCTION A construction crew is pouring cement for sidewalk slabs. Each slab is a square that has sides that measure 2 70 centimeters. What is the area of each slab? 4,900 cm Check off each step.
12
✔
Understand
✔
Plan
✔
Solve
✔
Check
ART Mrs. Brady asked her class to use an entire sheet of paper to finger paint. Each sheet of paper measured 8.5 inches by 11 inches. What was the area of the finger painting?
93.5 in2 13
Can two rectangles have the same area but different lengths and widths? Explain.
Sample answer: Yes; A rectangle with side lengths of 1 inch and 6 inches has an area of 6 in2. A rectangle with side lengths of 2 inches and 3 inches also has an area of 6 in2.
Draw a rectangle that has the given area. 14
8
6 square units Sample answer:
Chapter 3 Exact Measures in Geometry
15
24 square units Sample answer:
Copyright © by The McGraw-Hill Companies, Inc.
Skills, Concepts, and Problem Solving
Find the area of each rectangle. 16
16 m2
A=
17
A=
N
28 mi2 NJ
N NJ
18
18 yd2
A=
19
A=
36 in2 JO
ZE ZE
JO
Solve. 20
28 cm
PHOTOS At the portrait studio, Ines ordered the picture of her family shown. What was the area of Ines’s family portrait?
1,008 cm2 21
DOORS The screen door at Ethan’s house is 32 inches wide and 85 inches tall. What is the area of Ethan’s screen door?
36 cm
Copyright © by The McGraw-Hill Companies, Inc.
2,720 in2 Vocabulary Check Write the vocabulary word that completes each sentence.
Area
22
is the number of square units needed to cover the inside of a region or a plane figure.
23
A(n) sides.
square
is a rectangle with four congruent
24
rectangle A(n) has opposite sides that are equal and parallel. It is a quadrilateral with four right angles.
25
Writing in Math Explain how to find the area of a rectangle.
Sample answer: Identify the length and width of the rectangle. Substitute values of the length and width into the formula for the area of a rectangle. Multiply to find the area of the rectangle. Express the answer in square units.
GO ON
Lesson 3-1 Area of a Rectangle (frame) Getty Images; (inset) Ryan McVay/Getty Images
9
Spiral Review 26
PACKAGES A package is 8 feet long, 5 feet wide, and 4 feet tall. What is the volume of the package?
160 cubic feet 27
(Lesson 2-4, p. 77)
FOOD A cereal box is 8 inches by 10 inches by 3 inches. What is maximum volume of cereal the box can hold?
240 cubic inches 28
GARDENING A flower bed is 3 meters long, 1 meter wide, and 1 meter deep. How many cubic feet of dirt fills the flower bed? __ 2
1.5 cubic meters
Find the perimeter of each polygon. 29
(Lesson 2-2, p. 63)
The perimeter of the square is 28 feet.
30
GU
The perimeter of the square is 8 meters. N
N
GU
The perimeter of the rectangle is 16 inches.
32
The perimeter of the rectangle is 16 inches. DN
JO
DN
JO
Convert.
(Lesson 1-4, p. 25)
33
5,000 lb = 2.5 T
34
52 fl oz = 6.5 c
35
10 c =
5
pt
36
20 qt =
5
gal
37
3 pt =
6
c
38
48 oz =
3
lb
10
Chapter 3 Exact Measures in Geometry
Copyright © by The McGraw-Hill Companies, Inc.
31
Lesson
3-2 Area of a Parallelogram
4MG1.1 Measure the area of rectangular shapes by using appropriate units, such as square centimeter (cm2), square meter (m2), square kilometer (km2), square inch (in2), square yard (yd2), or square mile (mi2). 5MG1.1 Derive and use the formula for area of a triangle and of a parallelogram by comparing each with the formula for the area of a rectangle.
KEY Concept
parallelogram
VOCABULARY Cut a triangle from the parallelogram along the dashed line. Move the triangle and place it on the other side, next to the right edge of the parallelogram. In the parallelogram, b represents the base, and h represents the height. C
area the number of square units needed to cover a region or a plane figure (Lesson 2-3, p. 71)
C
I
square unit a unit for measuring area
I
(Lesson 2-3, p. 71)
Notice that the new shape formed from the parallelogram is a rectangle. So, the formula for the area of a parallelogram is similar to the formula for the area of a rectangle: The formula A = b × h is the area of a parallelogram. " ŰX
(Lesson 3-1, p. 4)
5IJTJTMJLFUIFBSFBPGB SFDUBOHMF FYDFQUUIFMFOHUI JTCBOEUIFXJEUIJTI
parallelogram a quadrilateral in which each pair of opposite sides is parallel and equal in length
Copyright © by The McGraw-Hill Companies, Inc.
" CI b is the length of the base.
rectangle a quadrilateral with four right angles; opposite sides are parallel and equal in length
h is the height.
Example 1 What is the area of the parallelogram? 1. The base of the parallelogram is 8 inches, and the height is 7 inches.
GL
2. Substitute these values into the formula. Multiply. A=b×h A = 8 in. × 7 in. A = 56 in2 The area of the parallelogram is 56 square inches.
GL
GO ON
Lesson 3-2 Area of a Parallelogram
11
YOUR TURN! What is the area of the parallelogram?
4
1. The base of the parallelogram is height is 9 feet.
feet, and the
DR
2. Substitute these values into the formula. Multiply. A=b×h A = 4 ft × A = 36 ft2
9
DR
ft
The area of the parallelogram is
36
Example 2
square feet.
YOUR TURN!
What is the area of the parallelogram? 1. The base of the DN parallelogram is 9 centimeters and the DN height is 9 centimeters. 2. Substitute these values into the formula. Multiply.
What is the area of the parallelogram? 1. The base of the parallelogram is 7 yards, and the height is 7 yards.
ZE
ZE
2. Substitute these values into the formula. Multiply.
The area of the parallelogram is 81 square centimeters.
7
yd
The area of the parallelogram is square yards.
Who is Correct? What is the area of the parallelogram?
Jen 2 A = 6 × 7 = 42 m
Emilio
Jade
2 A = 6 × 7 = 48 m
2 A = 6 + 7 = 13 m
Circle correct answer(s). Cross out incorrect answer(s).
12
Chapter 3 Exact Measures in Geometry
N
N
49
Copyright © by The McGraw-Hill Companies, Inc.
A=b×h A = 7 yd × A = 49 yd2
A=b×h A = 9 cm × 9 cm A = 81 cm2
Guided Practice Draw a parallelogram that has the given area. 1
30 cm2
Sample answer:
2
200 mm2
Sample answer:
KK AK KK AK
Step by Step Practice Find the area of the parallelogram. 3
Copyright © by The McGraw-Hill Companies, Inc.
GL
GL
Step 1 The base of the parallelogram is height is 9 inches.
7
inches, and the
Step 2 Substitute these values into the formula. Multiply. A=b×h A = 7 in. × A = 63 in2
9
in.
The area of the parallelogram is
63
square inches.
GO ON Lesson 3-2 Area of a Parallelogram
13
Find the area of each parallelogram. 4
The base of the parallelogram is is 9 inches. A=b×h A = 5 in. × A = 45 in2
9
5
inches, and the height
in.
The area of the parallelogram is
GL
45
square inches. GL
5
LN LN
The base is
4
A = 4 km × 2 A = 8 km
kilometers, and the height is
2
2
kilometers.
km
6
7 NJ DN NJ
72 mi2
DN
A=
80 cm2
8
9 N N
GU
A=
GU
A=
14
100 ft2
Chapter 3 Exact Measures in Geometry
9 m2
Copyright © by The McGraw-Hill Companies, Inc.
A=
Step by Step Problem-Solving Practice
Problem-Solving Strategies Draw a diagram.
Solve. 10
✓ Use a formula.
HOBBIES Tony bought a sail for his boat. The sail is in the shape of a parallelogram and is 21 feet wide at the base and 42 feet tall. What is the area of the new sail? Understand
Guess and check. Solve a simpler problem. Work backward.
Read the problem. Write what you know. The base of the sail is 21 feet and the height of the sail is 42 feet. Pick a strategy. One strategy is to use a formula.
Plan
Substitute values for base and height into the area formula. Use the formula.
Solve
A=b×h A = 21 × A = 882
HOBBIES A sailboat can have sails shaped like parallelograms.
42
The area of the sail is 882 ft2. Use a calculator to check your multiplication.
Check
Copyright © by The McGraw-Hill Companies, Inc.
11
ART Part of the sculpture that sits in the middle of Jacob Park is shaped like a parallelogram. The front of this piece is 13 feet tall and has a base of 8 feet. What is the area of the front of the 2 sculpture? 104 ft Check off each step.
12
✔
Understand
✔
Plan
✔
Solve
✔
Check
ART Camila’s class is making cardboard ornaments. Each ornament is shaped like a parallelogram with a height of 16 centimeters and a base of 9 centimeters. What is the area of each ornament?
144 cm2
GO ON Lesson 3-2 Area of a Parallelogram
Corbis
15
13
Compare the area of a parallelogram with a base of 10 feet and a height of 5 feet to the area of a rectangle with a length of 25 feet and a width of 2 feet. Explain.
See TWE margin.
Skills, Concepts, and Problem Solving Draw a parallelogram that has the area given. 14
Sample answer:
220 mm2
15
Sample answer:
5 in2
KK
GL
GL
KK
Find the area of each parallelogram. 16
17 N
GU GU
N
44 m2
A=
18
45 ft2
19 DN ZE DN ZE 2
A = 128 cm
A=
96 yd2
Solve. 20
FARMING Mrs. Rockwell’s cornfield is in the shape of a parallelogram. Refer to the photo caption at the right. What is the area of the cornfield?
4,108 m2
16 Corbis
Chapter 3 Exact Measures in Geometry
FARMING The height of Mrs. Rockwell’s cornfield is 79 meters and the base is 52 meters.
Copyright © by The McGraw-Hill Companies, Inc.
A=
21
PARTIES Toby made a sandwich. She cut the bread in the shape of a parallelogram. Each piece of bread is 45 millimeters at the base and 65 millimeters tall. What is the area of each piece of bread?
2,925 mm2 Vocabulary Check Write the vocabulary word that completes each sentence. 22 23
24
Square unit
is a unit for measuring area.
parallelogram A(n) is a quadrilateral in which each pair of opposite sides is parallel and equal in length. Writing in Math Explain how to find the area of a parallelogram with a base of 9 inches and a height of 10 inches.
See TWE margin.
Spiral Review 25
FITNESS A trampoline has a mat that is 10 feet wide and 14 feet long. What is the area of the trampoline mat? (Lesson 3-1, p. 4)
Copyright © by The McGraw-Hill Companies, Inc.
140 ft2 26
Draw a figure that has an area of 18 square units. (Lesson 2-3, p. 71)
Sample answer:
Convert using a table. 27
28
(Lesson 1-5, p. 33)
3 min = 180 s minutes
1
2
3
seconds
60
120
180
days
1
2
3
4
hours
24
48
72
96
96 h =
4
d
Lesson 3-2 Area of a Parallelogram
17
Chapter
Progress Check 1
3
(Lessons 3-1 and 3-2)
Draw each figure that has the given area. 3MG1.2, 4MG1.1, 5MG1.1 1
rectangle, 28 cm2
Sample answer:
2
parallelogram, 60 cm2
Sample answer:
AK
AK AK
AK
Find the area of each rectangle. 4MG1.1 3
4 N ZE N
A=
25 m2
ZE 2 A = 108 yd
Find the area of each parallelogram. 4MG1.1; 5MG1.1 6 DN
GU DN GU
A=
70 cm2
A=
66 ft2
Solve. 4MG1.1 7
DESIGN Lena hung a rectangular mirror on her bathroom wall. The mirror was 85 centimeters high and 67 centimeters wide. What 5,695 cm2 was the area of Lena’s mirror?
8
FOOD Lamont decorated cookies shaped like parallelograms. Each cookie was 84 millimeters tall and 61 millimeters at the base. 5,124 mm2 What was the area of each cookie?
18
Chapter 3 Exact Measures in Geometry
Copyright © by The McGraw-Hill Companies, Inc.
5
Lesson
3-3 Area of a Triangle 3MG1.2 Estimate or determine the area and volume of solid figures by covering them with squares or by counting the number of cubes that would fill them. 5MG1.1 Derive and use the formula for area of a triangle and of a parallelogram by comparing each with the formula for the area of a rectangle.
KEY Concept
triangle
VOCABULARY
Cut the parallelogram along the dashed line. Notice you now have two triangles, each one-half the size of the parallelogram.
area the number of square units needed to cover a region or a plane figure (Lesson 2-3, p. 71)
square unit a unit for measuring area I
(Lesson 2-3, p. 71)
I
C
C
h is the height.
_
A=1×b×h 2
A=b×h
Copyright © by The McGraw-Hill Companies, Inc.
A is the area of the triangle.
b is the length of the base.
The location of the height of a triangle can vary. There are three possibilities.
I
I C
C
The height is one side of a triangle.
The height is inside the triangle.
rectangle a quadrilateral with four right angles; opposite sides are equal and parallel and equal in length (Lesson 3-1, p. 4) parallelogram a quadrilateral in which each pair of opposite sides is parallel and equal in length (Lesson 3-2, p. 11)
triangle a polygon with three sides and three angles
I C
The height is outside the triangle.
GO ON Lesson 3-3 Area of a Triangle
19
Example 1
YOUR TURN!
What is the area of the triangle?
What is the area of the triangle?
JO
N
JO
1. The base of the triangle is 6 inches, and the height of the triangle is 4 inches. 2. Substitute values of the base and height into the area of a triangle formula. 1×b×h A = __ 2 1 × 6 in. × 4 in. A = __ 2 3. Multiply to find the area of the triangle. A = 12 in2 The area of the triangle is 12 square inches.
N
1. The base of the triangle is 8 and the height of the triangle is meters.
meters,
7
2. Substitute values of the base and height into the area of a triangle formula. 1×b×h A = __ 2 1× 8 m× 7 m A = __ 2 3. Multiply to find the area of the triangle. A=
28
m2
The area of the triangle is meters.
28
square
What is the area of the triangle? 1. The base of the triangle has endpoints at (2, 1) and (5, 1). It is 5 - 2 or 3 units long.
2. The height of the triangle has endpoints at (1, 5) and (1, 1). It is 5 - 1 or 4 units long. 3. Substitute values of the base and height into the area of a triangle formula. 1×b×h A = __ 2 1×3×4 A = __ 2 4. Multiply to find the area of the triangle. A=6 The area of the triangle is 6 square units, or 6 units2. 20
Chapter 3 Exact Measures in Geometry
Copyright © by The McGraw-Hill Companies, Inc.
Example 2
YOUR TURN! What is the area of the triangle? 1. The base of the triangle has endpoints at (1, 2) and (7, 2) . It is
6
units long.
2. The height of the triangle has endpoints at (4, 2) and
(4, 4) . Its height is
2
units long.
3. Substitute values of the base and height into the area of a triangle formula. 1×b×h A = __ 2 1× 6 × 2 A = __ 2 4. Multiply to find the area of the triangle. A=
6
The area of the triangle is
6
square units, or
6
units2.
Who is Correct?
Copyright © by The McGraw-Hill Companies, Inc.
What is the area of the triangle?
Chen
1 __ cm2 A = (9 + 16) = 12.5 2
Darby
Amelia
2 A = 9 × 16 = 144 cm
1 cm2 __ A = 2 × 9 × 16 = 72
DN
Circle correct answer(s). Cross out incorrect answer(s). DN
Guided Practice Draw a triangle that has the given area. 1
15 units2
Sample answer:
2
100 units2
Sample answer:
GO ON Lesson 3-3 Area of a Triangle
21
Step by Step Practice What is the area of the triangle? 3
The base of the triangle is 5 the triangle is 16 miles.
Step 1
miles, and the height of
Step 2 Substitute values of the base and height into the area of a triangle formula. 1×b×h A = __ 2 1 × 5 mi × A = __ 2
NJ
16
mi
Step 3 Multiply to find the area of the triangle. A=
40
mi2
The area of the triangle is
40
NJ
square miles.
Find the area of each triangle. 4
The base of the triangle is triangle is
10
N
m
The area of the triangle is
35
35
44
N
m2
1×b×h A = __ 2 1 × 8 ft × A = __ 2 A=
11
square meters.
ft
ft2
The area of the triangle is
GU
44
square feet.
GU
22
Copyright © by The McGraw-Hill Companies, Inc.
10
A=
meters, and the height of the
meters.
1×b×h A = __ 2 1 A = __ × 7 m × 2
5
7
Chapter 3 Exact Measures in Geometry
6
The area of the triangle is
45
7
square inches.
The area of the triangle is square units.
20
JO
JO
SLGRQ SLGRQ
Step by Step Problem-Solving Practice
Problem-Solving Strategies
Solve. 8
VOLUNTEERING Pam earned a club patch for her jacket by volunteering last week. The patch is shaped like a triangle. It is 66 millimeters tall and has a base of 53 millimeters. What is the area of Pam’s patch? Understand
Draw a diagram. Look for a pattern. Guess and check. ✓ Use a formula. Solve a simpler problem.
Read the problem. Write what you know. The base of the patch is 53 millimeters, and the height of the patch is 66 millimeters.
Copyright © by The McGraw-Hill Companies, Inc.
Plan
Pick a strategy. One strategy is to use a formula. Substitute values of the base and height into the formula for the area of a triangle.
Solve
Use the formula. 1×b×h A = __ 2 1 × 53 mm × A = __ 2 A=
1,749
mm
mm2
The area of the patch is millimeters. Check
66
1,749
square
Use a calculator to check your multiplication.
GO ON Lesson 3-3 Area of a Triangle
23
9
FLAGS The Coleman Camp flag is raised every morning. The flag is in the shape of a triangle. It is 36 inches long at its base and has a 2 height of 49 inches. What is the area of the camp flag? 882 in Check off each step.
10
✔
Understand
✔
Plan
✔
Solve
✔
Check
COSTUMES Manuel bought a triangular-shaped bandanna for his costume in the school play. The bandanna is 62 centimeters tall and has a base of 67 centimeters. What is the area of the bandanna?
2,077 cm2 11
Compare the area of a triangle with a base of 12 feet and a height of 6 feet to the area of a parallelogram with a base of 12 feet and a height of 6 feet.
See TWE margin.
Draw a triangle that has the given area. 12
12 units2 Sample answer:
24
Chapter 3 Exact Measures in Geometry
13
6 units2
Sample answer:
Copyright © by The McGraw-Hill Companies, Inc.
Skills, Concepts, and Problem Solving
Find the area of each triangle. 14
The area of the triangle is 1,000 square
15
millimeters, or 1,000 mm2.
The area of the triangle is yards, or
63
square
yd2.
NN
ZE
NN
16
63
The area of the triangle is
ZE
20
units2.
17
The area of the triangle is
9
units2.
SLGRQ SLGRQ SLGRQ
SLGRQ
Copyright © by The McGraw-Hill Companies, Inc.
Solve. 18
FOOD For a picnic, Wayne sliced cheese in the shape of triangles. Each slice was 48 millimeters at the base and 51 millimeters tall. What was the area of each slice of cheese?
1,224 mm2 19
HORSES Earl searched for a missing horse near his uncle’s ranch. He searched in an area that was shaped like a triangle. Refer to the photo caption at the right. What was the area of the piece of land that Earl searched?
306,556 yd2
HORSES The area in which Earl searched for the horse had a base of 886 yards and a height of 692 yards.
Vocabulary Check Write the vocabulary word that completes each sentence. 20
21
Area
is the number of square units needed to cover the inside of a region or a plane figure. A(n) triangle is a polygon with three sides and three angles.
GO ON
Lesson 3-3 Area of a Triangle Jerry Irwin/Photo Researchers
25
22
Writing in Math Explain how the area of a triangle is related to the area of a rectangle.
Answers will vary. Sample answer: A rectangle can be cut into two equal triangles. So, the area of a triangle is half the area of a rectangle.
Spiral Review 23
WEATHER Zing turned on the weather channel. She saw a region that had a tornado watch. The region was shaped like a parallelogram and measured 82 miles across at the base and 56 miles high. What was the area of the tornado watch? (Lesson 3-2, p. 11)
4,592 mi2 24
Find the area of the figure. (Lesson 2-3, p. 71)
SLGRQ
SLGRQ
20
The area of the rectangle is
25
DN
2 inches
JO
JO
26
(Lesson 2-1, p. 56)
7 centimeters DN
26
Copyright © by The McGraw-Hill Companies, Inc.
Draw a line segment of each length.
square units.
Chapter 3 Exact Measures in Geometry
Lesson
3-4 Surface Area of Rectangular Solids KEY Concept Ű TJEF
I
X
CBDL
I
CPUUPN
GSPOU
UPQ
X
3MG1.2 Estimate or determine the area and volume of solid figures by covering them with squares or by counting the number of cubes that would fill them. 4MG1.1 Measure the area of rectangular shapes by using appropriate units, such as square centimeter (cm2), square meter (m2), square kilometer (km2), square inch (in2), square yard (yd2), or square mile (mi2). 5MG1.2 Construct a cube and rectangular box from twodimensional patterns to compute the surface area for these objects.
VOCABULARY TJEF
The net can be folded to make a rectangular prism.
surface area the area of the surface of a three-dimensional figure net a flat pattern that can be folded to make a threedimensional figure
I X
Ű
The surface area of a rectangular prism is the sum of the areas of all the faces of the figure. Surface area is measured in square units.
face the flat part of a threedimensional figure that is considered one of the sides square unit a unit for measuring area
Copyright © by The McGraw-Hill Companies, Inc.
A rectangular prism has six faces.
(Lesson 2-3, p. 71)
Example 1 What is the surface area of the rectangular prism? 4. Find the area of faces C and E.
1. Draw a net of the rectangular prism. Label the faces A, B, C, D, E, and F.
A=×w A = 2 × 6 = 12
2. Find the area of faces A and F.
5. Find the sum of all the areas of all the faces.
A=×w A = 2 × 5 = 10 3. Find the area of faces B and D.
10 + 30 + 12 + 10 + 30 + 12 = 104 !
"
#
$
The surface area of the rectangular prism is 104 square units.
A=×w A = 5 × 6 = 30
%
GO ON
Lesson 3-4 Surface Area of Rectangular Solids
27
YOUR TURN! What is the surface area of the rectangular prism? 1. Draw a net of the rectangular prism. Label the faces A, B, C, D, E, and F. 2. Find the area of faces A and F. A=×w A= 3 ×
8
24
=
3. Find the area of faces B and D. A=×w A= 8 ×
7
56
=
!
"
#
$
4. Find the area of faces C and E. A=×w A= 3 ×
7
%
21
=
5. Find the sum of the areas of all the faces.
24
+
56
+
21
+
24
+
21
56
+
= 202
The surface area of the rectangular prism is 202 square units.
Example 2
YOUR TURN! What is the surface area of the cube?
What is the surface area of the cube?
1. Draw a net of the cube. 2. Find the area of each face.
!
"
#
$
%
A=×w A = 4 × 4 = 16 3. There are six faces on the cube. Find the sum of the areas of all six faces. A B C D E F 16 + 16 + 16 + 16 + 16 + 16 = 96 The surface area of the cube is 96 square units. 28
Chapter 3 Exact Measures in Geometry
!
2. Find the area of each face. A=×w A= 3 ×
3
"
#
$
%
=
9
3. Find the sum of the areas of all six faces.
9 9
+ +
9 9
+ =
9 54
+
The surface area of the cube is square units.
9 54
+
Copyright © by The McGraw-Hill Companies, Inc.
1. Draw a net of the cube.
Who is Correct? What is the surface area of the rectangular prism?
Jermaine
Chad 2
A = 3 × 3 = 9 units 2 A = 5 × 5 = 25 units 2 A = 9 × 9 = 81 units A = 9 + 25 + 81 2 = 115 units
Suja 2
A = 3 × 9 = 21 units 2 A = 3 × 5 = 15 units
2 A = 3 × 9 = 27 units 2 A = 3 × 5 = 15 units
2 A = 5 × 9 = 45 units A = 21 + 15 + 45 + 21 + 15 + 45 2 = 162 units
2 A = 5 × 9 = 45 units A = 27 + 15 + 45 + 27 + 15 + 45 2 = 174 units
Circle correct answer(s). Cross out incorrect answer(s).
Guided Practice Draw a net for a rectangular prism with the given length, width, and height. 2 × 8 × 9 Sample answer:
2
5 × 4 × 7 Sample answer:
Copyright © by The McGraw-Hill Companies, Inc.
1
GO ON Lesson 3-4 Surface Area of Rectangular Solids
29
Step by Step Practice What is the surface area of the rectangular prism? 3
Step 1 Draw a net of the rectangular prism. Label the faces A, B, C, D, E, and F.
!
"
Step 2 Find the area of faces A and F. A=×w A= 2 ×
4
=
#
$
%
8
Step 3 Find the area of faces B and D. A=×w A= 4 ×
3
=
12
Step 4 Find the area of faces C and E. A=×w A= 2 ×
3
=
6
Step 5 Find the sum of the areas of all the faces.
8
12
+
6
The surface area of the rectangular prism is
52
square units.
+
12
+
6
+
8
+
52
=
Sample answer:
4
Draw a net of the rectangular prism. Label the faces A, B, C, D, E, and F. Follow the steps at the top of page 31 to find the surface area.
30
Chapter 3 Exact Measures in Geometry
!
"
%
#
$
Copyright © by The McGraw-Hill Companies, Inc.
Find the surface area of each rectangular prism.
Find the area of faces A and F.
Find the area of faces B and D.
A=×w A=
10
A=×w ×
5
=
50
A=
5
×
7
=
35
Find the area of faces C and E. A=×w A=
10
×
7
=
70
Find the sum of the area of all the faces.
50
+
A
35
+
B
70
+
C
35
70
+
D
50
+
E
= 310
F
The surface area of the rectangular prism is 310 square units. Find the area of faces A and F.
5
A=×w
9
A=
4
×
=
36
Find the area of faces B and D. A=ℓ×w
4
A=
6
×
=
24
Find the area of faces C and E. Copyright © by The McGraw-Hill Companies, Inc.
A=ℓ×w
6
A=
9
×
=
54
Find the sum of the area of all the faces.
36
+
24
+
54
+
36
+
24
54
+
= 228
The surface area of the rectangular prism is 228 square units. 6
The surface area of the rectangular prism is 174 square units.
7
The surface area of the rectangular prism is 136 square units.
GO ON Lesson 3-4 Surface Area of Rectangular Solids
31
Step by Step Problem-Solving Practice
Problem-Solving Strategies ✓ Draw a diagram.
Solve. 8
Look for a pattern. Guess and check. Solve a simpler problem. Work backward.
PRESENTS Selena wrapped a present that was shaped like a cube. Each side measured 9 inches. What is the least amount of wrapping paper that Selena could have used to wrap the present? Understand
Read the problem. Write what you know. Each side of the present is 9 inches.
Plan
Pick a strategy. One strategy is to draw a diagram. Draw a net of the cube.
!
"
#
$
%
Find the area of each face. A=×w A= 9 ×
9
=
81
Find the sum of the areas of all the faces.
81 81
+ 81 + = 486
81
+
81
+
81
+
Selena used at least 486 square inches of wrapping paper. Check
32
Use a calculator to check your multiplicaton and addition.
Chapter 3 Exact Measures in Geometry
Copyright © by The McGraw-Hill Companies, Inc.
Solve
9
DESIGN Eddie put carpet on the floor, ceiling, and all of the walls in his video room. The video room has a length of 12 feet, a width of 15 feet, and a height of 10 feet. How much 2 carpet did Eddie use for his video room? 900 ft Check off each step.
10
✔
Understand
✔
Plan
✔
Solve
✔
Check
GEOMETRY What is the surface area of a number cube that has 15 millimeter edges?
1,350 mm2 11
Use what you know about how to find the area of triangles and rectangles to find the surface area of this triangular prism. (Hint: This triangular prism has 2 sides that are triangles and 3 sides that are rectangles.)
JO JO JO JO
See TWE margin.
Copyright © by The McGraw-Hill Companies, Inc.
JO
Skills, Concepts, and Problem Solving Draw a net for a rectangular prism with the given length, width, and height. 12
3 × 7 × 5 Sample answer:
13
4 × 8 × 4 Sample answer:
GO ON Lesson 3-4 Surface Area of Rectangular Solids
33
Find the surface area of each rectangular prism. 14
The surface area of the rectangular 168 prism is square units.
15
The surface area of the rectangular 340 prism is square units.
16
The surface area of the rectangular 216 prism is square units.
17
The surface area of the rectangular 142 prism is square units.
Solve. 18
JEWELRY Daisy made the jewelry box shown. What is the surface area of the jewelry box?
1,944 cm2 ART Quinn decorated a rectangular-shaped chest with wallpaper. The length of the chest is 3 feet, the width is 5 feet, and the height is 2 feet. What is the least amount of wallpaper Quinn used?
62 ft2 Vocabulary Check Write the vocabulary word that completes each sentence. 20
21
22
net A(n) is a flat pattern that can be folded to make a three-dimensional figure. A(n) figure.
face
Surface area
is the flat side of a three-dimensional is the area of the surface of a three-
dimensional figure.
34 Corbis
Chapter 3 Exact Measures in Geometry
JEWELRY Daisy’s jewelry box is shaped like a cube. The sides measure 18 centimeters each.
Copyright © by The McGraw-Hill Companies, Inc.
19
23
Writing in Math Explain how to find the surface area of a rectangular prism.
Sample answer: To find the surface area of a rectangular prism, find the sum of the areas of all the faces of the rectangular prism.
Spiral Review 24
TABLES The top of a table measures 54 inches by 116 inches. What 6,264 in2 is the area of the top of the table? (Lesson 3-1, p. 4)
25
SCHOOL PROJECT Larry made a poster for a school project. He wants to make a big envelope to protect the poster when he carries it to school. The poster is 24 inches by 18 inches. Larry needs two identical sheets of plastic that he will tack together on three sides. What is the area of one of the sheets Larry needs?
432 square inches 26
DECORATING Belinda plans to paint one wall of her bedroom bright blue. She needs to know the area of the wall to decide how much paint she needs. The wall is 14 feet by 8 feet. What is the area of the wall Belinda will paint?
Copyright © by The McGraw-Hill Companies, Inc.
112 square feet 27
Draw a figure that has a perimeter of 20 centimeters. (Lesson 2-2, p. 63)
Sample answer: DN
DN
DN
DN
DN
Lesson 3-4 Surface Area of Rectangular Solids
35
Chapter
Progress Check 2
3
(Lessons 3-3 and 3-4)
Draw a triangle that has the given area. 3MG1.2, 5MG1.1 1
40 units2 Sample answer:
2
Sample answer:
4 units2
Draw a net for a rectangular prism with the given length, width, and height. 3MG1.2 3
3 × 7 × 6 Sample answer:
4
2 × 8 × 5 Sample answer:
5
The surface area of the rectangular prism is square units.
268
6
The surface area of the rectangular prism is square units.
178
Copyright © by The McGraw-Hill Companies, Inc.
Find the surface area of each rectangular prism. 3MG1.2, 4MG1.1
Solve. 3MG1.2, 4MG1.1, 5MG1.1 7
SIGNS Stella designed the triangular-shaped sign shown at the right for a travel agency. What is the area of Stella’s sign?
IN
434 in2 8
STORAGE A storage cabinet is 24 inches wide, 26 inches long, and 40 inches high. What is the surface area of the storage cabinet?
5,248 in2 36
Chapter 3 Exact Measures in Geometry
IN
Lesson
3-5 Volume of Rectangular Solids KEY Concept The amount of space inside a threedimensional figure is the volume of the figure.
I
The volume of a rectangular solid is the product of its length, width, and height. V is the volume of the solid figure.
V=×w×h is the length.
or
X
Ű
w is the width.
VOCABULARY volume the number of cubic units needed to fill a threedimensional figure or solid figure
V = wh
(Lesson 2-4, p. 77)
cube a three-dimensional figure with six congruent square faces (Lesson 2-4, p. 77)
h is the height.
cubic unit a unit for measuring volume (Lesson 2-4, p. 77)
Volume is measured in cubic units.
Copyright © by The McGraw-Hill Companies, Inc.
3MG1.2 Estimate or determine the area and volume of solid figures by covering them with squares or by counting the number of cubes that would fill them. 5MG1.3 Understand the concept of volume and use the appropriate units in common measuring systems to compute the volume of rectangular solids.
Example 1
YOUR TURN!
What is the volume of the rectangular prism?
What is the volume of the rectangular prism?
1. The length of the cube is 3 units. The width of the cube is 7 units. The height of the cube is 4 units. 2. Substitute the length, width, and height into the volume formula. V=×w×h V=3×7×4 3. Multiply. V = 84 The volume of the rectangular prism is 84 cubic units.
1. The length of the rectangular prism is 5 units. The width of the rectangular prism is 4 units. The height of the rectangular prism is 8 units. 2. Substitute the length, width, and height into the volume formula. V=×w×h V= 5 ×
4
×
8
3. Multiply. V = 160 The volume of the rectangular prism is 160 cubic units.
GO ON
Lesson 3-5 Volume of Rectangular Solids
37
Example 2
YOUR TURN!
What is the volume of the cube?
What is the volume of the cube? 1. The length of the cube is 6 units.
1. The length of the cube is 5 units.
The width of the cube is 6 units.
The width of the cube is 5 units. The height of the cube is 5 units. 2. Substitute the length, width, and height into the volume formula.
The height of the cube is
units.
2. Substitute the length, width, and height into the volume formula. V=×w×h V= 6 ×
V=×w×h V=5×5×5 V = 53
6
V=
6
×
6
63
3. Multiply.
3. Multiply.
V = 216
V = 125 The volume of the cube is 125 cubic units.
The volume of the cube is 216 cubic units.
Who is Correct? What is the volume of the rectangular prism?
V=7×9×5 = 315 cubic units
Kristen V = 63 + 45 + 35 + 63 + 45 + 35 = 286 cubic units
Circle correct answer(s). Cross out incorrect answer(s).
38
Chapter 3 Exact Measures in Geometry
Jamie V=7×9×5 = 385 cubic units
Copyright © by The McGraw-Hill Companies, Inc.
Ramiro
Check your answer. Remember, you can find the volume of a solid figure by counting the number of cubic units it contains.
Guided Practice 1
How many cubes are in this rectangular prism?
8
2
How many cubes are in this rectangular prism?
35
Step by Step Practice Find the volume of the rectangular prism. 3
Step 1 The length of the rectangular prism is
4
units.
The width of the rectangular prism is
8
units.
The height of the rectangular prism is
2
units.
Step 2 Substitute the length, width, and height into the volume formula. V=×w×h
Copyright © by The McGraw-Hill Companies, Inc.
V=
4
×
8
×
2
Step 3 Multiply. V=
64
The volume of the rectangular prism is
64
cubic units.
Find the volume of each rectangular prism. 4
Substitute the length, width, and height into the volume formula. Then multiply. V=×w×h V=
6
×
10
×
3
V = 180 The volume of the rectangular prism is 180 cubic units.
GO ON
Lesson 3-5 Volume of Rectangular Solids
39
5
V=×w×h V=
2
×
12
×
5
V = 120 The volume of the rectangular prism is 120 cubic units.
6
The volume of the rectangular prism is 288 cubic units.
7
The volume of the rectangular prism is 168 cubic units.
Step by Step Problem-Solving Practice
Problem-Solving Strategies ✓ Use a model.
Solve. 8
Look for a pattern. Guess and check. Act it out. Work backward.
HOUSES Hillary has a window air conditioner in her bedroom that is 14 inches long, 17 inches wide, and 15 inches tall. What is the volume of Hillary’s air conditioner? Understand
Read the problem. Write what you know.
a width of
15 Plan
17
14
inches,
inches, and a height of
inches.
Pick a strategy. One strategy is to use a model. Stack cubes to model the air conditioner.
Solve
Use the formula. V=×w×h V=
14
in. ×
17
in. ×
15
in.
V = 3,570 in3 The volume of Hillary’s air conditioner is 3,570 cubic inches. Check
40
Use a calculator to check your multiplication.
Chapter 3 Exact Measures in Geometry
Copyright © by The McGraw-Hill Companies, Inc.
The air conditioner has a length of
9
CONSTRUCTION Yamin’s family has a storage shed that is 4.5 yards wide, 28 yards long, and 4.5 yards high. What is the 3 volume of the shed? 567 yd Check off each step.
10
11
✔
Understand
✔
Plan
✔
Solve
✔
Check
PACKAGING Mrs. Reynolds put together a card box for her daughter’s graduation party. The card box was shaped like a cube. Each side measured 48 centimeters. What was the 110,592 cm3 volume of the card box? Compare the volume of the rectangular prism shown at the right to its surface area.
See TWE margin.
Skills, Concepts, and Problem Solving Copyright © by The McGraw-Hill Companies, Inc.
Find the volume of each rectangular prism.
30
12
The volume of the rectangular prism is
13
The volume of the rectangular prism is 162 cubic units.
14
The volume of the rectangular prism is 80 cubic units.
cubic units.
15
The volume of the rectangular prism is 225 cubic units.
GO ON Lesson 3-5 Volume of Rectangular Solids
41
Solve. 16
CONSTRUCTION Marco’s father built the tree house shown at the right. What was the volume of the tree house?
420 ft3 17
COLLECTIONS Lydia used a shoe box for her rock collection. The shoe box was 12 inches long, 6 inches wide, and 4.5 inches high. What was the volume of the shoe box?
324 in3 Vocabulary Check Write the vocabulary word that completes each sentence.
CONSTRUCTION The tree house that Marco’s father built had a height of 7 feet, a length of 10 feet, and a width of 6 feet.
18
Cubic unit
19
Volume
is the number of cubic units needed to fill a three-dimensional figure or solid figure.
20
Writing in Math Explain how to find the volume of a rectangular prism.
is a unit for measuring volume.
Sample answer: Find the length, width, and height of the rectangular prism. Substitute the length, width, and height into the formula for the volume of a rectangular prism. Then multiply.
21
MUSIC Marika’s stereo speakers are cubed-shaped. Each side measures 20 millimeters. What is the surface area of each speaker? (Lesson 3-4, p. 27)
2,400 mm2 22
Draw a triangle that has an area of 320 square units. (Lesson 3-3, p. 19)
Sample answer:
42 Corbis
Chapter 3 Exact Measures in Geometry
Copyright © by The McGraw-Hill Companies, Inc.
Spiral Review
Chapter
Study Guide
3
Vocabulary and Concept Check face, p. 27
Write the vocabulary word that completes each sentence.
net, p. 27
1
parallelogram, p. 11 2
rectangle, p. 4 square, p. 4
3
surface area, p. 27 triangle, p. 19
4
5
A(n)
square is a rectangle with four congruent sides.
net A(n) is a flat pattern that can be folded to make a three-dimensional figure. parallelogram A(n) is a quadrilateral in which each pair of opposite sides is parallel and equal in length. rectangle A(n) right angles.
is a quadrilateral with four
Surface area
is the area of the surface of a three-dimensional figure.
Copyright © by The McGraw-Hill Companies, Inc.
Label each diagram below. Write the correct vocabulary term in each blank. 6
triangle
8
The net shown is of a
!
"
7
#
cube
and has
6
face
faces.
$
%
Chapter 3 Study Guide
43
Lesson Review
3-1
Area of a Rectangle
(pp. 4–10)
Find the area of each rectangle. 9
DR
Example 1 Find the area of the rectangle using the formula A = × w.
GLAFCQ GLAFCQ
is the length of the rectangle. A is the area of the rectangle.
DR 2 A = 99 square feet or 99 ft
10
w is the width of the rectangle.
A=×w
The length of the rectangle is 6 inches, and the width is 3 inches. Substitute these values into the formula. Multiply.
AK
AK
A=×w A = 6 in. × 3 in. The area of the rectangle is 18 square inches. A = 18 in2
A = 120 square centimeters or
120 cm2
Find the area of each square.
What is the area of the square? K WB
K 2 A = 49 square meters or 49 m
WB
The length of the square is 5 yards, and the width is 5 yards.
12
Substitute these values into the formula. Multiply. AK
AK
A = 100 square centimeters
or 100 cm2
44
Chapter 3 Study Guide
A=l×w A = 5 yd × 5 yd The area of the square is 25 square yards. A = 25 yd2
Copyright © by The McGraw-Hill Companies, Inc.
11
Example 2
3-2
Area of a Parallelogram
(pp. 11–17)
Find the area of each parallelogram.
Example 3 What is the area of the parallelogram?
13 AK
GL
AK
A = 75 square centimeters
or 75 cm2
The base of the parallelogram is 12 inches, and the height is 10 inches.
14
Substitute these values into the formula. Multiply.
DR DR 2 A = 24 square feet or 24 ft
3-3
Area of a Triangle
(pp. 19–26)
Find the area of each triangle. Copyright © by The McGraw-Hill Companies, Inc.
GL
15 DR DR
2 A = 1.5 square feet, or 1.5 ft
16
AK
AK
A = 99 square centimeters, or 99 cm2
A=b×h A = 12 in. × 10 in. A = 120 in2
The area of the parallelogram is 120 square inches.
Example 4 What is the area of the triangle? The base of the triangle is 10 inches, and the height of the triangle is 16 inches. Substitute values of the base and height into the area of a triangle formula. 1×b×h A = __ 2 1 × 10 in. × 16 in. A = __ 2
GL
GL
Multiply to find the area of the triangle. A = 80 in2 The area of the triangle is 80 square inches.
Chapter 3 Study Guide
45
3-4
Surface Area of Rectangular Solids
(pp. 27–35)
Find the surface area of each rectangular prism.
Example 5 What is the surface area of the rectangular prism? Draw a net of the rectangular prism. Label the faces A, B, C, D, E, and F.
17
Find the area of faces A and F. The surface area of the cube is 216 square units.
A=l×w A=2×4=8
!
Find the area of faces B and D.
18
"
#
$
%
A=l×w A = 4 × 5 = 20 Find the area of faces C and E. A=l×w A = 2 × 5 = 10 Find the sum of the areas of all the faces. The surface area of the cube is 486 square units.
8 + 20 + 10 + 8 + 20 + 10 = 76
Find the surface area of each rectangular prism. 19
20
The surface area of the rectangular prism is 248 square units. The surface area of the rectangular prism is 150 square units.
46
Chapter 3 Study Guide
Copyright © by The McGraw-Hill Companies, Inc.
The surface area of the rectangular prism is 76 square units.
3-5
Volume of Rectangular Solids
(pp. 37–42)
Find the volume of each rectangular prism.
Example 6 Find the volume of the rectangular solid using the formula below:
21
is the length.
The volume of the rectangular prism is 48 cubic units.
V=×w×h V is the volume of the solid figure.
22
h is the height.
w is the width.
The length of the cube is 4 units. The width of the cube is 9 units. The volume of the rectangular prism is 135 cubic centimeters.
The height of the cube is 2 units. Substitute the length, width, and height into the volume formula.
23
Copyright © by The McGraw-Hill Companies, Inc.
V=×w×h V=4×9×2 V = 72 The volume of the rectangular prism is 216 cubic units.
The volume of the rectangular prism is 72 cubic units.
24
The volume of the rectangular prism is 729 cubic units.
Chapter 3 Study Guide
47
Chapter
Chapter Test
3
Find the area of each rectangle. 4MG1.1 1
2 AK
GL AK GL
A=
56 in2
90 cm2
A=
Draw a rectangle that has the given area. 3MG1.2 3
35 units2 Sample answer:
4
49 units2
Sample answer:
Find the area of each parallelogram. 5MG1.1 6
Copyright © by The McGraw-Hill Companies, Inc.
5 ZE ZE
A=
JO
2
60 yd
JO
A=
792 in2
Find the area of each triangle. 3MG1.2, 4MG1.1, 5MG1.1 7
8
DN
DN 2 A = 6 units
48
Chapter 3 Test
A=
81 cm2
GO ON
Find the surface area of each rectangular prism. 3MG1.2, 4MG1.1 9
The surface area of the rectangular prism is units.
122
square
10
The surface area of the rectangular prism is units.
234
square
Find the volume of each rectangular prism. 3MG1.2 11
The volume of the rectangular prism is
180
cubic units.
12
The volume of the rectangular prism is
168
cubic units.
Copyright © by The McGraw-Hill Companies, Inc.
Solve. 4MG1.2, 5MG1.1 13
PETS The floor of Jet’s doghouse is a rectangle with a length of 32 inches and a width of 20 inches. What is the area of the floor of 2 Jet’s doghouse? 640 in
14
BAKING Betsy made dough cutouts in the form of triangles with a base of 11 centimeters and a height of 8 centimeters. 2 What was the area of each triangular dough cutout? 44 cm
Correct the mistakes. 15
Margaret bought a rug that was a square with 8 feet on each side. Later that day Margaret e-mailed her sister to describe the rug. What mistake did Margaret make?
I DN
C DN
'SPN.BSHBSFU 4VCKFDU/FXSVH 5P4JTUFS
.YNEWRUGISSQUAREFEET FT SQUAREFEET
.ARGARET
Sample answer: The mistake Margaret made was that she found the perimeter of the rug rather than the area. 16
Show how Margaret should have found the area of the rug.
Sample answer: To find the area of the rug, you multiply the length and width. 8 × 8 = 64 square feet, or 64 ft2. Chapter 3 Test
49
Chapter
Standards Practice
3
Choose the best answer and fill in the corresponding circle on the sheet at right. What is the area of the rectangle?
1
4MG1.1
4
N
N
Malik has a parallelogram-shaped mouse pad. It has a base length of 10 inches and a height of 8 inches. What is the area of the mouse pad?
5MG1.1
A 23 m2
F 9 in2
H 36 in2
G 18 in2
J 80 in2
B 46 m2 C 90 m2 5
D 100 m2
Use the formula for the area of a rectangle to find the area of one of the triangles in the figure. 5MG1.1
One wall in Mr. Avsharian’s classroom measures 35 feet by 14 feet. What is the area of this wall? 4MG1.1
2
JO
F 49 ft2 G 98 ft2
JO
A 16 in2
C 32 in2
J 560 ft2
B 24 in2
D 64 in2
Find the area of each figure. Which sentence is true? 5MG1.1
3
'JHVSF"
6
What is the area of the right triangle?
5MG1.1
'JHVSF#
DN
DN DN
LN
DN
A Area A > Area B B Area B > Area A C Area A < Area B D Area A = Area B
50
Chapter 3 Standards Practice
LN
F 21 km
H 54 km2
G 54 km
J 108 km2 GO ON
Copyright © by The McGraw-Hill Companies, Inc.
H 490 ft2
7
What is the perimeter of the rectangle?
10
3MG1.3
What is the area of the shaded figure?
3MG1.2 ZE ZE
F 7 square units
A 48 yards
G 8 square units
B 48 yd2
H 21 square units
C 63 yards
J 28 square units
D 63 yd2
ANSWER SHEET 8
Alyssa has a closed shoe box that measures 10 inches by 7 inches by 5 inches. What is the volume of the shoe box? 3MG1.2 F 22 in3
Copyright © by The McGraw-Hill Companies, Inc.
G 350 in
9
H 350 in2 3
J 350 in
What is the surface area of the rectangular solid? 3MG1.2
ZE
Directions: Fill in the circle of each correct answer. 1
A
B
C
D
2
F
G
H
J
3
A
B
C
D
4
F
G
H
J
5
A
B
C
D
6
F
G
H
J
7
A
B
C
D
8
F
G
H
J
9
A
B
C
D
10
F
G
H
J
ZE ZE
A 18 yd2 B 186 yd2 C 216 yd2
Success Strategy Try to answer every question. Work out the problem and eliminate answers you know are wrong. Do not change your answers unless you are very uncertain about your first answer choice.
D 248 yd2
Chapter 3 Standards Practice
51
Chapter
4
Angles and Circles Angles, circles, and shapes surround us. We must understand and measure angles, circles, and other shapes to build and use things like buildings, planes, maps, rockets, cars, and bicycles.
Copyright © by The McGraw-Hill Companies, Inc.
52
Chapter 4 Angles and Circles
Getty Images
STEP
STEP
1 Quiz
Are you ready for Chapter 4? Take the Online Readiness Quiz at ca.mathtriumphs.com to find out.
2 Preview
Get ready for Chapter 4. Review these skills and compare them with what you’ll learn in this chapter.
What You Know
What You Will Learn
You know that if you do a 180° turn, that you turn to face the opposite way.
Lesson 4-2 Angles are figures that are often measured in degrees. ¡
This angle measures 180°. It is a straight angle. You know how to describe and recognize some figures. TRY IT!
Copyright © by The McGraw-Hill Companies, Inc.
2
Triangles and quadrilaterals can be classified by their angles and sides.
See TWE margin.
A parallelogram has four sides. Each pair of opposite sides is parallel and equal in length.
See TWE margin.
An equilateral triangle has three sides equal in length and angles that are the same size.
Describe the figures. 1
Lesson 4-3
You know how to compare numbers.
Lesson 4-5
Example: 6 = 6
When two figures have the same size and shape, they are congruent .
TRY IT!
6
Use >, <, or = to make a true statement. 3
4 < 6
4
180° = 180°
5
10 > 9
6
1,234 = 1,234
)
¡
4
¡
¡
¡
5
'
¡
¡
(
These two triangles are congruent.
53 Corbis
Lesson
4-1 Lines 5MG2.1 Measure, identify, and draw angles, perpendicular and parallel lines, rectangles, and triangles by using appropriate tools.
KEY Concept ;
GH IJ
point Z A point is named using capital letters and is represented by dots. "
# N
$
%
, .
'
ray EF or EF
+
/
KL intersects MN. Intersecting lines meet or cross at a point. This symbol indicates a right angle.
0
2
1
3
OP ⊥ QR Perpendicular lines meet, or intersect, to form right angles . ⊥ means is perpendicular to.
You should know the meaning of the terms and their symbols.
Chapter 4 Angles and Circles
point an exact location in space line a set of points that goes straight in opposite directions without ending line segment a part of a line that has two endpoints ray
A ray is named by its endpoint and a point on the ray. The endpoint of the ray is listed first.
54
VOCABULARY
a part of a line that has one endpoint and extends indefinitely in one direction parallel lines lines that are the same distance apart; parallel lines do not meet intersecting lines lines that meet or cross each other perpendicular lines lines that meet or cross each other to form right angles right angle an angle with a measure of 90°
Copyright © by The McGraw-Hill Companies, Inc.
&
)
-
line segment −− CD or CD A line segment is named by its endpoints.
*
Parallel lines do not intersect. means is parallel to.
line AB or AB line m A line is named by two of the points on the line or by a single letter.
(
Example 1
YOUR TURN!
Draw a line. Name the line QR .
1. Use a ruler to draw a line.
1. Use a ruler to draw a line. 2. Draw arrows on each end of the line to show that the line does not end in that direction. 3. Place two points on the line. Label the points Q and R. 2
Draw a ray. Name the ray ST .
3
one end of the 2. Draw an arrow on line to show that the ray does not end in that direction. point on the other end. 3. Place a S Label the point because the endpoint of the ray is listed first. point
4. Draw a
Label the point
close to the arrow.
T
4
Example 2
. 5
YOUR TURN! Name each line. Identify the relationships.
Name each line. Identify the relationships. .
8 0
Copyright © by The McGraw-Hill Companies, Inc.
:
9
/
;
. 1. One line is WX . The other line is YZ 2. The lines cross, or intersect. 3. The lines intersect to form right angles. So, the lines are perpendicular lines. 4. Write the notation for the perpendicular lines.
1
MN . 1. One line is OP The other line is
.
2. The lines intersect . 3. Do the lines intersect to form right no angles? The lines are 4.
MN
intersecting
intersects
OP
lines. .
⊥ YZ WX
GO ON Lesson 4-1 Lines
55
Example 3
YOUR TURN!
Draw three lines GH , KN is , and TW . GH parallel to KN . and perpendicular to TW 5
(
Draw three lines AB , FH , and ZY . AB is parallel to ZY and perpendicular to FH .
,
;
"
)
'
)
/ #
8
. 1. Draw GH
. 1. Draw AB
2. Draw KN so that every point is equal . distance from line GH
2. Draw
so that it forms a right angle 3. Draw TW . with GH
3. Draw
:
ZY so that every point is equal . AB distance from line FH so that it forms a right . AB angle with
is also perpendicular Notice that TW to KN .
ZY is also perpendicular to FH .
Who is Correct? Identify the figure. 6
7
Mick
Erina
There are two endpoints. It is a ray. UV
There are no endpoints. It is a line. UV
There are two is a endpoints. It−− UV segment.
Copyright © by The McGraw-Hill Companies, Inc.
Mira
Circle correct answer(s). Cross out incorrect answer(s).
Guided Practice Draw each figure. 1
and UT intersecting lines WV
2
BC #
8 6
5
7
56
Chapter 4 Angles and Circles
$
Step by Step Practice 3
Name each line. Identify the relationships. '
)
(
*
FG
Step 1 Name the lines.
HI
;
Step 2 The lines do not intersect . The lines are
parallel
lines.
FG HI
Step 3 Write the notation for the lines.
Name each line. Identify the relationships. 4
AB
The lines are
and
YZ
.
The lines intersect . The lines are AB
Copyright © by The McGraw-Hill Companies, Inc.
5
⊥
perpendicular
" #
:
;
lines.
YZ
DE
The lines are
and
JK
.
The lines intersect .
,
+
intersecting
The lines are
%
&
lines.
DE intersects JK 6
MN
The lines are
and
parallel
The lines are
ST
.
.
4
/
5
lines.
ST MN 7
The lines are
WX
The lines are
perpendicular
⊥ YZ WX
and
YZ
8
.
lines.
:
9
;
GO ON Lesson 4-1 Lines
57
Draw each relationship described. 8
QR NO
/
0
2
3
$
⊥ KL CD
9
,
%
Step by Step Problem-Solving Practice
Problem-Solving Strategies ✓ Use a diagram.
Solve. 10
Look for a pattern. Act it out. Solve a simpler problem. Work backward.
MAPS Pia lives on Maple Street, and her aunt lives on Elm Street. On the map, what type of lines do the two streets form? Understand
Read the problem. Write what you know. The streets in the problem are Maple Street and Elm Street .
&MN4U
intersect
.BQMF4U
" WF SD I
The streets
#J
Solve
.
Do the streets intersect to form right angles? So the streets are
lines.
MAPS Keisha walked her dog from her house to the park. Ursula walked from her house to the library. Draw lines for the paths that Keisha and Ursula took. What type of lines do the intersecting two paths form? Check off each step.
✔
Understand
✔
Plan
✔
Solve
✔
Check
Chapter 4 Angles and Circles
,FJTIBT IPVTF -JCSBSZ
6STVMBT IPVTF
1BSL
Copyright © by The McGraw-Hill Companies, Inc.
58
perpendicular
yes
Use a protractor to measure the angle between Maple Street and Elm Street. Do the streets intersect yes to form 90° angles?
Check
11
U
Pick a strategy. One strategy is to use a diagram. Look at the diagram. Do the streets intersect?
L4
0B
Plan
12
DESIGN Mr. Inez bought bunk beds for his twin sons. In the picture to the right, what kind of lines are the two bases?
#BTF
parallel Is the pair of lines shown below intersecting lines? Explain.
13
#BTF
Yes; the lines will continue in both directions indefinitely, so the lines will eventually intersect.
Skills, Concepts, and Problem Solving Draw each figure. 14
Copyright © by The McGraw-Hill Companies, Inc.
16
parallel lines AB and FG "
#
'
(
15
-
WY
17
8
line segment LM
:
.
point X 9
Name each line. Identify the relationships. 18
The lines are
BC
parallel
The lines are BC 19
and
FG
The lines are
perpendicular
⊥
.
lines.
#
$
&
'
EF
The lines are
FG
EF
and
IJ
'
.
lines.
*
(
+
IJ GO ON Lesson 4-1 Lines
59
Name each line. Identify the relationships. 20
TU
The lines are
parallel
The lines are TU 21
and
XY
.
5
6
9
:
lines.
XY
CD
The lines are
and
intersecting
The lines are
WV
$
.
8
lines.
7
%
intersects WV CD 22
ST , CR , and UP .
The lines are
5
CR UP
6
1
ST ⊥ UP $
ST ⊥ CR
23
4
GT , DF , and
The lines are
JS .
24
The lines are
EF
DF ⊥
JS
TS OP
GT ⊥
JS
EF OP
%
TS , and OP .
TS
'
+
EF ,
Copyright © by The McGraw-Hill Companies, Inc.
GT DF
5
3
4 1
&
4
5
' 0
(
Draw each figure. 25 BD MN
#
/
26
⊥ HY AZ
:
" %
60
Chapter 4 Angles and Circles
.
; )
Draw each figure. 27 Parallel lines CD and LM . perpendicular to XY $
28
and VT Parallel lines QR . perpendicular to NO /
-
9
2
:
3
7 %
.
5 0
Solve.
"
29
HOBBIES Kelsey’s Kites designs kites. Classify the lines that perpendicular form the cross section of the kite.
30
CONSTRUCTION Doug has an iron fence that surrounds his house and yard. What type of lines do the vertical
#
$ 9
iron bars form?
" %
parallel
Copyright © by The McGraw-Hill Companies, Inc.
:
Vocabulary Check Write the vocabulary word that completes each sentence. 31
Perpendicular
lines are lines that meet or cross each other
to form right angles. 32
ray A(n) is a part of a line that has one endpoint and extends indefinitely in one direction.
33
A (n)
34
point
Intersecting
is an exact location in space. lines are lines that cross or meet at a point.
GO ON Lesson 4-1 Lines Ingram Publishing/SuperStock
61
35
Writing in Math Explain how to draw line segment JK.
Use a ruler to draw a line. Place point endpoints on both ends of the line. Label the points J and K.
Spiral Review 36
PACKAGING The tissue box that Tobias keeps on his desk at work is shaped like a cube. Each edge of the tissue box is 13 centimeters long. What is the volume of the tissue box? (Lesson 3-5, p. 37)
2,197 cm3
Find the area of each parallelogram. 37
The area of the parallelogram is
(Lesson 3-2, p. 11)
36
square meters or
56
square feet or
36
m2.
N N
38
The area of the parallelogram is
56
ft2.
Copyright © by The McGraw-Hill Companies, Inc.
GU
GU
Convert.
(Lesson 1-4, p. 25)
39
1.5 T = 3,000 lb
41
3 pt =
62
Chapter 4 Angles and Circles
48
fl oz
40
72 c =
42
2.5 gal =
18
20
qt
pt
Lesson
4-2 Angles 5MG2.1 Measure, identify, and draw angles, perpendicular and parallel lines, rectangles, and triangles by using appropriate tools.
KEY Concept " WFSUFY Ȝ# Ȝ
VOCABULARY
#
$
To name an angle, identify the vertex. The vertex of this angle is B. An angle can be named by using the point on one ray, the vertex, and then the point on the other ray, such as ∠ABC or ∠CBA. An angle can also be named using the letter of the vertex, such as ∠B. An angle is sometimes named by a number, such as ∠1. Angles are often measured in degrees. They can be classified, or grouped, according to their measures. You can use a protractor to measure an angle. %
protractor an instrument marked in degrees, used for measuring or drawing angles
degree a unit for measuring angles
&
'
Acute angles measure between 0° and 90°. ∠CEF is an acute angle with a measure of 45°. Right angles measure exactly 90°. ∠DEF is a right angle with a measure of exactly 90°. (
*
)
Copyright © by The McGraw-Hill Companies, Inc.
vertex a point of intersection of the sides of an angle (the plural is vertices)
$
angle a figure formed by two rays with the same endpoint
+
Obtuse angles measure between 90° and 180°. ∠GIJ is an obtuse angle with a measure of 135°. Straight angles measure exactly 180°. ∠HIJ is a straight angle with a measure of exactly 180°. Classify an angle as acute or obtuse to help you decide which scale of the protractor, the inner or outer, gives the angle’s measurement.
GO ON Lesson 4-2 Angles
63
Example 1
YOUR TURN! Identify the type of angle.
Identify the type of angle.
1. The angle does not have a square corner. It is not a right angle.
1. Does the angle look like a right angle?
2. The angle looks greater than 90°. It is an obtuse angle.
less than 2. The angle looks acute angle It is a(n) .
Example 2
YOUR TURN!
&
'
2. Place the center of the protractor at point E. Line up EF with the 0° mark on the protractor.
¡ 9
:
'
:
8
Chapter 4 Angles and Circles
9
. 4. Draw XW
%
'
Copyright © by The McGraw-Hill Companies, Inc.
8
4. Draw ED .
64
3. Use the inner scale. Draw a point W at 75 °
&
&
:
9
'
3. Use the inner scale. Draw a point D at 135°.
&
%
2. Place the center of the protractor at point X . Line up XY with the 0 ° mark on the protractor.
:
9
. 1. Draw XY
90°.
Draw ∠WXY that measures 75°.
Draw ∠DEF that measures 135°. 1. Draw EF .
no
Example 3
YOUR TURN! Measure and identify the angle.
Measure and identify the angle.
1. Place the center of the protractor at the vertex.
9
1. Place the center (hole or ⊥ symbol) of the protractor at the vertex, point Y.
:
;
with the line that extends 2. Line up YZ from 0° to 180° on the protractor.
140°
∠ABC measures
.
∠ABC is a(n)
obtuse
angle.
3. Look at point C. Use the 0° reading to know whether to read the inner scale or the outer scale on the protractor. Read the measure of the angle where BC passes through the inner scale.
with the line that extends 2. Line up AB from 0° to 180° on the protractor.
"
3. Look at point Z. Use the 0° reading to know whether to read the inner scale or the outer scale on the protractor. Read the measure of the angle where YX passes through the inner scale.
#
$
∠XYZ measures 60°. Copyright © by The McGraw-Hill Companies, Inc.
∠XYZ is an acute angle.
Who is Correct? What is the measure of ∠V?
6
Rachel
Ivan
Dion
90°
95°
100°
7
8
Circle correct answer(s). Cross out incorrect answer(s).
GO ON Lesson 4-2 Angles
65
Guided Practice Identify each type of angle.
acute
1
obtuse
2
Draw an angle with the given measurement. 90°
4
130°
3
Step by Step Practice 5
Measure and identify the angle. ' Copyright © by The McGraw-Hill Companies, Inc.
(
)
G
Step 1 Place the center of the protractor at point
.
Step 2 Line up the vertex and the line on the protractor that extends from 0° to 180° with Step 3 Read from the
outer
GH
.
scale.
'
(
)
Chapter 4 Angles and Circles
66
30 °. ∠FGH measures ∠FGH is a(n) acute angle.
Step 4 Read the measure of the angle passes through where GF the outer scale.
Measure and identify each angle. 6
Line up the vertex and the line on the protractor that extends
KL
from 0° to 180° with
inner
Read from the
.
+
scale.
KJ
Read the measure of the angle where through the inner scale.
100
∠JKL measures ∠JKL is a(n)
obtuse
passes ,
-
°. +
angle.
,
-
7
∠MNO measures ∠MNO is a(n)
90 right
°. angle.
.
Copyright © by The McGraw-Hill Companies, Inc.
/
8
∠TUV measures ∠TUV is a(n)
85 acute
°. angle.
5
6
9
∠RST measures
180
0
7
°.
∠RST is a(n) straight angle.
3
4
5
GO ON Lesson 4-2 Angles
67
Step by Step Problem-Solving Practice
Problem-Solving Strategies ✓ Use a diagram.
Solve. 10
Look for a pattern. Guess and check. Solve a simpler problem. Work backward.
BASEBALL While rounding the bases after hitting a home run, Quincy noticed the first and second bases formed an angle. What type of angle is formed by the line from first base to second base and the line from first base to home plate? Understand
Read the problem. Write what you know. Quincy was looking at the angle
first
formed by the line from base to
second base and the line first base
from
home plate
to . TFDPOE
Plan
Pick a strategy. One strategy is to use a diagram.
Solve
Place the center of the protractor on first base. Line up the vertex and the line on the protractor that extends from 0° to 180° with
the line from first base to second base
UIJSE
. IPNF
Read the measure of the angle where passes through the outer scale.
11
68
°.
right
angle.
Review the definition of a
right
The angle is a(n) Check
90
angle.
BILLIARDS How many acute angles does a pool rack have? 3 Check off each step.
✔
Understand
✔
Plan
✔
Solve
✔
Check
Chapter 4 Angles and Circles
(t)Rudi Von Briel/PhotoEdit, (b)Corbis
Copyright © by The McGraw-Hill Companies, Inc.
the line from first base to second base The angle measures
mSTU
12
TIME Justin eats lunch at school at 11:15. What type of angle do the hands on a clock form when the time is 11:15?
obtuse
Ruben said the measure of the angle is 105°. Explain Ruben’s error.
13
Sample answer: The measure of the angle is 75°. Ruben most likely read the outer scale of a protractor.
Skills, Concepts, and Problem Solving
14
straight
15
obtuse
16
acute
17
right
Draw an angle with the measurement given. 25°
19
90°
21
165°
20
105°
18
Copyright © by The McGraw-Hill Companies, Inc.
Identify each type of angle.
GO ON Lesson 4-2 Angles
69
Measure and identify each angle. 22
65
∠ABC measures
acute
∠ABC is a(n)
°. "
angle.
#
23
∠WXY measures
180
$
°.
∠WXY is a(n) straight angle.
24
∠PQR measures ∠PQR is a(n)
25
obtuse
∠STU measures ∠STU is a(n)
170
90 right
8
9
:
2
3
°. 1
angle.
°. angle.
4
5
6
Solve. 26
SAILING Val hoisted a sail to prepare for the start of a race. What types of angles are formed by the sides of the sail?
27
CARS After back surgery, Kono could not sit up straight while riding in a car. He had to recline the seat at an angle that measured 110°. What type of angle was Kono’s reclining position?
obtuse
SAILING This sailboat has triangular sails.
Vocabulary Check Write the vocabulary word that completes each sentence.
angle
28
A(n)
29
A(n) protractor is an instrument marked in degrees, used for measuring or drawing angles.
70
Chapter 4 Angles and Circles
Corbis
is formed by two rays with a common endpoint.
Copyright © by The McGraw-Hill Companies, Inc.
acute
30
A(n) obtuse angle is an angle that measures greater than 90° but less than 180°.
31
A(n) acute less than 90°.
32
Writing in Math Explain how to draw ∠XYZ measuring 125°.
angle is an angle that measures greater than 0° but
. Place the center of the protractor at point Y. Line up YZ with the 0° Draw YZ . mark on the protractor. Use the outer scale. Draw a point X at 125°. Draw YX
Spiral Review 33
FENCES Derek’s horse runs in a paddock that is surrounded by a fence. What type of lines do the slats on the fence form?
parallel
(Lesson 4-1, p. 54)
Find the area of each triangle. 34
Copyright © by The McGraw-Hill Companies, Inc.
35
(Lesson 3-3, p. 19)
The area of the triangle is 20 square yards or 20
ZE
yd2.
ZE
The area of the triangle is 22, 100 square millimeters or 22, 100 mm2.
FENCES Derek’s horse runs in the paddock surrounded by this fence.
NN
NN
Convert.
(Lesson 1-1, p. 4)
75
36
0.75 m =
38
54 m = 0.054 km
cm
37
0.2 cm = 0.02 dm
39
6.3 dm = 630 mm
Lesson 4-2 Angles Corbis
71
Chapter
Progress Check 1
4
(Lessons 4-1 and 4-2)
Draw each figure. 5MG2.1 1
⊥ CD AB
2
−− FG '
(
$ "
# %
Draw an angle with the measurement given. 5MG2.1 3
150°
4
40°
¡ ¡
Name each line. Identify the relationships. 5MG2.1 5
4
5
6
7
6
: 8
9 ;
The lines are ST ll UV
parallel
lines.
The lines are perpendicular ⊥ YZ WX
lines.
7
8
%
2
¡ &
'
¡ 3
∠DEF measures ∠DEF is a(n)
50 acute
°. angle.
∠QRS measures ∠QRS is a(n)
4
90 right
Solve. 5MG2.1
°. angle.
n
9
TIME Is the measure of the angle formed by the hands of the clock greater than, less than, or equal to 90°? See TWE margin.
n
72
Chapter 4 Angles and Circles
Copyright © by The McGraw-Hill Companies, Inc.
Measure and identify each angle. 5MG2.1
Lesson
4-3 Triangles and Quadrilaterals 5MG2.1 Measure, identify, and draw angles, perpendicular and parallel lines, rectangles, and triangles by using appropriate tools.
KEY Concept Classify Triangles by Angles
VOCABULARY
When you classify, you place items in groups. This symbol indicates a right angle.
congruent line segments that have the same length, or angles that have the same measure
right triangle one 90° angle
obtuse triangle one angle greater than 90° and less than 180°
acute triangle all three angles less than 90°
Classify Triangles by Sides
quadrilateral a polygon that has four sides and four angles trapezoid a quadrilateral with exactly one pair of parallel sides parallelogram a quadrilateral with four sides in which each pair of opposite sides is parallel and congruent
This symbol indicates congruent sides. Copyright © by The McGraw-Hill Companies, Inc.
triangle a polygon with three sides and three angles
rectangle a quadrilateral with four right angles; opposite sides are congruent and parallel
equilateral triangle three congruent sides and three congruent angles
rhombus a parallelogram with four congruent sides
This symbol indicates congruent angles.
isosceles triangle At least two congruent sides and two congruent angles
square a rectangle with four congruent sides
scalene triangle no congruent sides GO ON Lesson 4-3 Triangles and Quadrilaterals
73
Classify Quadrilaterals trapezoid quadrilateral with exactly one pair of parallel sides
This symbol indicates parallel sides.
parallelogram each pair of opposite sides is parallel and congruent
rectangle parallelogram with four right angles
opposite angles of each side are also congruent
square rectangle with four congruent sides
rhombus parallelogram with four congruent sides
Symbols
74
parallel lines
congruent sides
congruent angles
right angle
Chapter 4 Angles and Circles
Copyright © by The McGraw-Hill Companies, Inc.
Figures often have symbols that indicate congruence.
Example 1 Identify the figure.
1. The figure has three sides and three angles. The figure is a triangle. 2. Two sides are equal in length. The figure is an isosceles triangle, which means two angles are congruent. 3. One angle is a right angle. The figure is a right, isosceles triangle.
YOUR TURN! Identify the figure.
1. The figure has 3 sides and angles. The figure is a(n)
triangle
3
.
2. Look at the sides. Are any of the sides equal in length, or congruent?
no The figure is a(n)
scalene triangle .
3. Look at the angles in the figure. One angle has a measure
greater than
90°.
obtuse
The figure is a(n)
,
Copyright © by The McGraw-Hill Companies, Inc.
scalene triangle .
Example 2 Identify the figure.
1. The figure has four sides. The figure is a quadrilateral. 2. All sides are equal in length. The figure is a rhombus or a square. 3. All angles are not equal in size. The figure is a rhombus.
YOUR TURN! Identify the figure.
1. The figure has 4 The figure is a(n)
sides.
quadrilateral
.
2. Look at the sides in the figure. There is 1 pair of parallel sides. trapezoid The figure is a(n)
.
GO ON Lesson 4-3 Triangles and Quadrilaterals
75
Who is Correct? Sketch an equilateral triangle.
Aaron
Lakita
Lucinda
Circle correct answer(s). Cross out incorrect answer(s).
Guided Practice Sketch a figure with the description given. 1
obtuse, isosceles triangle
parallelogram
2
3
Identify the figure.
3 Step 1 The figure has sides and triangle The figure is a(n) .
3
angles.
Step 2 Look at the sides in the triangle. Are any of the sides no congruent? The figure is a(n) scalene triangle. Step 3 Look at the angles in the triangle.
90 °. One angle has a measure of right scalene The figure is a(n) , triangle. 76
Chapter 4 Angles and Circles
Copyright © by The McGraw-Hill Companies, Inc.
Step by Step Practice
Identify each figure. 4
number of sides:
3
number of sides equal in length:
5
number of sides:
0
less than
measure of angles: The figure is a(n)
3
number of angles:
90°
acute, scalene triangle 4
.
4
number of angles:
number of pairs of parallel sides: measure of angles:
90
The figure is a(n)
rectangle
2
° .
6
7
The figure is a(n)
The figure is a(n)
acute, isosceles triangle
Step by Step Problem-Solving Practice
Problem-Solving Strategies ✓ Use a diagram.
Copyright © by The McGraw-Hill Companies, Inc.
Solve. 8
SAFETY Miko saw a street sign that looked like the one at the right. Classify the shape of the sign. Understand
Solve
Look for a pattern. Guess and check. Act it out. Work backward.
Read the problem. Write what you know. Miko saw
Plan
square .
a street sign
.
Pick a strategy. One strategy is to use a diagram.
3 The street sign has sides and 3 angles. The figure is a(n) triangle . equal The sides appear to be in length. The figure is a(n) equilateral triangle .
Check
Review the definition of the figure you named.
GO ON
Lesson 4-3 Triangles and Quadrilaterals
77
9
10
FLAGS Delaware’s state flag is shown at the right. Classify rhombus the quadrilateral in the middle of the flag. Check off each step.
✔
Understand
✔
Plan
✔
Solve
✔
Check
HOBBIES Lucille’s vegetable garden is 12 feet long on each of its four sides. Each corner forms a right angle. What type of quadrilateral does Lucille’s garden represent?
square
Can an equilateral triangle be classified as equilateral and right? Explain.
11
No; an equilateral is a triangle with three congruent sides and three congruent angles. Since the 3 angles are congruent and there are 180° in a triangle, each angle must be 180° ÷ 3 = 60°.
Sketch a figure with the description given. 12
equilateral triangle
Chapter 4 Angles and Circles
Corbis
78
13
trapezoid
Copyright © by The McGraw-Hill Companies, Inc.
Skills, Concepts, and Problem Solving
Identify each figure. 14
15
The figure is a(n)
rectangle
The figure is a(n)
obtuse, isosceles triangle .
.
16
17
The figure is a(n)
right, scalene triangle
The figure is a(n)
rhombus
.
18
19
The figure is a(n)
parallelogram
The figure is a(n)
obtuse, scalene triangle .
.
20
Copyright © by The McGraw-Hill Companies, Inc.
.
21
The figure is a(n)
trapezoid
The figure is a(n) .
acute, isosceles triangle .
Solve. 22
SHOPPING At the Purse Pagoda, the evening purses were on sale. What type of quadrilateral is the purse shown at the right?
trapezoid 23
ART Ace made a model of the Great Pyramid in Egypt. Each triangle that made his pyramid had sides of length 30 centimeters, 26 centimeters, and 30 centimeters. What type of triangle did each side make?
isosceles
GO ON Lesson 4-3 Triangles and Quadrilaterals
Comstock/Imagestate
79
Vocabulary Check Write the vocabulary word that completes each sentence. 24
trapezoid A(n) of parallel sides.
is a quadrilateral with exactly one pair
25
A(n) isosceles triangle the same length.
is a triangle with at least two sides of
26
rhombus A(n) is a parallelogram with four congruent sides. This figure may or may not have right angles.
27
A(n)
28
obtuse triangle is a triangle with one angle greater A(n) than 90° and less than 180°.
29
Writing in Math Explain how to identify a figure.
scalene triangle
is a triangle with no congruent sides.
Find the number of sides and angles of the figure. Look at the sides and angles to compare the lengths and degrees of the figure so that you can classify the figure.
30
ARCHITECTURE Presently, the famed Tower of Pisa is leaning at about a 6° angle. This makes the angle of the leaning side to the ground measure about 84°. What type of an angle is the angle of the leaning side to the ground?
acute
(Lesson 4-2, p. 63)
Find the volume of the rectangular prism.
(Lesson 2-4, p. 77)
ARCHITECTURE Tower of Pisa
31
The volume of the rectangular prism is cubic units.
80
Chapter 4 Angles and Circles
Corbis
84
Copyright © by The McGraw-Hill Companies, Inc.
Spiral Review
Lesson
4-4 Add Angles KEY Concept Pairs of Angles
Descriptions
" ¡ #
%
%
N%#"
$
¡
¡ #
N%#$
"
This symbol means the “measure” of angle ABC.
¡ :
;
:
9
#
$
VOCABULARY complementary angles two angles that have measures with a sum of 90°
m∠ABC + m∠DBA 40° + 50° = 90°
supplementary angles two angles that have measures with a sum of 180°
9
¡
¡ ¡
complementary angles m∠ABC = 40° m∠DBA = 50°
3
9
"
5MG2.1 Measure, identify, and draw angles, perpendicular and parallel lines, rectangles, and triangles by using appropriate tools. 5MG2.2 Know that the sum of the angles of any triangle is 180° and the sum of the angles of any quadrilateral is 360° and use this information to solve problems. 6MG2.2 Use the properties of complementary and supplementary angles and the sum of the angles of a triangle to solve problems involving an unknown angle.
¡
¡ ¡ :
3
;
Copyright © by The McGraw-Hill Companies, Inc.
supplementary angles m∠XYZ = 115° m∠RYX = 65° m∠XYZ + m∠RYX 115° + 65° = 180° Measures of Figures
Descriptions ¡
¡
¡
¡
¡ ¡
The sum of the measures of the angles of a triangle is 180°.
4VNPGBOHMFT ¡
4VNPGBOHMFT ¡
The sum of the measures of the angles of a quadrilateral is 360°. 180° + 180° = 360° Complementary angles with a common ray form a right angle. Supplementary angles with a common ray form a straight angle.
GO ON Lesson 4-4 Add Angles
81
Example 1
YOUR TURN!
Identify the angles as complementary or supplementary. Find the missing angle. 1. The sum of the ( ¡ measures of ∠FHG ¡ Y ¡ and ∠GHJ equals 180. ' ) + The sum of the measures of complementary angles equals 180.
" # ¡ $
¡ %
m∠BCD + m∠ACB = 90 25° + ?° = 90° ?° = 65°
m∠FHG + m∠GHJ = 180 100° + ?° = 180° ?° = 80° ∠FHG and ∠GHJ are supplementary angles. m∠GHJ = 80
Example 2
3. 25° + 65° = 90° m∠ACB =
65
YOUR TURN!
Find the measure of the missing angle.
¡
)
1. Find the sum of the measures of the known angles. m∠G + m∠H = 75° + 50° = 125° 2. The sum of the measures of the angles of a triangle is 180°. Subtract the sum of the known measures from 180°. 180° - 125° = 55° The measure of the missing angle is 55°. Chapter 4 Angles and Circles
m∠Q + m∠R + m∠T
90 ° + = 245 ° =
75 ° +
5 ¡ 4
2
¡
¡
¡
3
80 °
2. The sum of the measures of the angles of a quadrilateral is 360 °. Subtract the sum of the known measures from 360 °.
360 ° - 245 ° = 115 ° The measure of the missing angle is 115 °.
Copyright © by The McGraw-Hill Companies, Inc.
¡
¡
Find the measure of the missing angle. 1. Find the sum of the measures of known angles.
'
82
1. The sum of the measures of angles equals 90 . The sum of the measures of complementary angles equals 90 .
2. Find the measure of ∠ACB.
2. Find the m∠GHJ
(
Identify the angles as complementary or supplementary. Find the missing angle.
Example 3
YOUR TURN!
Sketch supplementary angles when one angle’s measure is 45.
Sketch complementary angles when one angle’s measure is 70.
5. Find the missing angle by using the protractor’s scale. The supplementary angle is 135°.
¡
4. Find the given angle, 45°, inside the 180° angle and make a dot. Use a straightedge to draw the ray that connects the vertex and the dot at 45° from either of the two existing rays.
3. The sum of supplementary angles equals 180°. Find the angle on the protractor’s scale and make a dot. Use a straightedge to draw a second ray that connects at the vertex.
2. Draw one side of the angle and place an arrow at the end of the ray.
¡
1. Use the center point on the protractor as the common vertex of the supplementary angles.
7FSUFY
7FSUFY
Copyright © by The McGraw-Hill Companies, Inc.
¡
¡
¡
1. Use the center point on the protractor as the common vertex of the supplementary angles. 2. Draw one side of the angle and place an arrow at the end of the ray. 3. The sum of complementary angles equals 90° . Find the angle on the protractor’s scale and make a dot. Use a straightedge to draw a second ray that connects at the vertex. 4. Find the given angle, 70°, inside the 90° angle and make a dot. Use a straightedge to draw the ray that connects the vertex and the dot at 70° from either of the two existing rays. 5. Find the missing angle by using the protractor’s scale. The complementary angle is 20°.
Who is Correct? ∠MNO and ∠PNQ are supplementary angles. The measure of ∠MNO is 67°. What is the measure of ∠PNQ?
Diego
Patrick
90° - 67° = 23°
Malia
180° - 67° = 123°
180° - 67° = 113°
Circle correct answer(s). Cross out incorrect answer(s).
GO ON Lesson 4-4 Add Angles
83
Guided Practice Identify the angles as complementary or supplementary. Find the missing angle. 1
complementary . The angles are The measure of the missing angle is 75° .
2
supplementary . The angles are The measure of the missing angle is 105° . 0
,
¡
¡ /
¡
¡ ¡
.
2
1
4
Step by Step Practice 3
Find the measure of the missing angle. Step 1
Find the sum of the measures of known angles. m∠D + m∠E =
49 ° +
63 ° = 112 °
Step 2 The sum of the measures of the angles of a triangle is 180 °. Subtract the sum of measures of the known angles from 180 °.
180 ° - 112 ° =
" ¡ ¡ % ¡
¡ $
84 ° + 143 ° + 103 ° = 330 ° 360 ° - 330 ° = 30 ° The measure of the missing angle is 30 °. m∠A + m∠C + m∠D =
84
Chapter 4 Angles and Circles
&
¡
¡
'
Copyright © by The McGraw-Hill Companies, Inc.
68 °.
Find the measure of each missing angle.
#
¡
68 °
The measure of the missing angle is
4
%
Find the measure of each missing angle. 5
: ¡
¡
¡
9
;
58 ° + = 155 ° 180 ° - 155 ° = 25 °
97 °
m∠Y + m∠Z =
The measure of the missing angle is 6
25 °. 7
-
) ¡
¡
*
.
¡
¡
¡
, +
/
The measure of the missing angle is 90 °.
The measure of the missing angle is 79 °.
Sketch each type of angle given. Sketch supplementary angles when one angle’s measure is 85°.
9
Sketch complementary angles when. one angle’s measure is 41°.
8
3
¡ ¡
4
5
3
¡ ¡
8
Copyright © by The McGraw-Hill Companies, Inc.
¡ ¡
7
GO ON Lesson 4-4 Add Angles
85
Step by Step Problem-Solving Practice
Problem-Solving Strategies ✓ Draw a diagram.
Solve. 10
ART Samir drew his version of a pine tree. It was the shape of a triangle. It had a 36° angle and a 70° angle. What was the measure of the third angle of Samir’s pine tree? Understand
Look for a pattern. Guess and check. Act it out. Solve a simpler problem.
Read the problem. Write what you know. The measure of one angle of Samir’s pine tree was 36 °. The measure of the other angle was 70 °.
Plan
Pick a strategy. One strategy is to draw a diagram.
Solve
Find the sum of the measures of the known angles.
36 ° +
70 ° = 106 °
36°
Subtract this sum from 180°. 180° - 106 ° =
The measure of the third angle is
36° + 70° +
74 ° = 180 °
AGRICULTURE A cow pasture is shaped like a parallelogram. Two of the angles measure 68° and 112°. What are the measures of the 68° and 112° third and fourth angles? Check off each step.
✔
Understand
✔
Plan
✔
Solve
✔
Check
SPORTS Hughes Hardware engraves trophies. The trophies have sides in the shape of isosceles triangles. The base angles of each side measure 73°. How many degrees is the top angle?
34° 86
74 °.
Chapter 4 Angles and Circles
Copyright © by The McGraw-Hill Companies, Inc.
12
?
Add the three angle measures. The sum should equal 180°.
Check
11
70°
74 °
Explain how you can find the sum of the angle measures in a pentagon.
13
Divide the pentagon into triangles. Since the sum of the angles of each triangle equals 180°, the sum of the angle measures of a pentagon must equal 3(180), or 540°.
Skills, Concepts, and Problem Solving Identify the angles as complementary or supplementary. Find the missing angle. 14
complementary . supplementary . 15 The angles are The angles are 65° The measure of the missing angle is . The measure of the missing angle is 95° . ,
. ¡
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Copyright © by The McGraw-Hill Companies, Inc.
Find the measure of each missing angle. 4
16
17
1 ¡
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5
6
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2
The measure of the missing angle is 113 °. 18 ,
+
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The measure of the missing angle is 105 °. 19
-
4 ¡
:
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¡
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9
¡
¡
;
.
The measure of the missing angle is 110 °.
The measure of the missing angle is 50 °.
GO ON
Lesson 4-4 Add Angles
87
Sketch each type of angle given. Sketch complementary angles when one angle’s measure is 58°.
21
Sketch supplementary angles when one angle’s measure is 52°.
'
,
)
(
+
20
-
Solve. CROSS-COUNTRY A cross-country team ran on a path near the school that was in the shape of a quadrilateral. Three angles of the area measured 42°, 85°, and 102°. How many degrees was the fourth angle? 131°
23
SCULPTURE The geometric sculpture that sits at the entrance of Ugo Park has a triangle for its base. It has angles that measure 38° and 104°. What is the measure of the third angle? 38°
24
PETS The play area at Pooch Paradise is in the shape of a quadrilateral. Three angles of the area measure 25°, 112°, and 82°. How many degrees is the fourth angle? 141°
25
GARDENING The butterfly garden at Sweet Valley Academy is placed in the corner of the courtyard. It is in the shape of a triangle. The butterfly garden has angles that measure 62° and 47°. What is the measure of the third angle? 71°
Vocabulary Check Write the vocabulary word that completes each sentence. 26
Complementary
angles are two angles that have measures
with a sum of 90°. 27
Supplementary
angles are two angles that have measures
with a sum of 180°.
88
Chapter 4 Angles and Circles
Copyright © by The McGraw-Hill Companies, Inc.
22
28
Writing in Math Explain how to find the missing angle of a quadrilateral when the measures of three angles are given.
Sample answer: Find the sum of the known angle measures. The sum of the measures of the angles of a quadrilateral is 360°. Subtract the sum of the known angle measures from 360°.
Spiral Review 29
LETTERS Which letter in the name KEN forms perpendicular lines? E (Lesson 4-1, p. 54)
Find the volume of each rectangular prism.
(Lesson 3-5, p. 37)
30
31
Copyright © by The McGraw-Hill Companies, Inc.
The volume of the rectangular prism is 160 cubic units.
Find the area of each rectangle.
The volume of the rectangular prism is 126 cubic units.
(Lesson 3-1, p. 4)
32
33 GU DN
GU DN
The area of the rectangle is 108 square centimeters or 108 cm2.
The area of the square is 49 ft2. square feet or
49
Lesson 4-4 Add Angles
89
Chapter
Progress Check 2
4
(Lessons 4-3 and 4-4)
Draw a figure with the description given. 5MG2.1 1
acute, scalene triangle See TWE margin.
2
trapezoid See TWE margin.
4
Draw complementary angles.
Draw each type of angles given. 5MG2.1 Draw supplementary angles.
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3
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Classify each figure. 5MG2.1 5
6
The figure is a right, isosceles triangle .
rhombus
The figure is a
7
8
(
+
-
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, )
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The measure of the missing angle is 77 °.
Solve. 5MG2.1, 5MG2.2 REPAIRS A school bus was placed on a lift so that the repair workers could see under the bus. What type of quadrilateral does the side of the lift represent?
parallelogram 90
.
*
The measure of the missing angle is 71 °.
9
¡
Chapter 4 Angles and Circles
Copyright © by The McGraw-Hill Companies, Inc.
Find the measure of each missing angle. 5MG2.2, 6MG2.2
.
Lesson
4-5 Congruent Figures 7MG3.4 Demonstrate an understanding of conditions that indicate two geometrical figures are congruent and what congruence means about the relationships between the sides and angles of the two figures.
KEY Concept /
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VOCABULARY ¡
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corresponding sides −−− −− MN PQ −−− −− MO PR −−− −− NO QR
congruent figures figures having the same size and the same shape
3
1
corresponding angles
corresponding angles angles of congruent figures that have the same measure
m∠M m∠P m∠N m∠Q m∠O m∠R
MNO and PRQ are congruent because corresponding sides and corresponding angles are congruent.
corresponding sides sides of congruent figures that have the same length
5IFTZNCPMɁNFBOT iJTDPOHSVFOUUPw
The number of hash marks on a line segment indicates which sides are congruent.
Copyright © by The McGraw-Hill Companies, Inc.
Example 1
YOUR TURN!
STU and FGH are congruent. Identify corresponding sides and corresponding angles. )
5
Quadrilateral ABCD and quadrilateral WXYZ are congruent. Identify corresponding sides and corresponding angles. "
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1. Find the corresponding sides. −− −− ST corresponds to FG. −− −− SU corresponds to FH. −−− −− TU corresponds to GH. 2. Find the corresponding angles. ∠S corresponds to ∠F. ∠T corresponds to ∠G. ∠U corresponds to ∠H.
9
; ¡ ¡ 8
1. Find the corresponding sides. −− −− −− WX AB corresponds to corresponds to −− ; BC −−− −−− CD corresponds to YZ ; AD corresponds to
−− XY . −− WZ .
2. Find the corresponding angles.
∠A corresponds to ∠W ; ∠B corresponds to ∠X . ∠C corresponds to ∠Y ; ∠D corresponds to ∠Z .
Lesson 4-5 Congruent Figures
91
Example 2
YOUR TURN! Are the quadrilaterals congruent?
Are the triangles congruent?
0 4
7
¡
¡ 3
¡
3
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5
8
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4
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1
6 2
1. Find the corresponding sides. −− −−− RS WV
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5
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.
1. Find the corresponding sides. −− −− −−− −− MN QR −− RS NO −− −− −− ST OP QT MP
2. Find the corresponding angles. m∠S m∠V
2. Find the corresponding angles.
RST and WVU are not congruent because these triangles do not have 3 corresponding sides and 3 corresponding angles.
m∠Q m∠ M m∠R m∠ N m∠S m∠ O m∠T m∠ P Quadrilateral QRST and quadrilateral congruent MNOP are because corresponding sides and corresponding congruent angles are .
Copyright © by The McGraw-Hill Companies, Inc.
Who is Correct? Mrs. Johnson asked her students to draw two congruent triangles.
Andrea
Shannon
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Chapter 4 Angles and Circles
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Circle correct answer(s). Cross out incorrect answer(s). 92
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Guided Practice Sketch two congruent figures. 1
triangles 0
7
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quadrilaterals
2
3
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Step by Step Practice Are the triangles congruent? -
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+
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/
Step 1
Find the corresponding sides.
−− −− −− −−− HI and LM are corresponding sides; LM HI . −− −− − − IJ and MN are corresponding sides; IJ MN . −− −− −− −− JH and NL are corresponding sides; JH NL . Step 2
Find the corresponding angles. ∠H and ∠ L
are corresponding angles; m∠H m∠ L .
∠I and ∠ M are corresponding angles; m∠I m∠ M . ∠J and ∠ N
are corresponding angles; m∠J m∠ N .
congruent HIJ and LMN are because corresponding sides and corresponding congruent angles are . GO ON Lesson 4-5 Congruent Figures
93
%
4
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¡
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Find the corresponding sides. −− −−− LM BC −− −−− MN CD −− −− NL DB
Find the corresponding angles. ∠L ∠B ∠M ∠C ∠N ∠D
congruent LMN and BCD are because corresponding congruent sides and corresponding angles are . Are the figures congruent? Quadrilateral ABCD and quadrilateral not congruent because VWXY are corresponding sides and corresponding not congruent . angles are
congruent EFG and JKL are because corresponding sides and corresponding angles are congruent .
Chapter 4 Angles and Circles
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5
Step by Step Problem-Solving Practice
Problem-Solving Strategies Draw a diagram. Look for a pattern. Guess and check. ✓ Act it out. Solve a simpler problem.
Solve. OFFICE Lester saw two stacks of plain white paper by the copy machine. He decided that it did not matter which stack of paper he used because both have four right angles. Below is a picture of each sheet of paper Lester could use. Are the sheets of paper congruent?
7
Understand
Read the problem. Write what you know. Each sheet of paper has four right angles. Each sheet of paper is a rectangle .
Plan
Pick a strategy. One strategy is to act it out.
Solve
Measure the side lengths of each sheet of paper. Do the corresponding sides of each piece of paper no have the same length? The sheets of paper
Trace the sheets of paper. Cut them and place them at top of each other. Do the sheets of paper have the same shape and size? no Can they be rotated to have the same shape and size? no
Check
Copyright © by The McGraw-Hill Companies, Inc.
are not congruent.
. 8
9
TRIANGLES Are triangles ABC and XYZ congruent? yes Check off each step.
✔
Understand
✔
Plan
✔
Solve
✔
Check
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FLAGS An image of the United States flag is shown. Are the yes stars on the flag congruent? Are all triangles that have angle measures 30°, 60°, and 90° congruent? Explain.
10
No; the side lengths of the triangles could be different.
GO ON Lesson 4-5 Congruent Figures
Hespenheide Design
95
Skills, Concepts, and Problem Solving Sketch two congruent figures. 11
quadrilaterals "
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Are the figures congruent? +
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Find the corresponding sides.
−− HI − IJ −− JH
Find the corresponding angles.
∠D ∠H ∠E
∠I
∠F
∠J
congruent DEF and HIJ are sides and corresponding angles are
96
Chapter 4 Angles and Circles
because corresponding .
congruent
Copyright © by The McGraw-Hill Companies, Inc.
−− DE −− EF −− FD
14
¡
Are the figures congruent? 15
Quadrilateral PQRS and quadrilateral FGHI congruent are because corresponding sides and corresponding congruent angles are .
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Solve. 16
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POLYGONS There are some polygons that are congruent to each other. Which of the following polygons are congruent to pentagon ABCDE?
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pentagon UYXWV 17
BADGES Mrs. O’Shay is designing new badges for the safety patrol officers. She has to decide between the two designs pictured at the right. Are the figures congruent? Explain.
No; the figures
Copyright © by The McGraw-Hill Companies, Inc.
are not the same shape. Vocabulary Write the vocabulary word that completes each sentence. 18
Congruent figures have the same size and the same shape. angles
19
Congruent same size.
20
Congruent length.
21
Writing in Math Explain how to determine if two triangles are congruent.
sides
are angles that have the are sides that have the same
Find corresponding sides and angles. If the corresponding sides and angles are congruent, then the two triangles are congruent. GO ON Lesson 4-5 Congruent Figures
97
Spiral Review 22
TILE Each shower tile in Danica’s bathroom is shaped like a rhombus. One angle of each tile measures 115°. What are the measures of the other three angles? (Lesson 4-4, p. 81)
115°, 65°, and 65°
Measure and classify each angle. 23
(Lesson 4-2, p. 63)
30 ∠JKL measures °. ∠KL is a(n) acute angle because its angle measure is greater than 0° but less than 90°. +
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,
24
-
∠FGH measures
90
°. ∠FGH is a(n)
right
angle.
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25
The area of the square is square units.
98
Chapter 4 Angles and Circles
(Lesson 2-3, p. 71)
64
26
The area of the rectangle is square units.
42
Copyright © by The McGraw-Hill Companies, Inc.
Find the area of each figure.
Lesson
4-6 Pythagorean Theorem KEY Concept The Pythagorean Theorem 2 2 2 states that a +b =c
.
D
5MG2.1 Measure, identify, and draw angles, perpendicular and parallel lines, rectangles, and triangles by using appropriate tools. 7MG3.3 Know and understand the Pythagorean theorem and its converse and use it to find the length of the missing side of a right triangle and the lengths of other line segments and, in some situations, IZQPUFOVTF empirically verify the Pythagorean theorem by direct measurement.
B
VOCABULARY MFHT
2
2
a +b = c 2 2 3 + 4 = 52
2
(3×3) + (4×4) = (5×5) 9 + 16 = 25 25 = 25 Copyright © by The McGraw-Hill Companies, Inc.
legs of a triangle two sides that form the right angle in a right triangle
To prove the Pythagorean Theorem, count the number of squares beside each leg of the triangle to the left. The total is equal to the number of squares beside the hypotenuse. You can also use the formula.
C
hypotenuse side opposite the right angle in a right triangle Pythagorean Theorem the sum of the squares of the lengths of the legs in a right triangle is equal to the square of length of the hypotenuse
Use the formula. Substitute the measure of each side.
square root one of the two equal factors of a number
Simplify the exponents. Multiply. Add.
square number the product of a number multiplied by itself
You can use the Pythagorean Theorem to find the length of the hypotenuse given the lengths of the legs, or the length of one leg given the lengths of the hypotenuse and the other leg. A Note on Square Numbers and Square Roots: A square number, such as 42, indicates that 4 should be multiplied by itself 2 times. 2
4 = 4 × 4 = 16 To find a square root , such as √ 16 , think about what number times itself equals 16. √ 16 = √ 4×4 =4
(-4) × (-4) is another square root of 16. Every positive number has two different square roots. The symbol √ is used to show the positive square root of a positive number; for example, √ 16 = 4.
GO ON Lesson 4-6 Pythagorean Theorem
99
Example 1 Find the length of the hypotenuse of the right triangle. 1. From the figure, you know a = 6 and b = 8.
D
2. Substitute the values of a and b in the Pythagorean Theorem and solve for c. 2
2
2
Use the formula.
2
2
6 +8 =c
2
Substitute the measure of each side.
36 + 64 = c
2
Simplify the exponents.
2
Add 36 and 64.
a +b =c
100 = c √ 100 = √ c2 10 = c
Take the square root of both sides. Simplify the square root.
The length of the hypotenuse is 10 units. YOUR TURN! Find the length of the hypotenuse of the right triangle.
1. From the figure, you know a=
9
and b =
D
12 .
2. Substitute the values of a and b in the Pythagorean Theorem and solve for c. 2
2
a +b =c 2
81
+
12
+ 144
2
=c
2
=c
2
225 = c 2 225 = √ c2 15
=c
The length of the hypotenuse is
15
units.
Numbers like 4, 9, 25, 49, and 100 are called perfect squares because their square roots are whole numbers.
100
Chapter 4 Angles and Circles
Copyright © by The McGraw-Hill Companies, Inc.
9
2
Example 2 Find the length of the leg of the right triangle. 1. From the figure, you know b = 60 and c = 75. 2. Substitute the values of b and c in the Pythagorean Theorem and solve for a. a2 + b2 = c2 2
2
Use the formula. 2
a + 60 = 75
B
Substitute the measure of each side.
a2 + 3,600 = 5,625
Simplify the exponents.
a2 = 5,625 - 3,600
Subtract 3,600 from 5,625.
2
a = 2,025 √a2 = √ 2,025
Simplify. Take the square root of both sides.
a = 45
Simplify the square root.
The length of the leg is 45 units. YOUR TURN! Find the length of the leg of the right triangle. 1. From the figure, you know 12 20 a= and c = .
C
2. Substitute the values of a and c in the Pythagorean Theorem and solve for b.
Copyright © by The McGraw-Hill Companies, Inc.
a2 + b2 = c2
122 144
+ b2 = + b2 = b2 = b2 = √b2 = b=
202 400 400-144 256 √ 256 16
The length of the leg is
16
units.
Whenever you have a right triangle, then you know its sides follow the Pythagorean Theorem, a2 + b2 = c2. You can also do this in reverse. If you have a triangle and you know that the sum of the squares of the two smaller sides equals the square of the larger side (its sides follow the Pythagorean Theorem), then the triangle must be a right triangle.
GO ON
Lesson 4-6 Pythagorean Theorem
101
Example 3
YOUR TURN! Determine if a triangle with the following lengths is a right triangle.
Determine if a triangle with the following lengths is a right triangle. 13
5
15
8 12
12
Determine which side is the largest. 13
Determine which side is the largest.
Substitute the values into a2 + b2 = c2, ensuring that the largest number is c, and solve. a2 + b2 = c2 52 + 122 = 132 25 + 144 = 169 169 = 169
Use the formula. Substitute the measure of each side. Simplify the exponents. Add 25 and 144
Substitute the values into a2 + b2 = c2, ensuring that the largest number is c, and solve.
Since this is a true statement, the triangle is a right triangle.
a2 + b2 = c2
82 64
122 = 152 + 144 = 225 208 ≠ 225 +
Since this is not a true statement, the triangle is not a right triangle.
Mr. Fernandez told his students to draw and label the sides of a right triangle. Which student’s triangle is correct?
Rhonda
13 + 14 = 27
2 2 1002 - 60 = 80
Circle correct answer(s). Cross out incorrect answer(s).
102
Chapter 4 Angles and Circles
Sam
Pete
2 2 602 + 80 = 100
Copyright © by The McGraw-Hill Companies, Inc.
Who is Correct?
Guided Practice Use the measurements given to draw a right triangle or indicate if a right triangle cannot be drawn with the measurements. 1
4 cm, 3 cm, and 5 cm
2
A right triangle cannot be drawn with these measurements.
DN
DN
15 in., 20 in., and 30 in.
DN
Step by Step Practice 3
Find the length of the hypotenuse of the right triangle.
D
Step 1 Find the lengths of the legs, a and b. a=
8
15
and b =
Step 2 Substitute the values of a and b in the Pythagorean Theorem and solve for c. 2
2
a +b =c
8
2
Copyright © by The McGraw-Hill Companies, Inc.
64
2
15 2 = c 2 2 + 225 = c 289 = c 2 c2 √ 289 = √ 17 = c
+
The length of the hypotenuse is
17
units.
Find the length of the leg or hypotenuse of each right triangle. 2
2
a +b =c
4
2
2 a + 12 = 13 2 a + 144 = 169 2 a = 25 √ a 2 = √25 a= 5 2
B
2
The length of the leg is
5
units. GO ON Lesson 4-6 Pythagorean Theorem
103
5
B
The length of the leg is
7
units.
6 D
The length of the hypotenuse is
7
50
units.
B
The length of the leg is
9
units.
8
yes; 182 + 802 = 822
9
no; 32 + 52 ≠ 49
104
Chapter 4 Angles and Circles
Copyright © by The McGraw-Hill Companies, Inc.
Determine if each triangle is a right triangle.
Step by Step Problem-Solving Practice
Problem-Solving Strategies ✓ Draw a diagram.
Solve. 10
Understand
Read the problem. Write what you know. The ladder touches the wall at a height of feet. The ladder is wall.
9
12
feet away from the base of the
Plan
Pick a strategy. One strategy is to draw a diagram. Then, use the Pythagorean Theorem to find the length of the ladder.
Solve
Substitute 9 for a and 12 for b in the Pythagorean Theorem and solve for c. 2
2
a +b =c
Copyright © by The McGraw-Hill Companies, Inc.
Look for a pattern. Act it out. Solve a simpler problem. Work backward.
HOME IMPROVEMENT Jerry’s ladder is resting against a wall. The top of the ladder touches the wall at a height of 12 feet. The bottom of his ladder is 9 feet away from the base of the wall. How long is Jerry’s ladder? 15 feet
D
2
9 2 + 12 2 = c 2 81 + 144 = c 2 225 = c 2 √ 225 = √ c2 15 = c The height of the ladder is
GU
15
feet.
Use a calculator to check your answer.
Check
11
GU
CONSTRUCTION A staircase to the attic in Cassandra’s house has a length of 10 feet. The top of the stairs meets the wall at a height of 8 feet. How far away is the bottom of the staircase from the wall? 6 feet Check off each step.
✔
Understand
✔
Plan
✔
Solve
✔
Check
GO ON Lesson 4-6 Pythagorean Theorem
105
12
ADVERTISING Splash Village advertised a water slide that is 20 yards long and 12 yards from the base of the steps to the end of the slide. What is the height of the steps that reach the top of the slide? 16 yards Can line segments with lengths 30 inches, 30 inches, and 50 inches form a right triangle? Explain.
13
No; substitute 30 for a, 30 for b, and 50 for c in the Pythagorean Theorem: 2
2
2
30 + 30 50 ; 900 + 900 2,500; 1,800 ≠ 2,500.
Skills, Concepts, and Problem Solving Use the measurements given to draw a right triangle or indicate if a right triangle cannot be drawn with the measurements. 14
5 cm, 10 cm, and 15 cm
15
A right triangle cannot be drawn with these measurements.
6 cm, 8 cm, and 10 cm DN DN
DN
Find the length of the leg or hypotenuse of each right triangle. 16
17
D
The length of the leg is
12
units.
The length of the hypotenuse is 13 units.
Determine if each triangle is a right triangle. 18
19
20
16
12
no; 72 + 82 ≠ 122
106
Chapter 4 Angles and Circles
yes; 122 + 162 = 202
Copyright © by The McGraw-Hill Companies, Inc.
B
Solve. 20
TRAVEL Reggie left his house and drove 20 miles east and then 15 miles south. If Reggie follows a straight line to his house, how far is Reggie from his house?
NJFBTU
NJTPVUI
25 miles 21
DISTANCE A lighthouse on a cliff near Gina’s house is 12 meters tall. Gina stood 9 meters away from the base of the lighthouse. How far away was Gina from the top of the lighthouse?
15 meters Vocabulary Check Write the vocabulary word that completes each sentence. 22
hypotenuse A(n) right triangle.
is the side opposite the right angle in a
23
square number A(n) by itself.
is the product of a number multiplied
24
25
Pythagorean Theorem The states that the sum of the squares of the lengths of the legs in a right triangle is equal to the square of length of the hypotenuse. Writing in Math Explain how to find the length of the hypotenuse of a right triangle when the lengths of the legs are known. 2
Copyright © by The McGraw-Hill Companies, Inc.
Substitute the values of a and b in the Pythagorean Theorem (a 2 + b = c 2) and solve for c.
Spiral Review 26
POLYGONS Which two of the following polygons are congruent? (Lesson 4-5, p. 91) 120°
60° 60°
60°
120° 150°
60°
60°
120°
30°
30°
150°
60°
120° 30°
30°
60°
120°
The second and fifth polygons are congruent.
Lesson 4-6 Pythagorean Theorem
107
Chapter
Progress Check 3
4
(Lessons 4-5 and 4-6)
Draw two congruent figures. 5MG2.1, 5MG2.2 1
triangles
2
See TWE margin.
quadrilaterals
See TWE margin.
Use the measurements given to draw a right triangle or indicate if a right triangle cannot be drawn with the measurements. 5MG2.1, 7MG3.3 3
1.3 cm, 8.4 cm, and 8.5 cm
4
12 mm, 13 mm, and 20 mm
A right triangle cannot be drawn with these measurements.
DN DN DN
5
Are the figures congruent? 7MG3.4 -
6
2 ¡
¡
¡
1
¡
.
"
¡ ¡
Find the length of the leg of the right triangle. 7MG3.3
B
;
The length of the leg is
10
units.
Solve. 7MG3.3 7
FLAGS A flagpole stands in front of a city building. The base of the flagpole is 8 yards away from a spotlight that shines on the flag. The distance from the spotlight to the top of the flagpole is 10 6 yards yards. How tall is the flagpole? EJBHPOBM
8
RECTANGLES What is the length of the diagonal 13 of rectangle MNOP?
. 1
108
Chapter 4 Angles and Circles
/ D
0
Copyright © by The McGraw-Hill Companies, Inc.
PLM and QAZ are not congruent because corresponding sides and corresponding angles are not congruent .
Lesson
4-7 Circles
6MG1.2 Know common estimates of pi (3.14, 22 ) and 7 use these values to estimate and calculate the circumference and the area of circles; compare with actual measurements.
_
KEY Concept A circle is the set of all points in a plane that are the same distance from a point called the center. DFOUFS SBEJVT S
circle a closed figure in which all points are the same distance from a fixed point called the center of the circle
E EJBNFUFS
DJ
SD V
Formulas
NGFSFO
DF
Descriptions The length of the diameter is twice the length of the radius of a circle.
d = 2r diameter
S
radius
E S
circumference
Copyright © by The McGraw-Hill Companies, Inc.
C = πd
C = 2πr
or
diameter
radius
A = πr2 Area
radius
VOCABULARY
The ratio of the circumference of every circle to its diameter is always the same number: π C =π d
_
The area of any circle is always equal to π times the radius squared.
Pi (π) is an irrational number. It is not possible to write the exact value of π. To calculate using π, either use a calculator with a π button, or substitute one of the approximate values 22 . of π, 3.14 or ___ 7
chord a line segment with both endpoints on a circle diameter a chord that passes through the center of a circle radius a line segment that connects the center of a circle to a point on the circle circumference the distance around a circle pi (π) the ratio of the circumference of a circle to the diameter of the same circle The value of π is 22 approximately 3.14 or ___. 7
When you make a substitution for π, you have to change the = symbol to ≈ symbol to show that you are making an estimate. GO ON Lesson 4-7 Circles
109
Example 1
YOUR TURN! What is the circumference of the circle?
What is the circumference of the circle? 1. The diameter is 6 centimeters; d = 6.
GU DN
2. Substitute 6 for d and 3.14 for π in the circumference formula. C = πd C ≈ 3.14 × 6 C ≈ 18.84 The circumference of the circle is about 18.84 centimeters.
1. Find the length of the radius. The radius is so r = 8 .
8
feet,
2. Substitute 8 for r and 3.14 for π in the circumference formula. C = 2πr C ≈ 2 × 3.14 × C ≈ 50.24
8
The circumference of the circle is about
50.24 feet.
Example 2
YOUR TURN! What is the area of the circle?
What is the area of the circle? ZE
2. Substitute 10 for r and 3.14 for π in the area formula. A = πr2
1. Find the length of the radius.
The diameter is 14 inches. r = 14 ÷ 2 = 7 . 22 for π in 2. Substitute 7 for r and ___ 7 the area formula.
A ≈ 3.14 × 102
A = πr2
A ≈ 3.14 × 100
22 × A ≈ ___ 7 22 × A ≈ ___ 7
2200 A ≈ _____ or 314 7 The area of the circle is about 314 square yards.
7
2
49
A ≈ 154 The area of the circle is about
154 square inches.
110
Chapter 4 Angles and Circles
JO
Copyright © by The McGraw-Hill Companies, Inc.
1. The radius is 10 yards; r = 10.
Who is Correct? What is the circumference of a circle with a radius of 5 yards?
Tavio
Yang
Amanda
C = 2πr C ≈ 2 × 5 × 3.14
C = πr C ≈ 5 × 3.14 The circumference is about 15.7 yd.
C = πd 10 × 3.14 ≈ C
The circumference is about 31.4 yd.
The circumference is about 31.4 yd.
Circle correct answer(s). Cross out incorrect answer(s).
Guided Practice Identify the length of the radius and the diameter of each circle. 1
2 N
Copyright © by The McGraw-Hill Companies, Inc.
GU
radius: 2 ft diameter: 4 ft
3
radius: 14 m diameter: 28 m
4 N DN
radius: 32 cm diameter: 64 cm
33 in.
radius: 16.5 in. diameter: 33 in.
GO ON Lesson 4-7 Circles
111
Step by Step Practice Find the circumference and area of the circle. 5
DN
Step 1 Find the length of the diameter and radius. The diameter is
9
Step 2 For circumference, substitute formula. C = πd C ≈ 3.14 × C ≈ 28.26
r=
centimeters.
9
9
÷ 2 = 4.5 .
for d and 3.14 for π in the circumference
9 4.5
Step 3 For area, substitute A = πr2 4.5 A ≈ 3.14 × A ≈ 3.14 × 20.25 A ≈ 63.585
for r and 3.14 for π in the area formula.
2
Find the circumference and area of each circle. Use 3.14 for π. 6
26 d= C = πd C ≈ 3.14 × C ≈ 81.64
in.
26
13 r= in. 2 A = πr 13 A ≈ 3.14 × 169 A ≈ 3.14 × A ≈ 530.66
2
JO
The circumference of the circle is about 81.64 inches, and the area of the circle is about 530.66 square inches. 7
The circumference of the circle is about 21.98 meters, and the area of the circle is about 38.465 square meters. N
112
Chapter 4 Angles and Circles
Copyright © by The McGraw-Hill Companies, Inc.
The circumference of the circle is about 28.26 centimeters, and the area of the circle is about 63.585 square centimeters.
Find the circumference and area of each circle. Use 8
14 d= C = πd 22 × C ≈ ___ 7 C≈
in.
44
44
7 r= in. 2 A = πr 22 × 7 A ≈ ___ 7 22 × 49 A ≈ ___ 7 154 A≈
22 _ for π. 7
ZE 2
44 yards, and the The circumference of the circle is about 154 area of the circle is about square yards. 9
_
88 _
The circumference of the circle is about 7 88 area of the circle is about square feet.
feet, and the GU
7
Step by Step Problem-Solving Practice Solve.
Copyright © by The McGraw-Hill Companies, Inc.
10
CARS Alana’s tires each have a diameter of 35 inches. What is the circumference of each tire? Understand
Read the problem. Write what you know. Alana’s tires have a diameter of 35 inches.
Plan
Pick a strategy. One strategy is to use a formula. Use the formula for the circumference of a circle.
Solve
Problem-Solving Strategies Draw a diagram. ✓ Use a formula. Guess and check. Act it out. Solve a simpler problem.
Substitute 35 for d and 3.14 for π in the circumference formula. C = πd C ≈ 3.14 × C ≈ 109.9
35
The circumference of each tire is about 109.9 inches. Check
Estimate the circumference by substituting 3 for π and 35 for d. C = πd C ≈ 3 × 35 C ≈ 105 The circumference of each tire is close to the estimate, so the answer is reasonable.
GO ON
Lesson 4-7 Circles
113
11
SPORTS The skating rink has a radius of 4 yards. What is the circumference 25.12 of the skating rink? Use 3.14 for π. Check off each step.
✔ ✔ ✔ ✔ 12
Understand Plan Solve Check
HOBBIES J.C. collects clocks. The face of his largest clock has a radius of 23 inches. What is the area of J.C.’s largest clock? Use 3.14 for π.
1,661.06 in2
13
Using an appropriate measuring tool and the centimeter grid shown, find the actual circumference and area of the circle at the right. Next, find the circumference and area using the formulas C = 2πr and A = πr2. Write down both sets of numbers. How do the two sets of numbers compare? Explain the differences.
DN
C = 2πr So, the circumference of this circle is 2 · π · 2, or 4π. The area of the circle is πr 2 or π22, which is also equal to 4π.
Skills, Concepts, and Problem-Solving Identify the length of the radius and diameter of each circle. 15 LN JO
radius: 25 in. diameter: 50 in. 16
radius: 90 km diameter: 180 km 17 DN
GU
radius: 3.5 ft diameter: 7 ft 114
Chapter 4 Angles and Circles
radius: 12.5 cm diameter: 25 cm
Copyright © by The McGraw-Hill Companies, Inc.
14
Find the circumference and area of the circle. Use 3.14 for π. 18
47.1 yards, and the The circumference of the circle is about area of the circle is about 176.625 square yards.
22 _ for π.
_
Find the circumference and area of the circle. Use 19
ZE
132 7 The circumference of the circle is about feet, and the 7 198 square feet. area of the circle is about
_
GU
7
Solve. 20
ASTRONOMY The diameter of an eye-piece lens for a telescope is 2.4 centimeters. What is the circumference of the lens? Use 3.14 for π.
7.536 cm
21
HOBBIES Dane’s grandmother used 5 circular canvases last week while painting a collage. Each canvas had a diameter of 14 inches. What was the area of each canvas? Use 3.14 for π.
153.86 in2
Vocabulary Check Write the vocabulary word that completes each sentence. Copyright © by The McGraw-Hill Companies, Inc.
22 23
Circumference
is the distance around a circle.
radius A(n) is a line segment that connects the center of a circle to a point on the circle. Pi (π)
24
is the ratio of the circumference of a circle to the diameter of the same circle. Its value is approximately 3.14 22 . or ___ 7
25
Writing in Math Explain how to find the area of a circle.
Find the length of the radius. Substitute the length of the radius for r and 3.14 or 22 for π in the area formula. 7
_
GO ON Lesson 4-7 Circles
115
Spiral Review 26
FOOD The delivery ramp that leads to the back door of Rey’s Wholesale Foods is 5 feet long. There is a height of 3 feet from the top of the ramp to the base of the wall. How far is it from the bottom of the ramp to the base of the wall? (Lesson 4-6, p. 99) 4 feet
Find the measure of the missing angle. 27
(Lesson 4-4, p. 81)
The measure of the missing angle is 59 °.
28
The measure of the missing angle is 93 °.
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¡ $ 8
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Draw a net for a rectangular prism with the dimensions given. (Lesson 3-4, p. 27)
29
7×3×6
"
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116
Chapter 4 Angles and Circles
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Copyright © by The McGraw-Hill Companies, Inc.
!
Lesson
4-8 Volume of Triangular Prisms
6MG1.3 Know and use the formulas for the volume of triangular prisms and cylinders (area of base × height); compare these formulas and explain the similarity between them and the formula for the volume of a rectangular solid.
and Cylinders KEY Concept To find the volume of a triangular prism, you must first know the area of its base. The base of a triangular prism is a triangle. 1 bh. The formula for the area of a triangle is A = __ 2 The volume of a triangular prism is the product of the area of its base, B, I and its height, h. C
V is the volume of the solid figure.
B is the area of the base.
volume the number of cubic units needed to fill a threedimensional figure or solid figure (Lesson 2-4, p. 77)
I
/PUJDFUIBUUIFSFBSFUXPI T JOUIFEJBHSBN5IFIFJHIUPG UIFCBTFUSJBOHMFJTEJGGFSFOUUIBO UIFIFJHIUPGUIFUSJBOHVMBSQSJTN
V=B×h
VOCABULARY
h is the height of the base of the triangular prism.
To find the volume of a cylinder, you must first know the area of its base. The base of a cylinder is a circle. The formula for 2 the area of a circle is A = π r .
triangular prism a prism whose bases are triangular with parallelograms for sides cylinder a three-dimensional figure having two parallel congruent circular bases and a curved surface connecting the two bases
Copyright © by The McGraw-Hill Companies, Inc.
The volume of a cylinder is the product of the area of its base, B, and its height, h. V is the volume of the solid figure.
S
r is the radius.
V = B × h or V = π r 2h B is the area of the base.
I
h is the height.
As with circles, when you are calculating the volume of a cylinder and you substitute approximate values in for π, change from using an = symbol to ≈ symbol.
GO ON Lesson 4-8 Volume of Triangular Prisms and Cylinders
117
Example 1
YOUR TURN!
What is the volume of the triangular prism?
What is the volume of the triangular prism?
DN
GU
DN
1. The length of the base of the triangle is 5 centimeters, and the height of the triangle is 6 centimeters. 1 bh A = __ Use the area of a triangle formula 2 to find the area of the base. 1 × 5 × 6 Substitute the base and height of A = __ 2 the triangle. A = 15 The area of the base is 15 centimeters. 2. Find the volume by using the area of the base. V=B×h Use the formula for the volume of a triangular prism. V = 15 × 8 Substitute 8 for the height of the prism. V = 120 Multiply.
1. The length of the base of the triangle is 9 feet, and the height of the triangle is 4 feet. 1 bh A = __ 2 1× 9 × 4 A = __ 2 A = 18 2. Find the volume by using the area of the base. V=B×h V=
18
×
17
V = 306 The volume of the triangular prism is 306 cubic feet.
Example 2 What is the volume of the cylinder? 1. The radius of the circle is 4 inches. Find the area of the base. Use 3.14 for π. 2 A = πr Use the formula for the area of a circle. 2 A ≈ 3.14 × 4 Substitute the values for pi and the radius. A ≈ 50.24 Multiply. The area of the base is about 50.24 inches. 2. Find the volume by using the area of the base. V=B×h Use the formula for the volume of a cylinder. V ≈ 50.24 × 10 Substitute the values of the area of the circle and the height of the cylinder. V ≈ 502.4 Multiply. The volume of the cylinder is about 502.4 cubic inches. 118
Chapter 4 Angles and Circles
JO
JO
Copyright © by The McGraw-Hill Companies, Inc.
The volume of the triangular prism is 120 cubic centimeters.
GU
GU
DN
YOUR TURN! What is the volume of the cylinder? 1. The diameter of the circle is 12 meters, so the radius is 12 ÷ 2 = 6 . Find the area of the base. Use 3.14 for π.
N N
2
A = πr A ≈ 3.14 × A ≈ 113.04
6
2
2. Find the volume by using the area of the base. V=B×h V ≈ 113.04 V ≈ 904.32
×
8
The volume of the cylinder is about
904.32 cubic meters.
Example 3 What is the volume of the triangular prism?
DN C
Copyright © by The McGraw-Hill Companies, Inc.
1. Use the Pythagorean Theorem to find the leg of the base of the prism. This leg is the height of the triangular base. a2 + b2 = c2 82 + b2 = 172 64 + b2 = 289 b2 = 225 b = 15
Write the formula. Substitute the measure of the sides. Simplify the exponents. Subtract 64 from 289. Find the square root of 225.
2. Next, find the area of the base. 1 bh A = __ Use the formula for the area of a triangle. 2 1 × 8 × 15 Substitute 8 for the base and 15 for the height. A = __ 2 A = 60
DN DN
DN DN DN
Multiply.
3. Then find the volume of the prism. V=B×h V = 60 × 10 V = 600
Use the formula for the volume of a prism. Substitute 60 for the area of the base and 10 for the height of the prism. Multiply.
The volume of the triangular prism is 600 cubic centimeters. GO ON Lesson 4-8 Volume of Triangular Prisms and Cylinders
119
YOUR TURN! What is the volume of the triangular prism whose base is a right triangle?
DN DN
1. Use the Pythagorean Theorem to find the length of the other leg of the triangular base of the prism. a2 + b2 = c2 a2 +
16
2
=
20
DN
2
B
a2 + 256 = 400 a2 = 144 a=
12
2. Next, find the area of the base. 1 bh A = __ 2 1 × 12 × 16 A = __ 2 A = 96 3. Then find the volume of the prism. V=B×h V = 96 × V = 864
9
The volume of the triangular prism is 864 cubic centimeters.
What is the volume of the triangular prism whose base is a right triangle?
DN
DN
Nicole
Pedro
7x10 V = 2 × 15 = 375 The volume is 375 cubic centimeters
V = (7 × 10) ÷ 2 × 15 = 525 The volume is 525 cubic centimeters
_
Circle correct answer(s). Cross out incorrect answer(s). 120
Chapter 4 Angles and Circles
DN
Todd V = 7 × 10 × 15 = 1,050 The volume is 1,050 cubic centimeters
Copyright © by The McGraw-Hill Companies, Inc.
Who is Correct?
Guided Practice Name each solid figure. 1
cylinder
triangular prism
2
Step by Step Practice 3
JO
Find the volume of the solid figure whose base is an isosceles triangle. Step 1 Find the length of the base and height of the triangle. of Use the Pythagorean Theorem to find the height the triangle.
JO
Copyright © by The McGraw-Hill Companies, Inc.
The base of the triangular prism is an isosceles triangle. The length of the base of the isosceles triangle is 6 inches. Therefore, the length of half the base is 6 ÷ 2 = 3 inches. for a and 5 for c in the Substitute for 3 Pythagorean Theorem. a2 + b2 = c2 3 2 + b2 = 5 9 + b2 = 25 b2 = 16 b= 4
2
Remember that the Pythagorean Theorem states that in a right triangle a2 + b2 = c2.
JO
JO
JO
JO JO
Step 2 Find the area of the base of the triangular prism. Use the formula for the area of a triangle. The length of the base of the triangle is and the height of the triangle is
JO
4
6 inches,
inches.
Step 3 Substitute the area of the base and the height into the formula for the volume. Then multiply. The volume of the triangular prism is 144 cubic inches.
1 bh A = __ 2 1× A = __ 2 A=
6 × 4
12
V=B×h V = 12 × V = 144
12
GO ON
Lesson 4-8 Volume of Triangular Prisms and Cylinders
121
Find the volume of each solid figure. Use 3.14 for π. 4
The diameter of the circle is 20 centimeters, so the radius is 20 ÷ 2 = 10 . Use 3.14 for π. A = πr2 A ≈ 3.14 × A ≈ 314
10
DN
DN
2
Substitute the area of the base and the height into the formula for the volume. Then multiply. V=B×h V ≈ 314 × V ≈ 1,570
5
The volume of the cylinder is about 1,570 cubic centimeters. 5
The length of the base of the triangle is of the triangle is 3 yards. 1 bh A = __ 2 __ A=1× 2 A=
6
yards and the height
ZE
6
×
3
9
ZE
ZE
Substitute the area of the base and the height into the volume formula. Then multiply.
13
The volume of the triangular prism is 117 cubic yards. 6
7
DN
N DN
N
N
The volume of the triangular prism is 6 cubic meters.
122
Chapter 4 Angles and Circles
The volume of the cylinder is about 141,300 cubic centimeters.
Copyright © by The McGraw-Hill Companies, Inc.
V=B×h V= 9 × V = 117
Step by Step Problem-Solving Practice
Problem-Solving Strategies
Solve. 8
CONTAINERS Marta has a cylinder-shaped container. The diameter of the base is 14 centimeters, and the height is 25 centimeters. What is the volume of the container? Understand
Draw a diagram. Look for a pattern. Act it out. ✓ Use a formula. Work backward.
Read the problem. Write what you know. The diameter of the base of the container is 14 centimeters. The height of the container is 25 centimeters. Pick a strategy. One strategy is to use a formula.
Plan
Think of the problem as two parts. Use a formula to find the area of the circular base. Then use a formula to find the volume. Use 3.14 for π. The diameter of the circle is 14 7 . radius is 14 ÷ 2 =
Solve
centimeters, so the
DN
DN
2
Copyright © by The McGraw-Hill Companies, Inc.
A = πr A ≈ 3.14 × A ≈ 153.86
7
2
Substitute the area of the base and the height into the formula for the volume. Then multiply. V=B×h 25 V ≈ 153.86 × V ≈ 3,846.5 The volume of the container is about 3,846.5 cubic centimeters. Use a calculator to check your answer.
Check
9
CANDLES Conrad’s Candle Shop has two dozen candles shaped like triangular prisms. The largest candle has a height of 7 inches and a triangular base. The length of the triangular base is 6 inches, and its height is 5 inches. What is the volume of the candle? 105 in3 Check off each step.
✔
Understand
✔
Solve
✔
Plan
✔
Check
GO ON Lesson 4-8 Volume of Triangular Prisms and Cylinders
123
10
JO
SCIENCE The cylindrical beakers in Mr. Peters’s biology class have a diameter of 5 inches and a height of 10 inches. What is the volume of each beaker? Use 3.14 for π.
196.25 in3
JO
A box of crackers is in the shape of a rectangular prism. Use what you know about finding the volume of a triangular prism to find the volume of the box.
11
Find the area of the base of the rectangular prism. The area of a rectangle is A = l × w = 4 × 3 = 12
JO
square inches. Multiply the area of the base by the height of the prism: 12 × 6 = 72 cubic inches.
JO JO
JO
JO
Skills, Concepts, and Problem Solving Name each solid figure. 12
13
triangular prism
Find the volume of each solid figure. Use 3.14 for π. 14
15 GU JO
GU GU
JO
The volume of the triangular prism 45 is cubic feet.
124
Chapter 4 Angles and Circles
The volume of the cylinder is about 3,416.32 cubic inches.
Copyright © by The McGraw-Hill Companies, Inc.
cylinder
Solve. 16
CONSTRUCTION To keep the wind from blowing his door shut, Lorenzo made a doorstop. The doorstop is shaped like a triangular prism. The base triangle has a length of 6 centimeters and a height of 15 centimeters. The height of the prism is 8 centimeters. What is the volume of Lorenzo’s doorstop?
360 cm 3 17
FOOD Belinda’s mother stores bagels in a cylinder-shaped container to keep them fresh. The diameter of the container is 6 inches, and its height is 11 inches. What is the volume of the container? Use 3.14 for π.
310.86 in 3
Vocabulary Check Write the vocabulary word that completes each sentence. 18
19
Copyright © by The McGraw-Hill Companies, Inc.
20
21
A(n) triangular prism is a prism whose bases are triangular with parallelograms for sides.
Volume
is the number of cubic units needed to fill a three-dimensional figure or solid figure.
cylinder A(n) is a three-dimensional figure having two parallel congruent circular bases and a curved surface connecting the two bases. Writing in Math Explain how to find the volume of a triangular prism.
Find the area of the base using the formula for the area of a triangle. Substitute the area of the base and height into the formula for the volume. Then multiply.
Spiral Review 22
SPORTS The radius of Phoebe’s circular-shaped swimming pool is 5.5 feet. She swam across the pool 4 times. How many feet did Phoebe swim? (Lesson 4-7, p. 109)
44 feet
GO ON
Lesson 4-8 Volume of Triangular Prisms and Cylinders
125
Are the figures congruent? 23
(Lesson 4-5, p. 91)
9
.
¡
¡
7
¡ ¡ -
¡
¡
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8
congruent LMN and VWX are because corresponding sides and corresponding angles are congruent .
Draw a figure with the description given. 24
right, isosceles triangle
5×3×4
"
#
%
126
Chapter 4 Angles and Circles
$
(Lesson 3-4, p. 27)
Copyright © by The McGraw-Hill Companies, Inc.
Draw a net for the dimensions given.
!
trapezoid
25
26
(Lesson 4-3, p. 73)
Chapter
4
Progress Check 4
(Lessons 4-7 and 4-8)
Identify the length of the radius and diameter of each circle. 6MG1.2 1
3.5 radius: 7 diameter:
in.
2.75 cm radius: 5.5 diameter: cm
2
in.
DN JO
3
Find the circumference and area of the circle. Use 3.14 for π. 6MG1.2 The circumference of the circle is about 56.52 yards, and the area of the circle is about 254.34 square yards.
4
ZE
_
Find the circumference and area of the circle. Use 22 for π. 6MG1.2 7 The circumference of the circle is about The area of the circle is about
44 or 6_ 2 _ 7
7
44 or 6_ 2 _ 7
7
feet.
square feet.
GU
Find the volume of each solid figure. Use 3.14 for π. 5MG1.3, 6MG1.3
Copyright © by The McGraw-Hill Companies, Inc.
5
6
JO
N
N
JO
N
The volume of the triangular prism 75 is cubic meters.
The volume of the cylinder is about 4,615.8 cubic inches.
Solve. 5MG1.3, 6MG1.3, 6MG1.2 7
FOOD Kizzie purchased a triangular wedge of cheddar cheese. The cheese had a triangle base with length 7 centimeters and height 12 centimeters. The height of the cheese wedge was 9 centimeters. What 3 was the volume of the wedge of cheese? 378 cm
8
BASEBALL According to baseball regulations, the pitcher’s mound, which is circular, must have a diameter of 18 feet. What is 2 the area of a pitcher’s mound? 254.34 ft Lesson 4-8 Volume of Triangular Prisms and Cylinders
127
Chapter
Study Guide
4
Vocabulary and Concept Check cylinder, p. 117
Write the vocabulary word that completes each sentence.
degree, p. 63
1
A(n) protractor is an instrument marked in degrees, used for measuring or drawing angles.
2
A(n) degree angles.
3
A(n) diameter is a chord that passes through the center of a circle.
diameter, p. 109 hypotenuse, p. 99 parallel lines, p. 54 pi (π), p. 109 protractor, p. 63 Pythagorean Theorem, p. 99 4
radius, p. 109 trapezoid, p. 73 triangular prism, p. 117
5
is a unit for measuring
The Pythagorean Theorem states that the sum of the squares of the lengths of the legs in a right triangle is equal to the square of the length of the hypotenuse.
Pi (π)
is the ratio of the circumference of a circle to the diameter of the same circle. Its value is 22 . approximately 3.14 or ___ 7
Label each diagram below. Write the correct vocabulary term in each blank.
parallel lines
7
radius
8
cylinder
9
trapezoid
10
hypotenuse
11
triangular prism
128
Chapter 4 Study Guide
Copyright © by The McGraw-Hill Companies, Inc.
6
Lesson Review
4-1
Lines
Example 1
(pp. 54–62)
Name each line. Identify the relationships.
Name each line. Identify the relationships.
:
12
9 "
1
#
2
:
;
perpendicular
The lines are ⊥ lines. PQ 13
&
'
-
.
parallel
The lines are lines. EF
4-2
. YZ
Angles
(pp. 63–71)
;
105° . ∠XYZ measures ∠XYZ is a(n) obtuse angle.
Example 2 Measure and identify the angle. Place the center of the protractor at point B. with the Line up BC line that extends from 0° to 180° on the protractor.
.
15
#
#
35° . ∠MNP measures acute ∠MNP is a(n) angle.
"
1
$
/
"
Copyright © by The McGraw-Hill Companies, Inc.
9
:
intersects XY . AB
LM .
Measure and identify the angle. 14
. The other line is XY . One line is AB The lines cross, or intersect. The lines do not form right angles, so they are not perpendicular lines. The lines are intersecting lines.
$
Look at point C. Read the measure of the passes through the inner angle where BA scale. ∠ABC measures 45°.
Chapter 4 Study Guide
129
4-3
Triangles and Quadrilaterals
(pp. 73–80)
Identify each figure.
Example 3
16
Identify the figure.
The figure is a(n)
trapezoid
.
17
The figure has three sides and three angles. The figure is a triangle. None of its sides are equal in length. The figure is a scalene triangle. The figure is a(n)
acute, isosceles triangle
4-4
Add Angles
One angle is an obtuse angle. The figure is an obtuse, scalene triangle.
.
(pp. 81–89)
What types of angles are shown?
Example 4 (
18 %
What is the measure of the missing angle? ¡
¡
&
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3
4 ¡
&
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2
Find the measure of each missing angle. 9
19
¡
5
Find the sum of the measures of known angles.
: ¡
¡
¡
m∠Q + m∠R + m∠S = 300° 8
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¡
;
The measure of the missing angle is 145° . #
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$
The measure of the missing angle is 30° 130
The sum of the measures of the angles of a quadrilateral is 360°. Subtract the sum of the known measures from 360°. 360° - 300° = 60°
Chapter 4 Study Guide
.
Copyright © by The McGraw-Hill Companies, Inc.
See TWE margin.
Sketch complementary angles when one angle’s measure is 80°.
Example 5 Sketch supplementary angles when one angle’s measure is 50°.
21
¡ ¡
¡
¡
1. The sum of supplementary angles equals 180°. Using your protractor, sketch a 180° angle. 2. Find the given angle, 50°, inside the 180° angle and make a dot. Use a straightedge to draw the ray that connects the vertex and the dot at 50°. 3. Find the missing angle by using the protractor’s scale. The supplementary angle is 130°.
Copyright © by The McGraw-Hill Companies, Inc.
4-5 22
Congruent Figures
(pp. 91–98)
Example 6
Are the triangles congruent?
Are the quadrilaterals congruent?
-
/
¡
6 ¡
6
.
7
¡
-
¡
¡
¡
¡
.
¡
¡
5
¡ 4
Corresponding angles are congruent and corresponding sides are congruent. Therefore, the two triangles are congruent.
¡
¡
¡
5
Are corresponding angles congruent? m∠S m∠J. m∠T m∠K. m∠U m∠L. m∠V m∠M.
4
,
¡ +
Are corresponding sides congruent? −− −− ST JK. −− −− TU KL. −−− −−− UV LM. −− −− SV JM.
Quadrilaterals STUV and JKLM are congruent because all of the corresponding angles and sides are congruent. Chapter 4 Study Guide
131
4-6 23
Pythagorean Theorem
Find the length of the leg of the right triangle.
(pp. 99–107)
Example 7
B
Find the length of the leg of the right triangle. From the figure, you know a = 15 and c = 25.
The length of the leg is 24
8
units.
Find the length of the hypotenuse of the right triangle.
4-7 25
Circles
5
units.
What is the area of the circle? 22 for π. Use ___ 7 22 A ≈ ___ × 42 7
What is the area of the circle? JO
7
2
What is the area of the circle? 22 for π. Use ___ 7 22 A ≈ ___ × 62 7 22 × 36 A ≈ ___ 7
132
7
2
Chapter 4 Study Guide
GU
The radius is 7 cm; r = 7. 22 Substitute 7 for r and ___ 7 for π in the area formula.
DN
A = πr2 22 × 72 A ≈ ___ 7 22 A ≈ ___ × 49 7 A ≈ 22 × 7 A ≈ 154 The area of the circle is about 154 cm2.
Copyright © by The McGraw-Hill Companies, Inc.
352 in. _ A≈
792 ft _
The length of the leg is 20 units.
Example 8
7
A≈
C
(pp. 109–116)
22 × 16 A ≈ ___
26
Substitute the values of a and c in the Pythagorean Theorem and solve for b. a2 + b2 = c2 152 + b2 = 252 225 + b2 = 625 b2 = 400 b = 20
D
The length of the leg is
4-8 27
Volume of Triangular Prisms and Cylinders (pp. 117–126)
What is the volume of the triangular prism whose base is a right triangle? JO
JO
JO
The volume of the triangular prism 3 is 420 in . 28
What is the volume of the triangular prism? DN
DN
DN
The volume of the triangular prism is 3 about 600 cm .
Copyright © by The McGraw-Hill Companies, Inc.
29
Example 9 What is the volume of the triangular prism whose base is a right triangle? The length of the base of the triangle is 6 cm, and the height of the triangle is 4 cm.
DN
DN
DN
Use the area-of-atriangle formula to find the area of the base. 1 bh A = __ 2 1×6×4 A = __ 2 A = 12 Substitute the area of the base and the height into the volume formula. Then multiply. V=B×h V = 12 × 10 V = 120 The volume of the triangular prism is 120 cm3.
What is the volume of the cylinder? JO
Example 10 What is the volume of the cylinder?
JO
The volume of the cylinder is about 141.3 in3 . 30
What is the volume of the cylinder? DN
The radius of the circle is 1 ft. Use the formula for the area of a circle to find the area of the base. Use 3.14 for π.
GU
GU
A = πr2 A ≈ 3.14 × 12 A ≈ 3.14 Substitute the area of the base and 3 for the height into the volume formula. Then multiply.
DN
The volume of the cylinder is about 942 cm3 .
V=B×h V ≈ 3.14 × 3 V ≈ 9.42 The volume of the cylinder is about 9.42 ft3. Chapter 4 Study Guide
133
Chapter
Chapter Test
4
Draw each figure. 5MG2.1 1
perpendicular lines OP and XY
2
4
9 0
:
-
.
2
3
5MG2.1
Measure and identify the angle. 5MG2.1 4
"
#
The lines are LM
parallel QR
5MG2.1
rhombus
¡ $
75 ∠ABC measures acute angle ∠ABC is a(n)
lines.
Draw a figure with the description given. 5
5
1
Name each line. Identify the relationships. 3
line segment ST
°. .
Draw each type of angle given. 6
5MG 2.1, 5GM2.2
complementary angles
5
4
Find the measure of the missing angle. 7
5MG2.2, 6MG2.2
,
Are the figures congruent?
7MG3.4
8
¡
¡ 4 +
¡
(
5
¡
-
The measure of the missing angle 51 is °.
¡
¡ ¡
6
)
¡
¡
'
STU and FGH are
congruent
because corresponding sides and corresponding congruent angles are . GO ON
134
Chapter 4 Test
Copyright © by The McGraw-Hill Companies, Inc.
3
Find the length of the hypotenuse of the right triangle. 7MG3.3 9
The length of the hypotenuse is
15
D
units.
Use the circle for Exercises 10–13. Use 3.14 for π. 6MG1.2
50 m
10
diameter =
11
radius =
12
circumference ≈
13
area ≈
14
N
Find the volume. Use 3.14 for π.
The volume of the cylinder 3 is about 231.732 ft .
6MG1.3 GU
25 m
GU
157 m
1,962.50 m
Solve. 5MG2.1, 7MG3.4, 6MG1.3 15
CONSTRUCTION Marcel had to fit two pieces of crown molding together at a 90° angle. What type of angle did the molding form?
Copyright © by The McGraw-Hill Companies, Inc.
right
16
COOKING Annie made some fried corn tortillas like the one pictured at the right. What type of triangle do Annie’s tortillas equilateral triangles represent?
17
SHAPES Miss Rustin is making a collage for her club. The two shapes she is using are shown at the right. Are these shapes congruent?
T JODIFT
No, they are not congruent. 18
FOOD The soup kitchen uses cans of soup that have a diameter of 8 inches and a height of 6 inches. What is the volume of a cylindrical soup can? Use 3.14 for π.
about 301.44 in3 Correct the mistakes. 7MG3.3 19
When Aida took her geometry test, this was how she answered the following question: Mr. Hauser has a ladder resting against his bedroom wall. The top of the ladder touches the wall at a height of 12 feet. The ladder is 15 feet in length. How far is the bottom of Mr. Hauser’s ladder from the base of his bedroom wall?
n 5HEDISTANCE OFTHELADDER FROMTHEBASE OFTHEWALLIS FT
See TWE margin. Chapter 4 Test
135
Chapter
4
Standards Practice
Choose the best answer and fill in the corresponding circle on the sheet at right. 1
Which best describes these lines?
5MG2.1
5
Use the Pythagorean Theorem to find the triangle’s missing side length.
7MG3.3
A perpendicular
C bisecting
B intersecting
D parallel
D
ZE
ZE
2
What is the approximate measure of this angle? 5MG2.1
6
F 45°
H 120°
G 90°
J 175°
A 11 yd
C 17 yd
B 15 yd
D 18 yd
What is the circumference of the circle? Use 3.14 for π. 6MG1.2
JO
3
What is the measure of x in the figure?
5MG2.2
H 64 in.
G 50.24 in.
J 200.96 in.
¡
A 30°
C 60°
B 45°
D 150°
7
4
Find the area of the circle. Use 3.14 for π. 6MG1.2
GU
If the two parallelograms are congruent, what is the measure of x? 7MG3.4 Y
A 9.42 ft2 ¡
B 18.84 ft2
¡
C 28.26 ft2
136
F 67°
H 180°
G 113°
J 360°
Chapter 4 Standards Practice
D 113.04 ft2 GO ON
Copyright © by The McGraw-Hill Companies, Inc.
Y
F 9.14 in.
8
Gavin has a container with the following dimensions. What is its volume? Use 3.14 for π. 6MG1.3
11
What is measure of ∠a and ∠b? 3MG1.2
DN
¡
DN
F 60 cm3 G 188.4 cm3
¡
¡ B
D
¡
E
C
A 135°, 45°
C 60°, 75°
B 135°, 75°
D 45°, 75°
ANSWER SHEET Directions: Fill in the circle of each correct answer.
H 300 cm3 J 942 cm3
9
Find the length of the line to the nearest centimeter. 2MG1.3
Copyright © by The McGraw-Hill Companies, Inc.
A 3 cm B 5 cm C 8 cm D 12 cm
10
1
A
B
C
D
2
F
G
H
J
3
A
B
C
D
4
F
G
H
J
5
A
B
C
D
6
F
G
H
J
7
A
B
C
D
8
F
G
H
J
9
A
B
C
D
10
F
G
H
J
11
A
B
C
D
What is the volume of the solid figure?
3MG1.2
F 12 units3
Success Strategy If you do not know the answer to a question, go on to the next question. Come back to the problem, if you have time. You might find another question later in the test that will help you figure out the skipped problem.
G 24 units3 H 30 units3 J 48 units3 Chapter 4 Standards Practice
137
Index A angle acute, 63–71 complementary, 81–87 corresponding, 89–96 right, 63–71 obtuse, 63–71 straight, 63–71 supplementary, 81–87
D degree angle measure, 63–71
Step-by-Step Problem Solving Measurement and Geometry, 4, 11, 19, 27, 37, 54, 63, 73, 81, 91, 99, 109, 117
diameter, 109–116
F
N net, 27–35
face, 27–35
P
Answer sheet, 51, 137 area parallelogram, 11–17 rectangle, 4–10 triangle, 19–26 Assessment, 48–49, 132–133
G grid paper, 4–14, 16–21, 23–30, 32, 33, 36, 42–46, 48, 66, 67, 69, 76, 78, 80, 83, 85, 88, 90, 93, 96, 98, 116, 126, 129, 134
C California Mathematics Content Standards, 4, 11, 19, 27, 37, 54, 63, 73, 81, 91, 99, 109, 117 Chapter Preview, 3, 53 Chapter Test, 48–49, 134–135 chord, 109–116
circumference, 109–116
coordinate grid, 59 Correct the Mistakes, 49, 135 corresponding angles, 91–98 corresponding sides, 91–98 cube, 37–42 cubic unit, 37–42 cylinder volume, 117–126
H hypotenuse, 99–107
I intersecting lines, 54–62
Index
pi, 109–116 Problem-Solving, see Step-byStep Problem Solving Progress Check, 18, 36, 72, 90, 108, 127 protractor, 63–67, 69, 76, 78, 85, 88, 90, 126, 129, 135 Pythagorean Theorem, 99–105
Q
K Key Concept, 4, 11, 19, 27, 37, 54, 63, 73, 81, 91, 99, 109, 117
quadrilateral area, 73–80 sum of angles, 81–89 types of, 73–80
L R
legs of a triangle, 99–107 line intersecting, 54–62 parallel, 54–62 perpendicular, 54–62
M Manipulatives protractor, 63–67, 69, 76, 78, 85, 88, 90, 126, 129, 135 ruler, 3, 26, 42 Mathematical Reasoning, see
138
perpendicular lines, 54–62 point, 54–62
complementary angles, 81–87 congruent angles, 73–80 figures, 91–98
parallelogram, 19, 73 area 11–17
radius, 109–116 ray, 54–62 Real-World Applications advertising, 106 agriculture, 86 architecture, 80 art, 8, 15, 34, 79, 86 astronomy, 115 badges, 97 baking, 49 baseball, 68, 127 basketball, 7 billiards, 68
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
circle area, 109–116 circumference, 109–116 diameter, 109–116 radius, 109–116
parallel lines, 54–62
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
candles, 123 cars, 113 collections, 42 construction, 8, 41, 42, 61, 105, 125, 135 containers, 123 cooking, 135 costumes, 24 cross-country, 88 decorating, 35 design, 18, 33, 59 distance, 107 doors, 9 farming, 16 fences, 71 fitness, 17 flags, 78, 96, 108 food, 10, 18, 25, 116, 125, 127, 135 gardening, 10 geometry, 33 hobbies, 15, 61, 78, 114, 115 home-improvement, 105 horses, 25 houses, 40 jewelry, 34 letters, 89 maps, 58 music, 42 office, 95 packages, 10 packaging, 41, 62 parties, 17 pets, 49 photos, 9 polygons, 97, 107 presents, 32 rectangles, 108 repairs, 90 safety, 77 sailing, 70 school project, 35 science, 124 sculpture, 88 shapes, 135 shopping, 79 signs, 36 sports, 86, 114, 125 storage, 36 tables, 35 tile, 98 time, 69, 72 travel, 107 triangles, 90, 96 volunteering, 23 weather, 26
rectangle, 4, 73–80 area, 4–10 sum of angles, 81–89 rectangular prism surface area, 27–35 volume, 37–42 Reflect, 8, 16, 24, 33, 41, 59, 69, 78, 87, 96, 106, 114, 124 rhombus, 73–80 right angle, 54, 63–71, 99–107 ruler, 3, 26, 42
S Spiral Review, 10, 17, 26, 35, 42, 62, 71, 80, 89, 98, 106, 116, 125 square area, 73–80 sum of angles, 81–89 square number, 99–107 square root, 99–107 square unit, 4–10, 11–17, 19–25 Standards Practice, 50–51, 136–137 Step-by-Step Practice, 6, 13, 22, 30, 39, 57, 66, 76, 83, 94, 103, 112, 121 Step-by-Step Problem Solving Practice, 7–8, 15, 23–24, 32–33, 40–41, 58, 68, 77–78, 86, 95–96, 105, 113–114, 123 Act it out, 95 Draw a diagram, 32, 86, 105 Use a diagram, 58, 68, 77 Use a formula, 7, 15, 23, 113, 123 Use a model, 40
T trapezoid, 73–80 triangle area, 19–26 sum of angles, 81–89 types of, 73–80 triangular prism volume, 117–125
V vertex angle, 63–71 Vocabulary, 4, 11, 19, 27, 37, 54, 63, 73, 81, 91, 99, 109, 117 Vocabulary and Concept Check, 43, 128 Vocabulary Check, 9, 17, 25, 34, 42, 61–62, 70–71, 80, 88, 97, 107, 115, 125 volume cylinder, 117–125 prism, 37–42, 117–125
W Who is Correct?, 5, 12, 21, 29, 38, 56, 65, 76, 83, 93, 102, 111, 120 Writing in Math, 9, 17, 26, 34, 42, 62, 71, 80, 89, 97, 107, 115, 125
Study Guide, 43–47, 128–133 Success Strategy, 51, 137 supplementary angles, 81–89 surface area prism, 27–35
Index
139
