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Skills Practice Workbook
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Geo Skill Practice cover
9/16/08
4:17 PM
Page 1
Skills Practice Workbook
Contents Include: 96 worksheets—one for each lesson
Geometry_0-07-869312-8
9/16/08
4:07 PM
Page C1
Skills Practice Workbook
Contents Include: 96 worksheets—one for each lesson
Geometry_0-07-869312-8
9/16/08
4:07 PM
Page ii
To The Student: This Skills Practice Workbook gives you additional problems for the concept exercises in each lesson. The exercises are designed to aid your study of geometry by reinforcing important mathematical skills needed to succeed in the everyday world. The material is organized by chapter and lesson, with one skills practice worksheet for every lesson in Geometry: Concepts and Applications.
To the Teacher: Answers to each worksheet are found in Geometry: Concepts and Applications Chapter Resource Masters and also in the Teacher Wraparound Edition of Geometry: Concepts and Applications.
Copyright © by The McGraw-Hill Companies, Inc. All rights reserved. Printed in the United States of America. Except as permitted under the United States Copyright Act, no part of this book may be reproduced in any form, electronic or mechanical, including photocopy, recording, or any information storage or retrieval system, without permission in writing from the publisher. Send all inquiries to: The McGraw-Hill Companies 8787 Orion Place Columbus, OH 43240 ISBN: 0-07-869312-8
Geometry: Concepts and Applications Skills Practice Workbook
1 2 3 4 5 6 7 8 9 10 045 09 08 07 06 05 04
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CONTENTS Lesson
1-1 1-2 1-3 1-4 1-5 1-6 2-1 2-2 2-3 2-4 2-5 3-1 3-2 3-3 3-4 3-5 3-6 3-7 4-1 4-2 4-3 4-4 4-5 4-6 5-1 5-2 5-3 5-4 5-5 5-6 6-1 6-2 6-3 6-4 6-5 6-6 ©
Title
Page
Patterns and Inductive Reasoning . . . . . . . . . . . . . . . Points, Lines, and Planes . . . . . Postulates . . . . . . . . . . . . . . . . . Conditional Statements and Their Converses . . . . . . . . . . Tools of the Trade . . . . . . . . . . . A Plan for Problem Solving . . . Real Numbers and Number Lines . . . . . . . . . . . . . . . . . . . Segments and Properties of Real Numbers . . . . . . . . . . . . Congruent Segments . . . . . . . . . The Coordinate Plane . . . . . . . . Midpoints . . . . . . . . . . . . . . . . . Angles . . . . . . . . . . . . . . . . . . . Angle Measure . . . . . . . . . . . . . The Angle Addition Postulate . . . . . . . . . . . . . . . Adjacent Angles and Linear Pairs of Angles . . . . . . . . . . . Complementary and Supplementary Angles . . . . . Congruent Angles . . . . . . . . . . . Perpendicular Lines . . . . . . . . . Parallel Lines and Planes . . . . . Parallel Lines and Transversals . . . . . . . . . . . . . Transversals and Corresponding Angles . . . . . . . . . . . . . . . . . Proving Lines Parallel . . . . . . . . Slope . . . . . . . . . . . . . . . . . . . . Equations of Lines . . . . . . . . . . Classifying Triangles . . . . . . . . Angles of a Triangle . . . . . . . . . Geometry in Motion . . . . . . . . . Congruent Triangles . . . . . . . . . SSS and SAS . . . . . . . . . . . . . . ASA and AAS . . . . . . . . . . . . . Medians . . . . . . . . . . . . . . . . . . Altitudes and Perpendicular Bisectors . . . . . . . . . . . . . . . . Angle Bisectors of Triangles . . . Isosceles Triangles . . . . . . . . . . Right Triangles . . . . . . . . . . . . . The Pythagorean Theorem . . . .
Glencoe/McGraw-Hill
Lesson
6-7 1 2 3
7-1 7-2 7-3 7-4 8-1 8-2 8-3 8-4
4 5 6 7 8 9 10 11 12 13
8-5 9-1 9-2 9-3 9-4
14
9-5 9-6
15 9-7 10-1 10-2 10-3 10-4
16 17 18 19 20
10-5 10-6 10-7 11-1 11-2 11-3 11-4 11-5 11-6 12-1 12-2
21 22 23 24 25 26 27 28 29 30 31
12-3
32 33 34 35 36
12-4 12-5 12-6 iii
Title
Page
Distance on the Coordinate Plane . . . . . . . . . . . . . . . . . . . Segments, Angles, and Inequalities . . . . . . . . . . . . . . Exterior Angle Theorem . . . . . . Inequalities Within a Triangle . . Triangle Inequality Theorem . . . Quadrilaterals . . . . . . . . . . . . . . Parallelograms . . . . . . . . . . . . . Tests for Parallelograms . . . . . . Rectangles, Rhombi, and Squares . . . . . . . . . . . . . . . . . Trapezoids . . . . . . . . . . . . . . . . Using Ratios and Proportions . . Similar Polygons . . . . . . . . . . . . Similar Triangles . . . . . . . . . . . Proportional Parts and Triangles . . . . . . . . . . . . . . . . Triangles and Parallel Lines . . . Proportional Parts and Parallel Lines . . . . . . . . . . . . . . . . . . . Perimeters and Similarity . . . . . Naming Polygons . . . . . . . . . . . Diagonals and Angle Measure . Areas of Polygons . . . . . . . . . . . Areas of Triangles and Trapezoids . . . . . . . . . . . . . . Areas of Regular Polygons . . . . Symmetry . . . . . . . . . . . . . . . . . Tessellations . . . . . . . . . . . . . . . Parts of a Circle . . . . . . . . . . . . Arcs and Central Angles . . . . . . Arcs and Chords . . . . . . . . . . . . Inscribed Polygons . . . . . . . . . . Circumference of a Circle . . . . . Area of a Circle . . . . . . . . . . . . Solid Figures . . . . . . . . . . . . . . Surface Areas of Prisms and Cylinders . . . . . . . . . . . . . . . Volumes of Prisms and Cylinders . . . . . . . . . . . . . . . Surface Areas of Pyramids and Cones . . . . . . . . . . . . . . . Volumes of Pyramids and Cones . . . . . . . . . . . . . . . . . . Spheres . . . . . . . . . . . . . . . . . . .
37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72
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Page iv
Page
Similarity of Solid Figures . . . . Simplifying Square Roots . . . . . 45°-45°-90° Triangles . . . . . . . . 30°-60°-90° Triangles . . . . . . . . The Tangent Ratio . . . . . . . . . . . Sine and Cosine Ratios . . . . . . . Inscribed Angles . . . . . . . . . . . . Tangents to a Circle . . . . . . . . . Secant Angles . . . . . . . . . . . . . . Secant-Tangent Angles . . . . . . . Segment Measures . . . . . . . . . . Equations of Circles . . . . . . . . . Logic and Truth Tables . . . . . . . Deductive Reasoning . . . . . . . .
Glencoe/McGraw-Hill
Lesson
73 74 75 76 77 78 79 80 81 82 83 84 85 86
15-3 15-4 15-5 15-6 16-1 16-2 16-3 16-4 16-5 16-6
iv
Title
Page
Paragraph Proofs . . . . . . . . . . . . Preparing for Two-Column Proofs . . . . . . . . . . . . . . . . . . Two-Column Proofs . . . . . . . . . Coordinate Proofs . . . . . . . . . . . Solving Systems of Equations by Graphing . . . . . . . . . . . . . Solving Systems of Equations by Using Algebra . . . . . . . . . Translations . . . . . . . . . . . . . . . Reflections . . . . . . . . . . . . . . . . Rotations . . . . . . . . . . . . . . . . . Dilations . . . . . . . . . . . . . . . . . .
87 88 89 90 91 92 93 94 95 96
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Page 1
NAME
DATE
PERIOD
Skills Practice
Patterns and Inductive Reasoning Tell how to find the next term in each pattern. 1. 7, 12, 17, 22, . . .
Add 5.
2. 1, 3, 9, 27, . . .
Multiply by 3.
3. 50, 46, 42, 38, . . . Subtract 4.
4. 10, 20, 40, 80, . . . Multiply by 2.
5. 4, 1, 2, 5, 8, . . . Add 3.
6. 144, 134, 124, 114, . . . Subtract 10.
Find the next three terms of each sequence. 7. 2, 7, 12, 17, . . . 22, 27, 32
8. 100, 93, 86, 79, . . . 72, 65, 58
9. 11, 22, 33, 44, . . . 55, 66, 77 11. 2, 4, 8, 16, 32, . . .
10. 1, 4, 14, 56, . . . 224, 896, 3584
64, 128, 256 12. 0, 6, 12, 18, . . . 24, 30, 36
13. 99, 86, 73, 60, . . . 47, 34, 21
14. 25, 26, 24, 25, . . . 23, 24, 22, 23
15. 30, 31, 33, 36, . . . 40, 45, 51
16. 5, 10, 20, 40, . . . 80, 160, 320
17. 220, 210, 190, 160, . . . 120, 70, 10
18. 7, 7, 14, 42, . . . 168, 840, 5040
Draw the next figure in each pattern. 19.
20.
21.
©
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NAME
1–2
DATE
PERIOD
Skills Practice
Points, Lines, and Planes Use the figure at the right to name examples of each term. 1-10. Samples answers are given. 1. four points four of the following: S, T, M, R, X, Y 2. two lines SX , YR
T S
3. four segments M R , M S , MT , MX
R M
4. one ray whose endpoint is M MY Y
5. three collinear points S, M, and X
X
6. one point that is not on YR S
M T
7. a segment with points T and M as its endpoints 8. a line that does not contain R SX 9. a line containing M RY 10. a segment that lies on YR
R M
Determine whether each model suggests a point, a line, a ray, a segment, or a plane. 11. a toothpick segment
12. a floor plane
13. the tip of a pin point
14. the surface of the water in a swimming pool plane
15. a beam of light from a laser ray
16. fence pole segment
Draw and label a figure for each situation described. 17-20. Sample answers are given. 18. plane H contains line a 17. point K lies on RT R
K
T
H a
20. AX and AY such that point A is the only point common to both rays
19. AB lies in plane M containing point R not on AB
X
M A R
B A
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2
Y Geometry: Concepts and Applications
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NAME
1–3
DATE
PERIOD
Skills Practice
Postulates Refer to the figure at the right. 1. Name all of the different lines that can be drawn through the set of points.
X A
AX , AM , AC , XM , XC , MC
M
2. Name the intersection of AX and AM . point A
C
Name all of the planes that are represented in each figure. W
3. R V
P
4.
X
S
E
G
Y T
X
A
planes RSTV, SXYT, WXYS, RWSV, RWXS, VSYT
planes GXA, XEA, DEA, GDA, DEXG J
Refer to the figure at the right.
JL 5. Name the intersection of plane JLM and plane JKL. 6. Name the intersection of plane JKO and plane JOM. JO
K
O
7. Name two planes that intersect in ML . planes JLM and MLKO M
L
8. Name two planes that intersect in JM . planes JOM and JLM Determine whether each statement is true or false. If a statement is false, explain why. 9. If you have two points, then there is only one line that contains both points. true 10. The intersection of two distinct lines is two points.
False; two distinct lines intersect at one point. 11. If you have three noncollinear points, then you have two different planes.
False; three noncollinear points determine one plane. 12. A line is the intersection of two distinct planes. true 13. One point can be the only intersection of two planes.
False; two planes intersect in a line. 14. Three planes can intersect in one line. true ©
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NAME
DATE
PERIOD
Skills Practice
Conditional Statements and Their Converses Identify the hypothesis and the conclusion of each statement. 1. If you purchase a computer and do not like it, then you can return it within 30 days.
Hypothesis—you purchase a computer and do not like it Conclusion—you can return it within 30 days
2. If x 8 15, then x 7
Hypothesis—x 8 15 Conclusion—x 7
3. If the drama club raises $2000, then they will go on tour.
Hypothesis—the drama club raises $2000 Conclusion—they will go on tour 4. If the temperature today is 80° or more, then you will go swimming.
Hypothesis—the temperature today is 80° or more Conclusion—you will go swimming 5. If two lines intersect, then the intersection is a point.
Hypothesis—two lines intersect Conclusion—the intersection is a point Write two other forms of each statement. 6. If two planes intersect, then the intersection is a line.
The intersection of two planes is a line. All planes that intersect will intersect in a line. 7. If it snows, then you will go sledding.
You will go sledding if it snows. All people will go sledding on days that it snows. 8. Your dog will be happy if you feed him Doggy Chow.
If you feed your dog Doggy Chow, then he will be happy. All dogs that eat Doggy Chow are happy. 9. Hiking will be easier if you have hiking boots.
If you have hiking boots, then hiking will be easier. All people with hiking boots can hike easier. 10. All squares have four sides of equal length and four right angles..
If a figure has four sides of equal length and four right angles, then it is a square. A figure is a square if it has four sides of equal length and four right angles. Write the converse of each statement. 11. If a figure is a triangle, then it has three sides.
If a figure has three sides, then it is a triangle. 12. If you find a penny, then you will have good luck.
If you have good luck, then you found a penny. 13. If you ride your bicycle recklessly, then you can get hurt.
If you get hurt, then you rode your bicycle recklessly. 14. If two distinct lines intersect, then their intersection is one point.
If the intersection of two lines is one point, then the lines are distinct. 15. If your cat purrs, then it is contented.
If your cat is contented, then it purrs. ©
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NAME
1–5
DATE
PERIOD
Skills Practice
Tools of the Trade Use a straightedge or compass to answer each question. 1. Which segment is longest? G
E
F
2. Which of the three segments at the upper right forms a straight line with the segment at the lower left? b ab
G
c
e
4. Is C the midpoint of AB ? no
3. Does the inner arc or the outer arc on the right side of the segments go with the arc on the left side to form part of a circle? outer arc
A
C B
5. If the circle with center C is drawn completely, will point X lie on the circle?
6. If extended, will V X intersect ZY at Y ?
no
no V X
X C
Y Z
©
Glencoe/McGraw-Hill
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NAME
1–6
DATE
PERIOD
Skills Practice
A Plan for Problem Solving Find the perimeter and area of each rectangle. 30 ft
1.
3 cm
2.
1 mi
3.
1 mi 18 ft 9 cm
P 96 ft, A 540 ft2 4.
P 24 cm, A 27 cm2 5.
25 in.
P 4 mi, A 1 mi2 6.
22 mm
4m 4m
10 in. 27 mm
P 70 in., A 250 in2
P 98 mm, A 594 mm2
P 16 m, A 16 m2
Find the perimeter and area of each rectangle described. 7. 7 ft, w 2 ft
8. 2 in., w 1 in.
P 18 ft, A 14 ft2
P 6 in., A 2 in2
9. 26 cm, w 23 ft
10. 9 mi, w 1 mi
P 98 cm, A 598 cm2
P 20 mi, A 9 mi2
11. 7 m, w 7 m
12. 5 yd, w 25 yd
P 28 m, A 49 m2
P 60 yd, A 125 yd2
Find the area of each parallelogram. 13. 14.
15. 3 in.
5 in.
14 ft
16 ft
8 mi
7 mi
9 in. 22 ft 5 mi
308
ft2
27
16.
in2
35 mi2
17. 20 cm
18.
36 cm
15 mm 17 mm
54 cm 16 mm
5m 1m 2m
1080 ©
cm2
Glencoe/McGraw-Hill
5 m2
240 mm2 6
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NAME
2–1
DATE
PERIOD
Skills Practice
Real Numbers and Number Lines Write a real number with five digits to the right of the decimal point.
1-8. Sample answers given.
1. an irrational number between 2 and 3 2.43473. . . 2. a rational number between 2 and 3 with a 2-digit repeating pattern
2.57575. . .
3. two rational numbers between 3 and 4 3.01265 and 3.78659 4. an irrational number greater than 10 11.12123. . . 5. a rational number greater than 15 but less than 20 18.97531 6. an irrational number less than 30 26.54854. . . 7. an irrational number less than 5 6.10100. . . 8. a rational number greater than 8 with a 2-digit repeating pattern
6.13131. . .
Use a number line to find each measure. L
A
5
4
Z 3
Q
2
M 1
S 0
T
B
1
2
Y 3
X 4
5
9. LX
10
10. TX
4
11. ZT
3
12. LZ
3
13. QM
1
14. MS
1
15. BX
1 2 2
16. BT
1 1 2
17. ST
1 2
18. LQ
1 3 2
19. BY
1 2
20. AX
1 8 2
Consider 0.27, 0.27, and 0.2 7 . 21. How are these numbers alike? All three numbers are rational. 22. How are they different? 0.27 0.27; 0.27 0.277777. . . ; 0.2 7 0.27272727. . . 23. Which number is greatest? 0.27 24. How would you read each number? Sample answers: Read 0.27 as
as zero point two seven repeating. Read 0.2 7 27 hundredths. Read 0.27 as zero point two seven all repeating.
©
Glencoe/McGraw-Hill
7
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DATE
PERIOD
Skills Practice
Segments and Properties of Real Numbers Three segment measures are given. The three points named are collinear. Determine which point is between the other two. 1. XY 15, AY 31, AX 46 Y
2. AB 12, BC 20, AC 32 B
3. MO 75, MC 34, OC 41 C
4. DE 58, GE 12, DG 70 E
5. HM 2, JM 1, HJ 3 M
6. WX 8, WA 4, AX 4 A
Use the line to find each measure. A
C
7. If AC 10 and CG 21, find AG. 8. If AI 72 and GI 11, find AG.
E
G I
K
31 61
9. If CG 24 and EG 14, find CE. 10 10. If AK 80 and IK 24, find AI. 56 11. If AC 18 and CK 72, find AK. 90 12. If CI 65 and GI 13, find CG. 52
Find the length of each segment in centimeters and in inches. 13.
13 2 cm; in. 16
14.
7.6 cm; 3 in. 15.
7 1 cm; in. 16
16.
3 4.5 cm; 1 in. 4 ©
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Page 9
NAME
DATE
PERIOD
Skills Practice
Congruent Segments Use the number line to determine whether each statement is true or false. Explain your reasoning. K
L
M
N
O
9 8 7 6 5 4 3 2 1
P
0
1
2
S
3
4
5
T
6
7
8
9
M is congruent to N O . 1. L
2. O S is congruent to O L .
3. M is the midpoint of L N .
4. P S is congruent to N O .
5. O is the midpoint of LS .
6. K S is congruent to L T .
7. P S is congruent to K L .
8. The origin is the midpoint of KT .
false; LM 1 and NO 3
false; OS 6 and OL 5
true; LM 1 and MN 1
true; PS 3 and NO 3
false; OS 6 and OL 5
false; KS 14 and LT 15
true; PS 3 and KL 3
true; the distance from K to the origin is 9 and the distance from T to the origin is 9
Determine whether each statement is true or false. Explain your reasoning. H AZ , then GH AZ. true; definition of congruent segments 9. If G
true; definition of midpoint
10. Every segment has only one midpoint.
11. A ray cannot bisect a segment. False; a ray is a geometric figure that can
bisect a segment. 12. If D E WX and W X SP , then D E SP . True; segment congruence is
transitive.
13. If a segment has been bisected, then it is separated into two congruent segments.
true; definition of segment bisector 14. If M is the midpoint of A B , then AM MB . true; definition of midpoint 15. A plane cannot bisect a segment. False; a plane can bisect a segment. 16. If points A, B, and C are collinear, then B lies between A and C. False; the order
of the points may not be A, then B, then C. 17. A segment can have several midpoints. False; by definition, the midpoint is unique, meaning only one. 18. If Y is between X and Z, then XY = YZ. False; Y does not have to be the midpoint of XZ .
©
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9
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NAME
DATE
PERIOD
Skills Practice
The Coordinate Plane Graph and label each point on the coordinate plane. 1. Z(0, 5) 2. T(5, 5) 3. B(5, 2)
y
M
Z Q
4. Q(3, 3)
5. D(4, 4)
6. X(0, 4)
F
8. H(4, 0)
9. F(3, 1)
B
R A
H
7. M(2, 5)
C
x
O
D X
10. R(1, 1)
11. C(3, 4)
T
12. A(2, 0)
Name the ordered pair for each point. 13. R (0, 6) 14. T (0, 5)
y R
15. Z (5, 2)
M
16. B (4, 0)
Z D B
17. D (1, 1)
18. Q (0, 1)
AO
x
Q Y
19. Y (1, 3)
20. M (3, 3)
21. A (1, 0)
22. C (2, 5)
23. Graph x 2.
T
C
y x2
x
O
24. Graph y 1.
y
y 1
©
Glencoe/McGraw-Hill
O
x
10
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NAME
2–5
DATE
PERIOD
Skills Practice
Midpoints Use the number line to find the coordinate of the midpoint of each segment. A 6
5
D
H
4
3
F 2
C
1
0
1
E
2
3
4
G
B
5
6
E 2 1. C
2. FC 0
3. HF 2
4. H C 1
5. AD 5
6. DB 1
7. C G 3
8. GB 5
1 2
1 2
9. DH 3
The coordinates of the endpoints of a segment are given. Find the coordinates of the midpoint of each segment. 10. (0, 10), (0, 0)
(0,5)
11. (8, 0), (0, 0) (4, 0)
12. (0, 6), (0, 0)
(0, 3)
13. (20, 0), (0, 0)
14. (4, 12), (8, 20)
(6, 16)
15. (2, 3), (2, 5)
(10, 0)
(0, 4)
16. (1, 6), (5, 10) (2, 8)
17. (5, 14), (1, 8)
18. (1, 15), (9, 15) (4, 0)
19. (7, 2), (7, 2)
20. (7, 5), (3, 15) (5, 5)
21. (9, 7), (3, 5)
22. (1, 3), (4, 8)
©
Glencoe/McGraw-Hill
2 12 , 5 21
23. (6, 7), (9, 10)
11
(3, 11)
(0, 0)
(3, 6)
1 21 , 8 12
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3–1
DATE
PERIOD
Skills Practice
Angles Name each angle in four ways. Then identify its vertex and its sides. A
1.
2.
3. M
4
1
C
D
R
6
X
F
H
O
AXC, CXA, X, 1; X; XA , XC
MOR, ROM, O, 6; DFH, HFD, F, 4; O; OR , OM F; FH , FD
Name all angles having A as their vertex. 4.
5.
X
A
5
6.
M
6
B 7
9 8
A
Y
E P
1
Z
2
A
Z
XAZ, 1, 2
D
C
MAZ, 5, 6
ABC, ABD, EBC, 7, 8, 9
Tell whether each point is in the interior, exterior or on the angle. 7.
8.
L
9. A X
on 10.
exterior B
interior
11.
12. Z
C
on
exterior
exterior
Determine whether each statement is true or false. 13. The figure formed by opposite rays is sometimes referred to as a straight angle. true 14. The vertex is in the exterior of an angle. false 15. An angle separates the plane into two parts: the interior and the exterior of the angle.
false ©
Glencoe/McGraw-Hill
12
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3–2
Page 13
NAME
DATE
PERIOD
Skills Practice
Angle Measure Use a protractor to find the measure of each angle. Then classify each angle as acute, obtuse, or right. 2. BAX 90; right
1. TAR 30; acute
B
Q
3. CAX
25; acute
4. TAX
150; obtuse
5. BAR
90; right
6. QAB
20; acute
7. RAC
155; obtuse
8. TAC 125; obtuse
9. QAC
85; acute
10. QAR 65; acute
115; obtuse
12. TAB 60; acute
11. QAX
T
R
C A
Use a protractor to draw an angle having each measurement. Then classify each angle as acute, obtuse, or right. 13. 50°
acute
14. 120° obtuse
15. 90° right
16. 25°
acute
17. 100° obtuse
18. 140° obtuse
19. 10°
acute
20. 135° obtuse
21. 85° acute
©
Glencoe/McGraw-Hill
13
Geometry: Concepts and Applications
X
Geometry_0-07-869312-8
3–3
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4:08 PM
Page 14
NAME
DATE
PERIOD
Skills Practice
The Angle Addition Postulate Refer to the figure at the right. 1. If mABE 100 and mABF 65, find m2. 35
A
2. Find m4 if mEBC 80 and mEBD 44. 36
C
B 1
F
23
4
D E
3. Find m3 if mFBD 85 and mFBE 42. 43 4. If mABE 105 and mEBD 46, find mABD. 151 5. If mABF 46 and mFBE 54, find mABE. 100 6. Find mFBC if m2 45 and mEBC 78. 123 7. If mFBD 102 and BE bisects FBD, find FBE. 51 Refer to the figure at the right. 8. If mWAZ 95 and mZAO 40, find mWAO. 135
W
9. If WAZ is a right angle and mYAZ 35, find mWAY. 55
X
Y Z
A
O
10. If mXAZ 82 and AY bisects XAZ, find mYAZ. 41 11. If mWAY 66 and AX bisects WAY, find mXAY. 33 12. Find mYAO if mWAO 130 and mWAY 70. 60 13. If mWAO 142, and AY bisects WAO, find mWAY. 71 14. Find mXAO if mXAY 35, mYAZ 40, and mZAO 42. 117 ©
Glencoe/McGraw-Hill
14
Geometry: Concepts and Applications
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3–4
Page 15
NAME
DATE
PERIOD
Skills Practice
Adjacent Angles and Linear Pairs of Angles Use the terms adjacent angles, linear pair, or neither to describe angles 1 and 2 in as many ways as possible. 1.
1
2
2.
3. 1
neither
1
2
adjacent angles
4.
adjacent angles; linear pair
5. 1
2
6.
2 1 1
neither 7.
2
adjacent angles 8.
1 2
adjacent angles
2
adjacent angles 9.
2
1
2
1
neither
adjacent angles; linear pair
In the figure at the right, MA and MG are opposite MJ are opposite rays. rays. Also, MC and
C
10. Which angle forms a linear pair with AMC?
A
CMG or AMJ
11. Do CME and EMJ form a linear pair? Justify your answer. Yes, their noncommon sides
are opposite rays.
M
E G
K
J
12. Name two angles that are adjacent to EMG.
Sample answer: CME, GMJ
13. Name two angles that form a linear pair with JMG.
AMJ, CMG
14. Name three angles that are adjacent to AMK. AMC, KMJ, KMG © Glencoe/McGraw-Hill 15
Geometry: Concepts and Applications
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4:08 PM
Page 16
NAME
DATE
PERIOD
Skills Practice
Complementary and Supplementary Angles Refer to the figures at the right. 1. Name a pair of complementary angles.
BOD, DOF
2. Name two right angles.
B
BOJ, BOF
D 57° O
3. Name three pairs of adjacent supplementary angles. JOB, BOF; JOD, DOF;
F
85° 95°
J
JOH, HOF
4. Find the measure of an angle that is complementary to JOH. 5
H
Exercises 1-6
5. Find the measure of an angle that is supplementary to DOF. 147 6. Find the measure of BOH.
33°
175
7. Name a pair of complementary angles.
X
WMR, RMS or SMT, TMZ WMS, ZMS
Y
8. Name two right angles.
9. Find the measure of an angle that is complementary to YMZ. 58 10. Find the measure of an angle that is supplementary to WMT. 60 11. Find the measure of XMY.
M 120° 32° 35° 60° 55° 30°
Z
W
R
T S
28
Exercises 7-13
12. Is YMT a right angle? Justify your answer.
No, its measure is 32 60, or 92.
13. Find the measure of an angle that is supplementary to XMR. 25
14. Find m3 if 3 and 4 form a linear pair and m4 55. 125 15. If 1 and 2 form a linear pair and m1 130, find m2. 50 16. Angles DEF and XYZ form a linear pair. If mDEF 170, what is mXYZ? 17. If 4 and 8 are complementary and m4 45, find m8.
45
18. If m3 10 and 3 and 7 are complementary, what is m7? ©
Glencoe/McGraw-Hill
16
10
80
Geometry: Concepts and Applications
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Page 17
NAME
3–6
DATE
PERIOD
Skills Practice
Congruent Angles Find the value of x in each figure. 1.
2.
3.
x° 152°
58° 122° (x 8)°
x°
152 4.
58
114
5.
6. 45°
(6x 15)°
78°
(3x)°
90° (4x 10)°
26
25
25 8. OA and OB are opposite rays OD are also and OC and opposite rays. If m2 90 and m1 and 45, what is m4?
7. What is the measure of an angle complementary to XYZ if MOR XYZ? 42 Y
45
X
A
M 1
Z
C
2 3 O 4
D
48°
O
B
R
Use the figure at the right. 9. If 1 3 and m1 64, find the measure of an angle that is supplementary to 3. 116
B A 1
10. If AOB is supplementary to BOC, BOC is supplementary to COD, and mAOB 58, find mBOC and mCOD. 122; 58
4
2
O3 C
D
11. Find the measure of an angle that is complementary to 1 if 1 2 and m2 75. 15 12. Find the measure of an angle that is supplementary to 4 if 4 9 and m9 24.
156
©
Glencoe/McGraw-Hill
17
Geometry: Concepts and Applications
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Page 18
NAME
DATE
PERIOD
Skills Practice
Perpendicular Lines AB ⊥ BE , FC ⊥ BE , and point X is the C . Determine whether each of the midpoint of A following is true or false. 1. XCB DCE true
F
A
X E
2. BCY is a right angle. true
W
3. FCE and FCX are supplementary. false
C
B
D
Z
Y
4. AB ⊥ BC true 5. FCD is a right angle. false 6. FCX and XCB are complementary. true 7. mWBZ mWBA
false
8. FC is the only line ⊥ to WE at C true 9. FCE and YCE are supplementary.
true
10. A X X C true 11. FCD WBA
false
12. A X F C false 13. AB ⊥ A C false 14. XCF DCY true BM ⊥ MC , MA and MD are opposite rays. 15. If mDMC 25, find mAMB.
65
16. If mAMB 72, find mDMC.
18
M C
17. If mDMC 2x 2 and mAMB 8x 2, find mDMC and mAMB. 20; 70 ©
Glencoe/McGraw-Hill
B
A
18
D
Geometry: Concepts and Applications
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Page 19
NAME
4–1
DATE
PERIOD
Skills Practice
Parallel Lines and Planes Describe each pair of segments in the prism as parallel, skew, or intersecting. , M L parallel 1. EG
2. L K , E G skew
3. L K , G H parallel
4. E G , G H intersecting
5. J N , M L skew
6. L K , N K
intersecting
7. N K , J H parallel
8. E G , H K
skew
9. M N , L K
M L E
G N K H
J
10. M N , G L skew
parallel
Use the figure for Exercises 1–10. Name the parts of the rectangular prism. 11. six planes
EGH, GLK, MLK, EMN, EMG, JNK
12. all pairs of parallel planes
EGH || MLK, MLG || NKH, EMN || GLK
13. all segments skew to J H M N , L K 14. all segments parallel to EG J H , ML , N K 15. all segments intersecting M L IE M , M N , L K , GL 16. all segments parallel to JN IH K , EM , G L Name the parts of the triangular prism. 17. all pairs of intersecting planes BFD and BFY, BFD and FDZ, BFD and BDZ,
XYZ and XZD, XYZ and XYF, XYZ and YZD X
18. all pairs of parallel segments
D F || Y Z ,
X || D Z , BX || F Y , DZ || F Y , BD || X Z , BF || X Y B
Y
B
F
D
Z
BF and X Z , BF and Y Z , FD and X Y , FD and X Z , BD and X Y , BD and Y Z , FY and X Z , FY and B D , BX and F D , BX and Y Z , DZ and X Y , DZ and B F
19. all pairs of skew segments
20. all points at which three segments intersect
©
Glencoe/McGraw-Hill
19
B, F, D, X, Y, Z
Geometry: Concepts and Applications
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4:08 PM
Page 20
NAME
DATE
PERIOD
Skills Practice
Parallel Lines and Transversals Identify each pair of angles as alternate interior, alternate exterior, consecutive interior, or vertical. 1. 1 and 6
vertical 3. 2 and 7
alternate interior 5. 2 and 5
vertical 7. 13 and 12
alternate exterior 9. 3 and 8
vertical 11. 9 and 14
vertical
2. 2 and 3
consecutive interior 4. 1 and 8
alternate exterior
12 56 13 14 9 10
6. 10 and 11
34 78 15 16 11 12
consecutive interior 8. 5 and 4
alternate exterior 10. 14 and 15
consecutive interior 12. 14 and 11
alternate interior
Find the measure of each angle. Give a reason for each answer. 13. 7 45; 14. 4 45;
Vertical angles are congruent.
Alternate interior angles are congruent. 15. 3 135; 16. 6 135; Consecutive interior Linear pairs are angles are supplementary. supplementary.
1 3 2 4 45° 5
6 7
Find the measure of each angle. Give a reason for each answer. 17. 1 125; Linear pairs are supplementary.
55° 1 2 3
18. 3 55; Vertical angles are congruent.
10 8 9 100°
4 5 11 12 6 7 13 14
19. 12 80; Consecutive interior angles are
supplementary. 20. 11 100; Alternate interior angles are congruent.
©
Glencoe/McGraw-Hill
20
Geometry: Concepts and Applications
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Page 21
NAME
DATE
PERIOD
Skills Practice
Transversals and Corresponding Angles In the figure, m p. Name all angles congruent to the given angle. Give a reason for each answer.
1 2 5 6
9 10 13 14
1. 1 6, vertical;
2. 7 4, vertical;
3. 13 10, vertical;
4. 8 3, vertical;
5. 9 14, vertical;
6. 16 11, vertical;
3 4 7 8
11 12 15 16
m p
12, alternate interior; 15, corresponding
9, corresponding; 14, alternate exterior
11, alternate interior; 16, corresponding
5, corresponding; 2, alternate exterior
8, corresponding; 3, alternate exterior
6, alternate interior; 1, corresponding
Find the measure of each numbered angle. 7.
8. 1 2 3 4
5 80° 2 6 1
40° 5
6
4 3 7 8 12 11 10 9
7
m1 40, m2 140, m3 140, m4 40, m5 140, m6 140, m7 40
m1 35, m2 80, m3 80, m4 100, m5 65, m6 65, m7 65, m8 115, m9 65, m10 115, m11 100, m12 80
9.
10. 70° 6 5 4 7 8 70° 3 50° 9 10 11 12 1 15 2 14 13
14 13 8 9
Glencoe/McGraw-Hill
12 10 11
78° 5 1 2
m1 70, m2 70, m3 60, m4 70, m5 110, m6 110, m7 110, m8 70, m9 40, m10 70, m11 60, m12 120, m13 60, m14 120, m15 70 ©
35°
21
6 7 3 4
m1 102, m2 78, m3 102, m4 78, m5 102, m6 78, m7 102, m8 102, m9 78, m10 102, m11 78, m12 102, m13 102, m14 78 Geometry: Concepts and Applications
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Page 22
NAME
4–4
DATE
PERIOD
Skills Practice
Proving Lines Parallel Find x so that c d. 1.
2.
3.
c
48°
c
(5x)°
(2x 8)°
d
x°
d
115°
48
41 c
4.
23
d
5.
6.
135° (7x 2)° 82°
(4x 3)°
d
12
c
d
24°
d
7
12
c
7.
c
(5x 11)°
c
8.
c
9.
(6x 1)°
d
c (6x 6)°
85° (3x 8)°
(9x 4)°
14
d
9
d
19
Name the pairs of parallel lines or segments. 10.
11.
A
12.
B
W
X 124°
56°
40° 75°
55°
124°
C
56°
Z
Y
75° 55°
D
ab 13. a 75°
A B D E 14.
b
E
C
D
15.
d
95°
100°
75°
70°
c d
85°
F
E
D E F C
cd ©
X W Z Y , WZ X Y
Glencoe/McGraw-Hill
70°
a b c
110°
abc 22
Geometry: Concepts and Applications
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Page 23
NAME
4–5
DATE
PERIOD
Skills Practice
Slope Find the slope of each line. 1.
2.
y (3, 2)
y
(2, 4)
(1, 3)
(1, 1)
x
O
3.
y
O
(1, 1)
x
x
O (2,2)
2
1 4
4.
undefined
5.
y
6.
y
y (3, 4)
(2, 4)
x
O
(0, 2)
x
O
x
O
(2, 3)
(2,3)
(2, 2)
1
0 7.
6
8.
y
9.
y
y
(1, 3) (1, 1)
(1, 2)
O
(3, 0)
x
x
O
(1, 2)
O
x
(1, 2)
12
undefined
1 2
Given each set of points, determine if AB and CD are parallel, perpendicular, or neither. 10. A(1, 1), B(2, 3), C(4, 1), D(6, 2) perpendicular 11. A(0, 5), B(5, 0), C(3, 2), D(4, 1)
parallel
12. A(2, 3), B(4, 5), C(0, 3), D(1, 0)
perpendicular
13. A(0, 0), B(4, 5), C(0, 3), D(5, 4)
neither
14. A(1, 1), B(3, 2), C(5, 0), D(3, 7) neither 15. A(2, 5), B(5, 2), C(3, 1), D(4, 0) parallel 16. A(2, 1), B(5, 3), C(2, 2), D(3, 3) neither 17. A(3, 0), B(6, 3), C(4, 3), D(5, 4) ©
Glencoe/McGraw-Hill
perpendicular 23
Geometry: Concepts and Applications
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Page 24
NAME
4–6
DATE
PERIOD
Skills Practice
Equations of Lines Name the slope and y-intercept of the graph of each equation.
4; 1 y 5x 1 5; 1 10x y 7 10; 7 y x 1;0 y 6 0; 6 2x 2y 14 1; 7 1 5x 10y 20 ;2 2
1. y 3x 2
2. y 4x 1
3.
4.
5. 7. 9. 11. 13.
3; 2 y 2x 5 2; 5 x 2 undefined; none y 3x 1 3; 1 2x y 3 2; 3 4x y 8 4; 8 9x 3y 12 3; 4
6. 8. 10. 12. 14.
Graph each equation using the slope and y-intercept. 15. y 2x 1
16. y x 3 y
y
y 2x 1 x
O
x
O
yx3
18. 3x y 2
17. x y 4 y
y
3x y 2 x
O
x
O
xy4
19. 2x y 1
20. 3x y 2 y
y
2x y 1 O
©
Glencoe/McGraw-Hill
3x y 2
x
O
24
x
Geometry: Concepts and Applications
Geometry_0-07-869312-8
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Page 25
NAME
5–1
DATE
PERIOD
Skills Practice
Classifying Triangles Classify each triangle by its angles and by its sides. 1.
60°
2.
6 in.
3 cm
60°
6 in.
16 ft 100° 16 ft 40°
35° 60°
4 cm
acute, equilateral
right, scalene
6.
50° 20 cm
11 m
isosceles, obtuse
5.
15 m 50°
72° 72°
38° 36°
11 m 80°
22 cm
isosceles, acute
42°
scalene, obtuse 8.
130°
30 cm
100°
32°
40° 25 ft
6 in.
4.
7.
3.
5 cm
55°
9.
60°
18°
isosceles, acute 45°
60°
45°
60°
obtuse, scalene
acute, equilateral
isosceles, right
Make a sketch of each triangle. If it is not possible to sketch the figure, write not possible. 10. right scalene
11. obtuse isosceles
12. right isosceles
13. right equilateral
not possible
©
Glencoe/McGraw-Hill
25
Geometry: Concepts and Applications
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Page 26
NAME
5–2
DATE
PERIOD
Skills Practice
Angles of a Triangle Find the value of each variable. 1.
2.
x°
3.
x°
110°
x°
x°
55° 70°
35
70°
40
4.
35
5.
x°
6. x°
50°
x°
52°
45°
44° 84°
85 7.
38
52
8.
18°
9.
x°
x°
60°
135° 75°
72 10.
45 11.
50°
x°
45 12.
x°
x° 45° 30°
48°
x°
40
42
105
Find the measure of each angle in each triangle. 13.
14.
x°
43°
(3x 10)° 64°
(2x)°
73
©
Glencoe/McGraw-Hill
32, 58
26
Geometry: Concepts and Applications
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5–3
Page 27
NAME
DATE
PERIOD
Skills Practice
Geometry in Motion Identify each motion as a translation, reflection, or rotation. 1.
2.
translation 4.
reflection 7.
reflection
3.
rotation
translation
5.
6.
translation
rotation
8.
9.
translation
rotation
In the figure at the right, quadrilateral ABCD → quadrilateral WXYZ.
B
C
10. Which angle corresponds to D? Z 11. Which side corresponds to X Y ? B C
D
A X W
12. Name the image of point A.
point W
Y
13. Name the image of A D . W Z 14. Which vertex of ABCD corresponds to Y? C
Z
15. Name the side that corresponds to A B . W X ©
Glencoe/McGraw-Hill
27
Geometry: Concepts and Applications
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Page 28
NAME
5–4
DATE
PERIOD
Skills Practice
Congruent Triangles Name the congruent angles and sides for each pair of congruent triangles. Then draw arcs and slash marks to show the congruent angles and sides. 1.
A
2.
X
O
Y
B Z
C
ACE XYZ
E
D
Z
N
C
M C, N B, O A, N M CB , NO BA , MO C A 4.
N
B
B
MNO CBA
A X, C Y, E Z, C XY , CE Y Z , AE X Z A 3.
A
M
Q
Z
T
A
R
C
I
BDE ZNQ
TRI ZAC
B Z, D N, E Q, D Z N, D EN Q, B EZ Q B
T Z, R A, I C, T R Z A, RI A C,T I Z C
Complete each congruence statement. 5.
B
C
W
Z
6.
O A
X
R A
CAX 7.
M
?
ZWO
BRX 8.
C
I
E
ARO
? O
Y
X
D
A
R
B
MAB 9.
?
EIX
MCD
B
A
L
10.
ABD ©
Glencoe/McGraw-Hill
A
M
C
O D
OIY
?
B
C
?
CDB 28
LMO
?
CBA
Geometry: Concepts and Applications
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Page 29
NAME
5–5
DATE
PERIOD
Skills Practice
SSS and SAS Write a congruence statement for each pair of triangles represented. C N O , CL O P , C O 1. A
2. W X A B , X Z B C , WZ A C
3. E G P S , EH P T , E P
4. H Y R P , E Y A P , Y P
5. Z A Q R , A P R S , Z P Q S
6. M L Z N , LR N B , L N
ACL NOP EGH PST
ZAP QRS
WXZ ABC HEY RAP MRL ZBN
Determine whether each pair of triangles is congruent. If so, write a congruence statement and explain why the triangles are congruent. 7.
A
B
C
8.
E
A
W
D R
D
ABD CBD; SAS 9.
H
G
AER WDG; SSS 10.
I
D
A
Q
G J U
E
HGE HIJ; SAS 11.
P
A
QAD QAU; SAS 12.
U X
W L
R
Z Y
PLR RAP; SSS
UWZ XWY; SAS
Use the given information to determine whether the two triangles are congruent by SAS. Write yes or no. LD M R , L O M A yes 13. L M,
L
14. L M, LD M R , O A, no 15. L D MR , L O M A , O A, no 16. L D MR , L O M A , D O R A no © Glencoe/McGraw-Hill 29
D
R
A
O
M
Geometry: Concepts and Applications
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Page 30
NAME
5–6
DATE
PERIOD
Skills Practice
ASA and AAS Write a congruence statement for each pair of triangles represented. Z R . ABC ZRX 1. In ABC and ZXR, C X, A Z, and AB 2. In DEF and BGO, D B, E O, and D E B O . DEF BOG 3. In TRI and GAN, T A, T I A G , and T R A N . TRI ANG 4. In ZIP and LOS, P O, I L, and P I O L . ZIP SLO Name the additional congruent parts needed so that the triangles are congruent by the postulate or theorem indicated. 5. AAS
D
T
S
A
R
M
6. ASA
A
F
B
C
C
R D
S B
7. AAS
8. ASA M
R
A
B Y
X O
B
A Z N
C
C
N O A C or M O B A
Z X B C
Determine whether each pair of triangles is congruent by SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write not possible. 9.
10.
F
S
T V
G
I
X
R
H
SSS
ASA
11.
12.
V B
N
A
K
C
not possible ©
Glencoe/McGraw-Hill
AAS 30
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Page 31
NAME
DATE
PERIOD
Skills Practice
Medians AD , BE , and CF are medians of ACE. 1. If AE 24, find AF. 12 2. Find AE, if FE 15.
3. What is BC if AC 36? 4. Find CE, if DE 7.
18
C M
D
F
14
E
5. What is CD if CE 68? 6. If AF 3, find AE.
B
A
30
34
6
R T , ZX , and SW are medians of TXS. 7. If TX 18, find TW.
9
T W
8. If TO 26, find OR. 9. If WO 5, find OS.
13
X
O
Z
10
R S
10. Find ZO if OX 50.
25 1
11. What is TZ if TS is 2? 12. What is OS if OW is 9?
18
RN , PM , and LO are medians of LNP. 13. What is LP if RL is 4? 8
L R
14. Find AO if LO 18
6
15. What is RA if AN is 42?
P
21
A
M
O N
16. If MA 13, find MP.
39
17. Find AN if RN 30.
20
18. If LO 15, find AO.
©
Glencoe/McGraw-Hill
5
31
Geometry: Concepts and Applications
Geometry_0-07-869312-8
6–2
9/16/08
4:08 PM
Page 32
NAME
DATE
PERIOD
Skills Practice
Altitudes and Perpendicular Bisectors Tell whether the bold segment or line is an altitude, a perpendicular bisector, both, or neither. 1. 2. 3.
both
neither
4.
5.
neither
6.
both
7.
perpendicular bisector
8.
both
9.
altitude
10.
perpendicular bisector
11.
neither
12.
perpendicular bisector
Use the figure at the right. 13. Name a segment in the triangle that is an altitude.
both
A
C A or B C
D
14. Name a segment in the triangle that is a D perpendicular bisector. F
E C
14. Name a segment in the triangle that is not an altitude and is not a perpendicular G bisector. A
©
altitude
Glencoe/McGraw-Hill
32
F
G
B
Geometry: Concepts and Applications
Geometry_0-07-869312-8
9/16/08
4:08 PM
6–3
Page 33
NAME
DATE
PERIOD
Skills Practice
Angle Bisectors of Triangles In ACD, DB bisects ADC, and C E bisects ACD. 1. If m1 40, what is m2? 40
A E
2. Find mACD if m4 25.
50 D
3. What is m3 if mACD 36?
B
1 2
18 3
4. If m1 45, what is mADC?
90
5. What is mDCA if mDCE 20? 6. Find mADB if mBDC 39.
39
7. What is mACD if m4 18?
36
8. Find m2 if m1 43.
4
C
40
43
9. If m3 21, what is m4?
21
10. What is mECD if mECA 24?
24
In MOR, MP bisects OMR, RN bisects MRO, and O S bisects MOR. 11. Find m6 if mMOR 34.
O
17
12. What is mOMR if m1 23?
46 M
13. If m3 55, what is m4?
55
15. Find m1 if m2 27.
16
17. What is mSOR if m6 15?
P 34
R
120 15
18. If mMRP 112, what is m3? 19. Find mOMP if mPMR 30.
56 30
20. What is m4 if MRO is a right angle?
Glencoe/McGraw-Hill
5
27
16. If m4 60, what is mMRO?
©
1 2
S
14. What is mMOS if mMOR 32?
6
N
45
33
Geometry: Concepts and Applications
Geometry_0-07-869312-8
9/16/08
4:08 PM
Page 34
NAME
6–4
DATE
PERIOD
Skills Practice
Isosceles Triangles Find the values of the variables for each triangle. 1. 2. A D a°
10
b
F 60°
3. X
Z
y°
Y x°
40°
54°
E d°
60°
a 60; b 10 4. B
B
A
B
C
x 40; y 100 5.
c 54; d 63 6.
C
R
x°
40°
a° P b60° °
y
A
45°
W
c°
4
O
x° y°
44°
C
A
M
x 45; y 4 7.
W
X
b°
a 60; b 68 8.
x 70; y 110 9.
S
a°
R y°
24°
A
x° 45°
10.
x°
a°
E
x 90; y 60 11.
52 °
b°
a°
T
Y
a 102; b 78
B
Z
60°
D
a 60; b 64 12.
d°
42°
a° b°
c°
88°
44°
a 45; x 90
c 92; d 44
a 46; b 96
Use the figure at the right. 13. In ADF, if AD x 6 and DF 3x 10, what is AD? 14
D
14. In ADH, if mADH 2x 4, find the value of x. 24
46°
15. If AH 5x 1 and FH 3x 21, what is AH? 54 16. In ADF, what is mADF?
©
Glencoe/McGraw-Hill
A
H
F
88 34
Geometry: Concepts and Applications
Geometry_0-07-869312-8
9/16/08
4:08 PM
Page 35
NAME
6–5
DATE
PERIOD
Skills Practice
Right Triangles Determine whether each pair of right triangles is congruent by LL, HA, LA, or HL. If it is not possible to prove that they are conguent, write not possible. 1. 2. M
F
E
W
X
Y
V D
Z
G
A
LL
HL
3.
4. A
R
C
A
M D
E
T
not possible 5.
Q
B
HA 6.
R
T
S C
A
B
C
D
S E
HL
LL
7.
8.
R
R A
M
Q T
S P
HL
HA
9.
10. F
C
D
I
G
LA ©
Glencoe/McGraw-Hill
A
B
E
D
C
not possible 35
Geometry: Concepts and Applications
Geometry_0-07-869312-8
9/16/08
4:08 PM
Page 36
NAME
6–6
DATE
PERIOD
Skills Practice
The Pythagorean Theorem Find the missing measure in each right triangle. Round to the nearest tenth, if necessary. 1.
2. 4 cm
c in.
10 in.
3. 33 ft
20 ft
c cm 25 in.
b ft
3 cm
5 4.
26.9
26.2
5.
8 in.
6.
am
b in. 9 in.
28 m
am
3m 4.5 m
10 m
4.1
26.2
7.
8. 3 in.
3.4 9.
25 m
24 ft
c in.
6 ft
am b ft
29 m 5 in.
5.8
14.7
23.2
Find each missing measure if c is the measure of the hypotenuse. Round to the nearest tenth, if necessary. 11. b 6, c 10, a ? 8 10. a 15, b 10, c ? 18.0 12. c 100, b 60, a ? 14. a 2, b 3, c ?
13. c 16, a 9, b ?
80
15. c 5, b 2, a ?
3.6
16. b 7, c 15, a ?
13.2 4.6
17. c 30, a 20, b ?
13.3
22.4
The lengths of three sides of a triangle are given. Determine whether each triangle is a right triangle. 18. 3 cm, 4 cm, 5 cm yes 19. 1 ft, 1 ft, 2 ft no 20. 2 in., 2 in., 4 in.
no
22. 5 in., 10 in., 15 in.
©
Glencoe/McGraw-Hill
21. 8 m, 15 m, 17 m
no
yes
23. 14 cm, 48 cm, 50 cm
36
yes
Geometry: Concepts and Applications
Geometry_0-07-869312-8
9/16/08
4:08 PM
6–7
Page 37
NAME
DATE
PERIOD
Skills Practice
Distance on the Coordinate Plane Find the distance between each pair of points. Round to the nearest tenth, if necessary. 1. X(4, 0), Y(2, 0)
2
6
3. F(5, 0), G(–1, 0)
5
2. C(–3, 0), D(–8, 0)
4. F(0, 8), G(0, 1)
5
5. M(0, –1), N(0, –6)
7. A(2, –1), B(5, 3)
5
9. H(–3, 4), I(5, 4)
8
7
6. R(0, 6), S(0, –2)
8
8. D(1, –5), E(1, 5)
10
10. G(0, 0), H(6, 8)
10
11. K(0, 0), M(–3, –4)
5
12. A(0, 7), C(24, 0)
25
13. E(1, 1), A(–1, –1)
2.8
14. M(0, 4), Z(–2, 5)
2.2
15. A(–1, –2), Z(2, 1)
4.2
16. Y(5, 6), B(–5, –6)
15.6
17. C(2, –3), X(2, 7)
10
18. D(6, 9), U(9, 6)
19. L(5, 3), M(–1, 0)
6.7
20. S(–2, –6), T(7, 6)
15
22. G(8, –9), X(–1, 0)
12.7
24. W(4, –2), N(1, 2)
5
21. H(1, –1), A(–5, 4)
7.8
23. B(–7, 4), T(–10, –1)
©
Glencoe/McGraw-Hill
5.8 37
4.2
Geometry: Concepts and Applications
Geometry_0-07-869312-8
7–1
9/16/08
4:08 PM
Page 38
NAME
DATE
PERIOD
Skills Practice
Segments, Angles, and Inequalities For Exercises 1–12, use the figure below. A
B
C
D
O
E
F
G
H
20
15
10
5
0
5
10
15
20
Replace each with , , or to make a true sentence. 1. AB FH
2. AD EH
3. AE HE
5. DF OG
6. CG BF
4. AE DE
Determine whether each statement is true or false.
true
8. OE DF true
10. BD FG true
11. OH OA true
7. AF BG
For Exercises 13–24, use the figure at the right. Lines AJ, DM, and GO intersect at P.
9. CG OH false 12. AG BH
true
D G
A 45° 55°
P
J
O M
Replace each with , , or to make a true sentence. 13. mAPD mMPJ
14. mDPG mAPD
15. mDPG mGPJ
16. mAPO mGPJ
18. mAPO mDPG
19. mOPJ mAPG true
20. mOPJ mGPJ
false
21. mOPJ mDPG true
22. mAPG mGPJ
false
23. mDPJ mAPM false
24. mAPO 2 • mAPD
17. mMPJ mOPM
Determine if each statement is true or false.
©
Glencoe/McGraw-Hill
38
false
Geometry: Concepts and Applications
Geometry_0-07-869312-8
9/16/08
4:08 PM
Page 39
NAME
7–2
DATE
PERIOD
Skills Practice
Exterior Angle Theorem Name the angles. 1. an exterior angle of ABX
2. an exterior angle of BEA
7 or 9
B
5
3. an interior angle of ABX
2 4
4. an interior angle of BEA
1, 6, or ABX
1, 2, or 3
5. a remote interior angle of ABX with respect to 7
X
6. a remote interior angle of BEA with respect to 5
1 or ABX
1
A
35
E
6 7 C 9 8
D
1 or 2
Find the measure of each angle. 7. C
32
1
8.
40
58°
A
9.
X
B
T 72°
C 105°
148
Y
4
11.
M
3
47°
4
93°
U
115
E
2
50°
65°
Z
10. 3
122
R
1
D
2
12.
5
55 R
G
T 5
135° 80°
55° R
68°
A
W
A
Use the figure at the right. 13. Find the value of x.
B x°
50
50
14. Find mB.
105 16. Find mBMZ. 75 15. Find mBMA.
(2x 5)°
A
55°
M
Z
Replace each with , , or to make a true sentence. 17.
A 1
B 2
18.
19.
E
3
1
4
F 1
J 2
2
3
5 6
C 3
Y
4 5
H
6
X
G 4
m3 m1 ©
Glencoe/McGraw-Hill
m5 m3 39
I
m2 m4 m6 Geometry: Concepts and Applications
Geometry_0-07-869312-8
9/16/08
4:08 PM
Page 40
NAME
7–3
DATE
PERIOD
Skills Practice
Inequalities Within a Triangle List the angles in order from least to greatest measure. 1.
2.
A 5 in.
3 in.
3.
W
A 4 ft
26 cm
18 cm
8 ft
M D
Z
C
4 in.
C, A, D 4.
6 ft
X, Z, W 5.
B
X
30 cm
6.
R
O, A, M
O V
13 in.
9 in. 27 m
10 cm
16 cm
24 m
Y
Z C
X
16 in.
8 cm 18 m
O
B, C, A
A
R, O, Z
X, Y, V
List the sides in order from least to greatest measure. 7.
8.
C
9.
R
M 62°
50°
35° 28°
70°
60°
B
L
Y A
70°
75°
B A , BC , AC
A
T A , RT , RA
T
Y M , YL , ML
Identify the angle with the greatest measure. 10.
12 in.
A
11.
B 11 in.
14 in.
12.
D 17 cm
11 cm
C
F
B
13 cm
G 19 mm
I
E
F
38 mm
27 mm
I
H
Identify the side with the greatest measure. 13.
14.
J
15.
M
59°
48°
53°
95° 55°
L
J L ©
P
37°
O
72°
K
Glencoe/McGraw-Hill
N M
N
40
61°
R
60°
Q
Q P Geometry: Concepts and Applications
Geometry_0-07-869312-8
9/16/08
4:08 PM
7–4
Page 41
NAME
DATE
PERIOD
Skills Practice
Triangle Inequality Theorem Determine if the three numbers can be measures of the sides of a triangle. Write yes or no. Explain.
1. 6, 7, 8
yes; 6 7 8, 7 8 6, 687
2. 1, 1, 2
no; 112
3. 2, 4, 6
4. 5, 8, 10 yes;
5. 10, 20, 30 no;
6. 3, 4, 5
8. 6, 12, 24 no;
9. 1, 7, 10
5 8 10, 8 10 5, 5 10 8 yes; 3 5 7, 5 7 3, 375
7. 3, 5, 7
10. 10, 15, 26 no;
10 15 26
10 20 30
6 12 24
yes; 8 12 19, 12 19 8, 8 19 12
11. 8, 12, 19
no; 246
yes; 3 4 5, 4 5 3, 354 no; 1 7 10
yes; 4 7 10, 7 10 4, 4 10 7
12. 4, 7, 10
Find the range of possible measures for the third side of a triangle with the measures given for two sides. 13. 7, 13 6 x 20
14. 20, 25
5 x 45
16. 32, 38 6 x 70
17. 50 ,70
20 x 120 18. 8, 20 12 x 28
19. 55, 10 45 x 65
20. 2, 10
22. 45, 70 25 x 115 23. 9, 19
©
Glencoe/McGraw-Hill
15. 1, 5
4x6
8 x 12
21. 60, 70
10 x 28
24. 100, 120 20 x 220
41
10 x 130
Geometry: Concepts and Applications
Geometry_0-07-869312-8
9/16/08
4:08 PM
Page 42
NAME
8–1
DATE
PERIOD
Skills Practice
Quadrilaterals Refer to quadrilaterals ABCD and MNOZ.
A
1. Name the side opposite B C . A D 2. Name a side that is consecutive with AB .
B
C B or A D
D
3. Name a pair of consecutive vertices in ABCD.
A and B or B and C or C and D or A and D 4. Name the vertex opposite B. D C
5. Name the two diagonals in ABCD.
C A and B D
Exercises 1 - 5
6. Name a pair of consecutive angles in MNOZ.
Sample answer: M and N 7. Name the vertex opposite O. M
N
M
8. Name the side opposite MZ . N O 9. Name a side that is consecutive with MN .
Z
O
O N or M Z
Exercises 6 - 10
10. Name the diagonals in MNOZ. M O and N Z Find the missing measure(s) in each figure. 11. x°
12.
x° 72°
110° 75°
120°
70°
70°
105 13.
98 14.
x°
x°
x°
x°
x°
110° 130° 80°
40 15.
90, 90, 90, 90
x°
16.
2x°
2x°
Glencoe/McGraw-Hill
50°
x°
50°
x°
60, 120, 60, 120 ©
x°
130, 130 42
Geometry: Concepts and Applications
Geometry_0-07-869312-8
9/16/08
4:08 PM
8–2
Page 43
NAME
DATE
PERIOD
Skills Practice
Parallelograms Find each measure. 1. mH
130
3. GH 25
2. mG 50
F
G
25
4. FG 40
50°
I
H
40
Find each measure. 5. mZ 60
6. mW 120
7. XY 7
8. WX 7
X
W 7 120° 7
Z
Y
In the figure, TQ 42 and SA 14. Find each measure. 9. TA 21
10. mQST
110
11. SR 28
12. mSTR
70
13. SQ 22
14. ST
30
15. AQ 21
16. AR
14
S
T
A Q
30
22 65° 45°
R
17. In a parallelogram, the measure of one side is 38. Find the measure of the opposite side. 38
18. The measure of one angle of a parallelogram is 45. Determine the measures of the other three angles. 45, 135, 135
©
Glencoe/McGraw-Hill
43
Geometry: Concepts and Applications
Geometry_0-07-869312-8
8–3
9/16/08
4:08 PM
Page 44
NAME
DATE
PERIOD
Skills Practice
Tests for Parallelograms Determine whether each quadrilateral is a parallelogram. Write yes or no. If yes, give a reason for your answer. 1.
2.
yes, Theorem 8-7
4.
3.
yes, Theorem 8-9
5.
no
yes, definition of parallelogram
6.
yes, definition of parallelogram
7.
8.
no
9.
90°
yes, Theorem 8-8
10.
110°
70°
70°
yes, Theorem 8-9
11.
no
12.
50° 130°
110°
90°
105° 75°
yes, definition of parallelogram
©
Glencoe/McGraw-Hill
no
yes, Theorem 8-8
44
Geometry: Concepts and Applications
Geometry_0-07-869312-8
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4:08 PM
Page 45
NAME
8–4
DATE
PERIOD
Skills Practice
Rectangles, Rhombi, and Squares Identify each parallelogram as a rectangle, rhombus, square, or none of these. 1.
2.
none of these 4.
3.
square
rhombus
5.
none of these
6.
rectangle
none of these
Find each measure. 7. BO
10
8. OC
10 A
9. AC 20
10. BD
B
20 10 cm
11. mAOB
90
12. mABC
90
13. mADB
45
14. mDCA
45
15. MG
18
16. HF
O
D
C
Exercises 7 - 14
20 E
17. mEHG
120
18. mFEH
60
F 18 in.
M
19. mEMF
90
20. FG
60° 10 in.
22
H
21. EF
©
22
Glencoe/McGraw-Hill
22. mFGE
45
30
22 in.
G
Exercises 15 - 22
Geometry: Concepts and Applications
Geometry_0-07-869312-8
9/16/08
4:08 PM
Page 46
NAME
8–5
DATE
PERIOD
Skills Practice
Trapezoids For each trapezoid, name the bases, the legs, and the base angles. 1. 2. 3. X A
C
T
R
W G
E
P Z
Y
Z , XY ; W X , ZY ; W W and Z, X and Y
A C , G E ; AG , CE ; A and C, G and E
Find the length of the median in each trapezoid. 3 ft 4. 5. 10 cm
T R , PA ; T P , A R ; T and R, P and A 6.
16 m
24 cm 14 m
7 ft
5 ft 7.
A
15 m
17 cm 8.
14 yd
9.
7 in.
5m
12 m
1 in.
38 yd
26 yd
4 in.
8.5 m
Find the missing angle measures in each isosceles trapezoid. 10. 11. 12. N A B
Q
U
M 105°
55°
C
D
P
124°
D
A
O
55, 125, 125 13.
H
124, 56, 56 B
14.
D
64°
Glencoe/McGraw-Hill
T
X 72°
O C
D
64, 116, 116
©
15.
M
150°
X F
105, 75, 75
150, 30, 30
46
A
72, 108, 108
Geometry: Concepts and Applications
Geometry_0-07-869312-8
9/16/08
4:08 PM
Page 47
NAME
9–1
DATE
PERIOD
Skills Practice
Using Ratios and Proportions Write each ratio in simplest form. 1.
15 20
3 4
2.
7 49
1 7
3.
10 15
2 3
4.
28 35
4 5
5.
11 22
1 2
6.
20 25
4 5
7.
3 15
1 5
8.
18 81
2 9
9.
36 27
4 3
10.
55 33
5 3
11.
12 2
6 or 6 1
12.
8 5
40 25
16 5
3 5
13. 15 feet to 25 feet
14. 48 centimeters to 15 centimeters
7 12
3 4
15. 45 meters to 60 meters
16. 14 inches to 24 inches
2 5
1 3
17. 12 inches to 3 feet
18. 80 centimeters to 2 meters
2 5
2 15
19. 4 feet to 10 yards
20. 8 quarts to 5 gallons
Solve each proportion. 21.
6 3 8 x
24.
8 x 28 21
27.
30. ©
22.
24 x 18 3
4
23.
7 14 12 x
24
6
25.
4 x 8 12
6
26.
32 16 6 x
3
9 x 5 20
36
28.
5 35 70 x
10
29.
22 x 18 27
33
2 14 21 x
3
31.
8 56 7 x
1
32.
x 6 5 30
16
Glencoe/McGraw-Hill
47
1
Geometry: Concepts and Applications
Geometry_0-07-869312-8
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Page 48
NAME
9–2
DATE
PERIOD
Skills Practice
Similar Polygons Determine whether each pair of polygons is similar. Justify your answer. 1.
2.
6
3
2
4
4
3.
5
2
2 5
5
2
5
2 1
1 2 3
3 5
3
8
8 5 6
Yes; corresponding angles are congruent 4 3 and . 8 6 4.
3 4
3
2 1 No; . 3 5
5.
4
3
Yes; corresponding angles are congruent 2 2 and . 5 5
6. 14
14
4
26
26
6
14
3
10 6
4
6 6
6
10
10
6 10
10 10
26
No; corresponding angles are not congruent.
Yes; corresponding angles are congruent 6 and is the ratio for 10 the lengths of all six sides.
Yes; corresponding angles are congruent 14 14 14 and . 26 26 26
If each pair of polygons is similar, find the values of x and y. 7.
30
8. 20
y
x
20
20
16
9.
20
x
20
6
16 4
y
y
12
6
x
x 8, y 10 10. 4
3 10
11.
x 4 6
x 30, y 30
y
4 4
4 4
4 4
x 6, y 24 12.
11
x y
6 10
14 16
y 9
x
x 8, y 20
©
Glencoe/McGraw-Hill
x 11, y 11
48
x 15, y 24
Geometry: Concepts and Applications
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Page 49
NAME
9–3
DATE
PERIOD
Skills Practice
Similar Triangles Determine whether each pair of triangles is similar. If so, tell which similarity test is used and complete the statement. 1.
2. Z
A 10 6
C
4
Y
B M
B
M
20
12
3. 6
5
R
A
A 8
10
O 3 N
Y
Z
Y
B
C
ABC ?
MNO ?
BAC ?
yes; ZYM; SAS
yes; YBA; SSS
yes; ZYR; AA
4.
5. D
6.
Q
F
A
A 5
7
R
M Y
D
R
4
5
Z I
M
D
G
E
F
D
DEF ?
DYA ?
FGI ?
not similar
yes; RMZ; AA
not similar
Find the value of each variable. 7.
8.
16
14
7 24
x 12
9.
16 24
36
3 7
x
x
y
y 24
x 16, y 24
x8 10.
x 3, y 7
11.
12. 30
8
36 24
y
20
15
x 6, y 18 ©
24
Glencoe/McGraw-Hill
y x
x
20
20
30
x 16, y 24 49
y
30
x
42
x 20, y 28 Geometry: Concepts and Applications
Geometry_0-07-869312-8
9/16/08
4:08 PM
Page 50
NAME
9–4
DATE
PERIOD
Skills Practice
Proportional Parts and Triangles Complete each proportion. 1.
MB MD = BA ?
DC
2.
MR MW = RS ?
WT S A
3.
BD MD = AC ?
MC
4.
MB ? = MA MC
MD
R
B
W C
5.
CD AB = DM ?
6.
MS MT = RM ?
BM
MW
7.
ST SM = RW ?
RM
8.
MW ? = WT RS
MR
9.
RW MW = ST ?
MT
10.
TW SR = WM ?
RM
11.
CM ? = MD BM
AM
12.
SM TM = SR ?
TW
T
M
D
Find the value of x. 13.
14. 12
42
15
50
15.
36
30 15
x
x
8
18
x
10
6
16.
17. 15
25
25 18.
15
x
21
12
7
15
x
9
x
8
24
7
9 20
19.
20.
5
21.
x
x 10
12 2x
5 ©
18
15
Glencoe/McGraw-Hill
30
x8
5
4
7.2 50
Geometry: Concepts and Applications
Geometry_0-07-869312-8
9/16/08
4:08 PM
Page 51
NAME
9–5
DATE
PERIOD
Skills Practice
Triangles and Parallel Lines In each figure, determine whether AB || RS . 1.
2.
C 15
A 8
3.
S
30
B 6S
26
9
8
32
13
R
C
R R C
24
S
20
21
A A
yes 4.
B
B
yes C 13 S
no
5.
23
B
7
18
R 14
R
5
6.
C 7
A 4
S
R 12
21
A
C
A
24
S 8
B
B
no
no
yes
M, O, and R are the midpoints of the sides of ABC. Complete each statement. 7. OR ||
?
B A
8. BC ||
?
M R
B
9. If MO 15, then AC
?
30
10. If BC 62, then MR
?
31 ?
75
12. If mBCA 52, then mBOM
?
52
?
60
?
?
C
28
15. If AB 50, then OR
?
25
16. If BC 74, then BO
?
37
17. If mCOR 60, then mCBA 18. If BO 19, then MR Glencoe/McGraw-Hill
R
O M
14. If BM 28, then AM
©
A
11. If mBMO 75, then mBAC
13. AC ||
O
M
?
19 51
Geometry: Concepts and Applications
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Page 52
NAME
9–6
DATE
PERIOD
Skills Practice
Proportional Parts and Parallel Lines Complete each proportion. ? TX 1. WY YA XZ YA XZ 3. TX WY ?
TX WY 2. TZ WA ?
W
T
Y
X
TZ WA 4. WY ? TX
Z
AY XZ 5. WY ? TX
A
WA TZ 6. TX ? WY
Find the value of x. 7.
8.
9. 9
10
10
15
x
12
15
x
x
15
20
10.
12. 30
4
30
8
x
26
5
x
60
13.
20
14.
15.
3
20
5 9
16
30
18
8
15
18
11.
x
10
12
x
5 88 7
x
x
60 21
20 ©
Glencoe/McGraw-Hill
22
20 52
Geometry: Concepts and Applications
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Page 53
NAME
9–7
DATE
PERIOD
Skills Practice
Perimeters and Similarity Find the value of each variable for each pair of similar triangles. 1.
2.
A
3 18
R
15
N
3.
A
M 5
P
4
C
x
x 8, y 10, z 12 A
Perimeter of ADE 24
25
x
x 3, y 3, z 3
B E 24
N
O
A
C
32
D
R 20
14
z y
15
Q
6.
46
y z
Perimeter of OPS 9
x 6, y 8, z 10 5.
M
G
10
S
Perimeter of RST 30
15
z
y
H T
C
S
D
y
y
x
10
10
B
12
4.
O
z
x
E z
F
x
z
x
T 14 S
F y
Z
P
B
Perimeter of MNO 11
x 5, y 3, z 3
Perimeter of DEF 51
x 16, y 12, z 23
Perimeter of QPZ 168
x 49, y 49, z 70
Determine the scale factor for each pair of similar triangles. F M
X
9 30
18
Z
38
A
D
15 27
B
19
28
9
15
A
Y
18
X
M
6 B
Y 5
A
B
12
H
24
N
42 18 36
L
2 3
7. XYZ to MBA
2 1
8. ADX to MYB
3 1
9. FHA to BLN
1 2
11. MYB to ADX
1 3
12. BLN to FHA
10. MBA to XYZ
3 2
13. The perimeter of BDE is 72 feet. If BDE FMR and the scale factor of 4 BDE to FMR is , find the perimeter of FMR. 90 ft 5
©
Glencoe/McGraw-Hill
53
Geometry: Concepts and Applications
P
Geometry_0-07-869312-8
10–1
9/16/08
4:08 PM
Page 54
NAME ______________________________________DATE __________PERIOD______
Skills Practice
Naming Polygons Identify each polygon by its sides. Then determine whether it appears to be regular or not regular. If not regular, explain why. 1.
2.
quadrilateral, not regular; sides are not all congruent
3.
triangle, regular
quadrilateral, not regular; sides not congruent, angles not congruent
Name each part of hexagon ABCDEF. 4. two consecutive vertices
A
Sample answer: A, B 5. two diagonals
B
F
Sample answer: AC , AD 6. all nonconsecutive sides of A B
C
E
D
Sample answer: CD , DE , EF
7. any three consecutive sides
Sample answer: AB , BC , CD 8. any four consecutive vertices
Sample answer: A, B, C, D Classify each polygon as convex or concave. 9. 10.
convex 12.
©
convex 13.
concave
Glencoe/McGraw-Hill
11.
concave 14.
convex
54
concave
Geometry: Concepts and Applications
Geometry_0-07-869312-8
9/16/08
4:08 PM
Page 55
NAME ______________________________________DATE __________PERIOD______
10–2
Skills Practice
Diagonals and Angle Measure Find the sum of the measures of the interior angles in each figure. 1. X
2.
A
3.
B
Z
R
T
C
F
Q
Y S E
180
D
720
4.
360 I
5.
E
6.
M
R
L
J
H P
K D L
O Z
540
L
A
N
M
1260
180
Find the measure of one interior angle and one exterior angle of each regular polygon. If necessary, round to the nearest degree. 7. triangle
60, 120 10. nonagon
140, 40
8. octagon
9. decagon
135, 45
144, 36
11. quadrilateral
90, 90
12. hexagon
120, 60
13. The sum of the measures of three interior angles of a quadrilateral is 245. What is the measure of the fourth interior angle? 115 14. The sum of the measures of six exterior angles of a heptagon is 310. What is the measure of the seventh exterior angle? 50 15. The sum of the measures of seven interior angles of an octagon is 1000. What is the measure of the eighth interior angle? 80 16. The sum of the measures of nine exterior angles of a decagon is 325. What is the measure of the tenth exterior angle? 35
©
Glencoe/McGraw-Hill
55
Geometry: Concepts and Applications
Geometry_0-07-869312-8
10–3
9/16/08
4:08 PM
Page 56
NAME ______________________________________DATE __________PERIOD______
Skills Practice
Areas of Polygons Find the area of each polygon in square units. 1.
2.
5 units2
3.
5 units2
4.
5.
4 units2
6.
10 units2
7.
8.
9 units2
14 units2 9.
5 units2
Estimate the area of each polygon in square units. 10–12. 10.
8 units2
11.
16 units2 13. Sketch two polygons that both have a perimeter of 20 units, but that have different areas.
10 units2 Sample answers given. 12.
17 units2
23 units2
Sample answer: 3 5
P 20
14. Sketch a pentagon with an area of 20 square units.
©
Glencoe/McGraw-Hill
Sample answer:
56
A 25
P 20
5
A 21
7
Geometry: Concepts and Applications
Geometry_0-07-869312-8
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Page 57
NAME ______________________________________DATE __________PERIOD______
10–4
Skills Practice
Areas of Triangles and Trapezoids Find the area of each triangle or trapezoid. 1. 2.
3.
14 cm
6m 5m
20 ft 20 cm
8m
37 ft
370 ft2
140 cm2
4.
5.
35 m2
5 ft
6.
5 in.
19 cm
7 ft 16 cm
12 in.
26 cm
15 ft
30 in2
70 ft2
7.
399 cm2
8.
11 m
9.
27 in. 5 in.
3m
19 mm
17 in.
3m 24 mm
228 mm2
21 m2
10.
297 in2
11. 28 cm
12. 2 cm 3 cm
22 cm
308 cm2
25 m 18 m
6 cm
12 cm2
225 m2
13. Find the area of a trapezoid whose altitude measures 8 feet and whose bases are 9 feet and 21 feet long. 120 ft2 14. Find the area of a triangle whose base measures 17 inches and whose altitude is 10 inches. 85 in2 15. Find the area of a trapezoid whose altitude measures 77 meters and whose bases are 200 meters and 300 meters long. 19,250 m2 16. The area of a triangle is 500 square feet. The height of the altitude is 25 feet. What is the length of the base? 40 ft ©
Glencoe/McGraw-Hill
57
Geometry: Concepts and Applications
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Page 58
NAME ______________________________________DATE __________PERIOD______
10–5
Skills Practice
Areas of Regular Polygons Find the area of each regular polygon. 1.
2.
3. 5 ft
12 cm
4 ft
18 cm
540 cm2 4.
10 in.
60 ft2 3 cm
9 in.
360 in2
5.
6.
8m
22 in. 2.5 cm
7 in. 7m
22.5 cm2
231 in2
168 m2
Find the area of the shaded region in each regular polygon. 7.
38 in.
8.
9.
15 ft
5m
6m
12 ft 24 in.
1824 in2
270 ft2
10.
80 m2
11.
18 m
28 in.
12.
18 cm 30 cm
24 in.
1344 in2
©
Glencoe/McGraw-Hill
13 m
351 m2
58
13 cm
17 cm
177 cm2
Geometry: Concepts and Applications
Geometry_0-07-869312-8
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4:08 PM
10–6
Page 59
NAME ______________________________________DATE __________PERIOD______
Skills Practice
Symmetry Determine whether each figure has line symmetry. If it does, draw all lines of symmetry. If not, write no. 1.
2.
yes 4.
3.
no
yes
5.
yes
6.
no
yes
Determine whether each figure has rotational symmetry. Write yes or no. 7.
8.
yes 10.
no
yes
11.
yes ©
9.
Glencoe/McGraw-Hill
12.
no
no 59
Geometry: Concepts and Applications
Geometry_0-07-869312-8
10–7
9/16/08
4:08 PM
Page 60
NAME ______________________________________DATE __________PERIOD______
Skills Practice
Tessellations Identify the figures used to create each tessellation. Then identify the tessellation as regular, semi–regular, or neither. 1. 2.
trapezoids; neither
rhombi; neither
3.
4.
equilateral triangles and right triangles; neither
regular hexagons; regular
5.
6.
squares and isosceles right triangles; neither
trapezoids and rectangles; neither
Create a tessellation using the given polygons. 7. right triangles
©
Glencoe/McGraw-Hill
8. equilateral triangles and rhombi
60
Geometry: Concepts and Applications
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Page 61
NAME
11–1
DATE
PERIOD
Skills Practice
Parts of a Circle Use A at the right to determine whether each statement is true or false. T is a radius of A. true 1. A S
2. R B is a chord of A. false 3. ZU 2(ZA)
R Z B
true
T
C A
4. SA SW
false
5. AT BX
false
Y U X W
6. S W is a diameter of A. true 7. S W is a chord of A. true 8. AT AZ true 9. A T is a chord of A. false 10. SU RX false 11. SA AU true 12. S Y is a chord of A. true 13. SC SA false 14. Z U is a chord of A. true 15. Z U is a radius of A. false 16. B U is a chord of A. false
Circle W has a radius of 15 units, and Z has a radius of 10 units. 17. If XY 7, find YZ. 3 18. If XY 7, find WX. 8 T
19. If XY 7, find TX. 23
Z
W X
Y
20. If XY 7, find WR. 28
©
Glencoe/McGraw-Hill
61
Geometry: Concepts and Applications
R
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Page 62
NAME
11–2
DATE
PERIOD
Skills Practice
Arcs and Central Angles
Find each measure in C if mACB 80, mAF 45, and AE and F D are diameters. 1. mACF
45
2. mAB 80
3. mFCE
135
4. mEF 135
B D
5. mABE 180
6. mBCE
100
7. mAFE 180
8. mDCE
45
10. mBCD
55
9. mDE 45 11. mBAE 260
A E
C F
12. mABF 315
In A, B D is a diameter, mBAE 85, and mCAD 120. Determine whether each statement is true or false. 13. mBAC 60 true 14. mCD mCAD true C
15. ABE is a central angle. false 16. mBAC mDAE false
B
17. mCED 220 false
A
18. mBCD 180 true
D
E
19. mCE 145 true 20. mDAE mDE true Q is the center of two circles with radii Q D and Q E. If mAQE 90 and mRE 115, find each measure. 21. mAE 90
22. mRQE
115
23. mAR 155
24. mRQA
155
25. mAER 205
26. mBSD 270
27. mDS 115
28. mBD 90
©
Glencoe/McGraw-Hill
R S
B Q
A
D E
62
Geometry: Concepts and Applications
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11–3
Page 63
NAME
DATE
PERIOD
Skills Practice
Arcs and Chords Complete each sentence. G 1. If SG RE, then S
E R
?
2. If ST ET, then SMT 3. If TM ⊥ R G, then RT 4. If ST TE, S T
T
Q
E T
?
R
B A
?
6. If RE SG, then RME
GMS
? ?
SG
8. If TM ⊥ RG , then ST
?
9. If TM ⊥ RG , then S Q
?
Q E
TE
10. If T M ⊥ RG and ST TE, then SQT 12. If SR AR, then R G ⊥
G
M
B
A
7. If RE S G, then RE
11. If S Q EQ , then T M ⊥
E
S
TG
?
5. If RG ⊥ AS , then S B
EMT
?
EQT
?
G R
?
A S
?
Use B, where B X ⊥W Y , to complete each sentence. 13. If BW 23, then BY
?
23
14. If WY 38, then WZ
?
19
15. If WZ 15, then WY
?
30
16. If BZ 6 and WZ 8, then WB 17. If WB 15 and BZ 9, then WZ
Z
12
?
X Y
18. If WY 40 and BZ 15, then WB
?
25
19. If BY 30 and BZ 18, then WY
?
48
20. If mWY 110, then mWX
55
©
Glencoe/McGraw-Hill
B
W
10
?
?
63
Geometry: Concepts and Applications
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9/16/08
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Page 64
NAME
DATE
PERIOD
Skills Practice
Inscribed Polygons Use O to find x. 1. WM x 8, ZN 2x 5
3
2. WM 3x 10, ZN 2x 15 3. WM x 6, ZN 2x 12
WZ
5
P
6
O
Q
M N
4. WM 2x 5, ZN x 5
10
5. WM 5x 1, ZN 2x 5
2
6. WM 4x 15, ZN 3x 19 7. WM x 8, ZN 2x 5 8. WM 4x, ZN 3x 1
4
3
1
9. WM x 1, ZN 2x 7
6
10. WM 20x 100, ZN 30x 80 2
Square SQUR is inscribed in C with a radius of 20 meters. 11. Find mSCQ. 90
S
A
Q
12. Find SQ to the nearest tenth. 28.3 m C
13. Find CA to the nearest tenth. 14.2 m
©
Glencoe/McGraw-Hill
64
R
U
Geometry: Concepts and Applications
Geometry_0-07-869312-8
9/16/08
4:08 PM
11–5
Page 65
NAME
DATE
PERIOD
Skills Practice
Circumference of a Circle Find the circumference of each object to the nearest tenth. 1. a round swimming pool with radius 12 feet
75.4 ft
2. a circular top of a trampoline with diameter 16 feet
3. the circular base of a paper weight with diameter 3 centimeters
50.3 ft
4. a CD with diameter 11 centimeters
34.6 cm
9.4 cm
5. circular garden with radius 10 feet
6. circular mirror with diameter 4 feet
62.8 ft
12.6 ft
Find the circumference of each circle to the nearest tenth. 7. r 7 cm
8. d 20 yd
9. r 1 m
10. d 6 ft
11. r 200 ft
12. d 5 in.
13. r 2 m
14. d 70 ft
15. r 3 in.
16. d 10 in.
17. r 19 m
18. d 35 yd
44.0 cm
18.8 ft
12.6 m
31.4 in.
62.8 yd
1256.6 ft
219.9 ft
119.4 m
6.3 m
15.7 in.
18.8 in.
110.0 yd
Find the radius of each circle to the nearest tenth for each circumference given. 19. 100 m
15.9 m 22. 28 cm
4.5 cm 25. 75 yd
11.9 yd ©
Glencoe/McGraw-Hill
20. 32 ft
21. 18 mi
5.1 ft
2.9 mi
23. 80 in.
24. 25 m
12.7 in. 26. 14 cm
4.0 m 27. 250 ft
2.2 cm 65
39.8 ft Geometry: Concepts and Applications
Geometry_0-07-869312-8
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Page 66
NAME
11–6
DATE
PERIOD
Skills Practice
Area of a Circle Find the area of each circle to the nearest hundredth. 1. r 10 in.
2. r 18 cm
2
314.16 in
1017.88 cm
4. d 50 ft
5. d 6 in.
1963.50 ft2
2
78.56 yd
1385.51 m
10. r 1 mi
11. d 90 m
3.14 mi2
6361.73 m2
2
70,685.83 ft
14. r 6 in.
50.27 mm2
706.86 m2
8. C 131.95 m
2
3. r 4 mm
6. d 30 m
28.27 in2
7. C 31.42 yd
13. d 300 ft
2
2
113.10 in
9. C 232.48 ft
4300.92 ft2
12. C 628.32 ft
31,416.07 ft2
15. C 150.80 m
1809.64 m2
A circle has a radius of 10 inches. Find the area of a sector whose central angle has the following measure. Round to the nearest hundredth. 16. 90°
17. 30°
78.54 in2
19. 45°
20. 60° 2
39.27 in
22. 100°
87.27 in2
©
18. 120°
26.18 in2
Glencoe/McGraw-Hill
104.72 in2
21. 135° 2
52.36 in
23. 150°
117.81 in2
24. 70°
130.90 in2
66
61.09 in2
Geometry: Concepts and Applications
Geometry_0-07-869312-8
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Page 67
NAME
12–1
DATE
PERIOD
Skills Practice
Solid Figures Name the faces, edges, and vertices of each polyhedron. 1.
2.
C
A
J
G
B
I
F
K
E
O
H M
N D
3.
L S
Faces: rectangles ABED, BCFE, ACFD, ACB, DEF Edges: C , DF , AD , BE , CF , A B , BC , DE , EF A Vertices:A, B, C, D, E, F
P
T R
Faces: OSR, ORP, OTP, OTS;
Faces: rectangles GHLK, HIML, JIMN, GJNK, JIHG, NMLK Edges: G H , HL , KL , K G , GJ , KN , JN , JI, IM , NM , HI, LM Vertices: G, H, K, L, J, I, M, N
rectangle STPR Edges: S O , RO , PO , O T , SR , RP , PT , TS Vertices: O, T, P, R, S
Name each solid. 4.
5.
cylinder
6.
cube or rectangular prism
triangular prism
Determine whether each statement is true or false for the solid. 7. The figure is a prism.
false
D
8. The base of the figure is a pentagon. true
I H
E
9. There are five lateral faces. true G
10. The figure has 6 edges. false
©
Glencoe/McGraw-Hill
67
F
Geometry: Concepts and Applications
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Page 68
NAME
12–2
DATE
PERIOD
Skills Practice
Surface Areas of Prisms and Cylinders Find the lateral area and the surface area for each solid. Round to the nearest hundredth, if necessary. 1.
2.
3. 10 ft
4 cm 8 ft 3m
30 ft
4 cm
5m
4m
4m
100.53 cm2; 201.06 cm2
760 ft2; 1240 ft2
4.
48 m2; 60 m2
5.
6. 3 in. 2 yd
18 in. 8 in. 2 yd 8 in.
2 yd
904.78 in2; 1306.90 in2 7.
16 yd2; 24 yd2
84 in2; 180 in2
8.
20 cm 12 cm
6 in.
16 cm 15 cm
9. 5 ft
15 ft
11 ft 3 ft 3 ft
720
cm2;
912
cm2
10.
471.24
ft2;
1884.96
11. 28 m
ft2
84 ft2; 150 ft2 12.
16 m 18 cm
12 cm
33 mm
24 cm
18 mm
1407.43 m2; 1809.56 m2 ©
Glencoe/McGraw-Hill
864 cm2; 1296 cm2 68
1866.11 mm2; 2375.04 mm2 Geometry: Concepts and Applications
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Page 69
NAME
12–3
DATE
PERIOD
Skills Practice
Volumes of Prisms and Cylinders Find the volume of each solid. Round to the nearest hundredth, if necessary. 1.
2.
3. 10 ft
20 in. 10 ft 3 in. 13 in.
5 in.
10 ft 4 in.
10,618.58 in3
1000 ft3
4.
30 in3
5.
6.
1 ft
42 ft 6 ft
4 ft 22 cm 7 ft 6 cm
4750.09 ft3
4 cm
1584 cm3
7.
28 ft3
8.
9. 6m
17 in. 20 in.
26 in.
3m 8 in.
9025.80 in3
6000 in3
10.
36 m3
11. 7 ft
4m
15 in.
12.
2 ft
32 ft
5m
20 ft 11 m
32 ft
3m
21.99 ft3 ©
Glencoe/McGraw-Hill
165 m3 69
20,480 ft3 Geometry: Concepts and Applications
Geometry_0-07-869312-8
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9/16/08
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Page 70
NAME
DATE
PERIOD
Skills Practice
Surface Areas of Pyramids and Cones Find the lateral area and the surface area of each regular pyramid or cone. Round to the nearest hundredth, if necessary. 1.
2.
3. 23 m
7 ft 26 cm 3 ft
4m
10 m
11 cm
42 ft2; 51 ft2 4.
898.50 cm2; 1278.63 cm2
575 m2; 675 m2
5.
2 ft
6.
180 in.
54 in.
7 ft
60 cm
32 cm
43.98 ft2; 56.55 ft2 7.
3840 cm2; 4864 cm2
30,536.28 in2; 36,697.16 in2
8.
9. 75 cm 11 in.
125 cm
28 ft 8 in. 7 ft
8 ft
672 ft2; 840 ft2 10.
276.46 in2; 477.52 in2 11.
18,750 cm2; 24,375 cm2 12. 4 km
10 mi
18 cm
9 cm
5 km
4 mi
62.83 mi2; 75.40 mi2 ©
Glencoe/McGraw-Hill
324 cm2; 405 cm2 70
31.42 km2; 43.98 km2 Geometry: Concepts and Applications
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NAME
12–5
DATE
PERIOD
Skills Practice
Volumes of Pyramids and Cones Find the volume of each solid. Round to the nearest hundredth, if necessary. 1.
2.
3.
3 in.
40 ft 20 cm
3 in.
4 in.
15 cm
6 in3
30 ft
4712.39 cm3
4.
9 mm
12,000 ft3
5.
6.
1 mi
8 yd
25 mm
7 mi
2 yd
6 yd
2120.58 mm3 7.
80 yd3
7.33 mi3
8.
9.
24 in. 21 ft
8 ft
15 in.
30 cm 14 cm
22 in.
2640 in3 10.
66 in. 36 in.
24 cm
1407.43 ft3
1680 cm3
11.
12.
2 cm
6 ft
4 ft
3 cm
3 ft
41,054.33 in3
©
Glencoe/McGraw-Hill
24 ft3
3.14 ft3
71
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12–6
DATE
PERIOD
Skills Practice
Spheres Find the surface area and volume of each sphere. Round to the nearest hundredth. 1.
2. 9m
1017.88 m2; 3053.63 m3 4.
2827.43 in2; 14,137.17 in3 5.
12.57 mi2; 4.19 mi3 7.
70 km
11,309.73 ft2; 113,097.34 ft3 Glencoe/McGraw-Hill
50.27 in2; 33.51 in3 9. 100 mi
28 ft
2463.01 ft2; 11,494.04 ft3 11.
60 ft
2 in.
61,575.22 km2; 1,436,755.04 km3
10 mm
10.
5026.55 ft2; 33,510.32 ft3 6.
8.
1256.64 mm2; 4188.79 mm3
40 ft
15 in.
2 mi
©
3.
31,415.93 mi2; 523,598.78 mi3 12.
6 cm
452.39 cm2; 904.78 cm3 72
400 m
502,654.82 m2; 33,510,321.64 m3 Geometry: Concepts and Applications
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Page 73
NAME
12–7
DATE
PERIOD
Skills Practice
Similarity of Solid Figures Determine whether each pair of solids is similar. 1.
2.
8m 6m 24 m 18 m
10 in. 15 in.
yes
yes
3.
4.
5 cm
4 cm
10 in. 5 in.
8 cm
10 in.
20 in.
13 cm
yes
no
5
6. 45 m
30 m
6 mm 3 mm
40 m
12 mm
60 m
20 m 30 m
5 mm
yes
no
Find the scale factor of the solid on the left to the solid on the right for each pair of similar solids. Then find the ratios of the surface areas and the volumes. 7.
8. 32 m
16 m
7 ft
2 4 8 ; ; 1 1 1 ©
Glencoe/McGraw-Hill
8 ft
7 49 343 ; ; 8 64 512 73
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Page 74
NAME
DATE
PERIOD
Skills Practice
Simplifying Square Roots Simplify each expression. 1. 9
3
5. 256 16
9. 44 211
13. 98 72
17. 250
510
2. 169 13
3. 400 20
4. 225 15
6. 900 30
7. 289 17
8. 2500
10. 18 32
11. 75 53
14. 90 310
15. 125
18. 12 5
21. What is the square root of 625?
55
215 19. 6 6 6
50
12. 300 103
16. 80
45
20. 10 2
25
25
22. Multiply 17 3 . 51
23. Write 3 15 in simplest form. 35
©
Glencoe/McGraw-Hill
74
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NAME
13–2
DATE
PERIOD
Skills Practice
45°-45°-90° Triangles Find the missing measures. Write all radicals in simplest form. 1.
2.
3. 2 45°
45°
16 45°
y
9
x
y
y
45°
x 45°
45°
x
x 9, y 92 4.
x 2, y 22 5.
6.
1 45°
32
x 45°
x 16, y 162
50
45°
y
45°
y x
x
45°
y 45°
x 2 , y 1 7.
x 322 , y 32 8.
9. 45°
10√2 x 45° 45° y
x
x
45°
y
14√2 45°
x 10, y 10
10.
x 502 , y 50
45°
y
x 14, y 14
11.
x
√2
45°
x 1, y 1
12.
7√2 45°
x y
y
y 45°
50√2
x 50, y 50
©
Glencoe/McGraw-Hill
x
45°
45° 5√2
x 72 , y 14
75
45°
x 5, y 5
Geometry: Concepts and Applications
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13–3
DATE
PERIOD
Skills Practice
30°-60°-90° Triangles Find the missing measures. Write all radicals in simplest form. 1.
2.
3.
9 60°
30°
30°
x
x
y
y
30°
60°
60° 21
5
x 10, y 53 4.
x 42, y 213 5.
x 17
6.
29
y 30°
x
x
y
y
60°
x 18, y 93
60°
30°
y
x
60°
30
30°
x 173 , y 34 7.
x 293 , y 58
x 15, y 153
8.
9. 60°
y 30° 42
y 60°
44
x 26
30°
y
60°
x 30°
x
x 21, y 213 10.
x 22, y 223 11.
y 30° 10√3
12. 19√3
30°
60° x
x 10, y 20 13.
x 133 , y 13
x 60° y 30° 12
y
y 60° 14√3 30° x
60° x
x 19, y 38
x 28, y 14
14.
15. 8√3 60° 33
30° y
x 30° y
60° x
x 43 , y 83 ©
Glencoe/McGraw-Hill
x 113 , y 223 76
x 163 , y 24
Geometry: Concepts and Applications
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NAME
13–4
DATE
PERIOD
Skills Practice
Tangent Ratio Find each tangent. Round to four decimal places, if necessary. X
A 15
E
50
16
14
Z
C
20
1.3333
1. tan A
R
34
25
4. tan C 0.75
Y
30
T
S
48
2. tan Y
0.5333
3. tan S
0.2917
5. tan X
1.875
6. tan R 3.4286
Find each missing measure. Round to the nearest tenth. 7.
8. x cm 26° 30 cm
20° 19 in.
x ft
14.6
10.
9.
13 ft 50°
15.5
6.9
11.
xm
12. 30 ft
48 m
x°
55°
68.6
13.
x°
90 m
60 ft
x° 30 m
26.6
5 yd
71.6
14.
15. 11 ft 78°
5m
5 yd
x°
x ft
45 ©
Glencoe/McGraw-Hill
x in.
12 m
51.8
22.6 77
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NAME
13–5
DATE
PERIOD
Skills Practice
Sine and Cosine Ratios Find each sine or cosine. Round to four decimal places, if necessary. D G 25
A
B
28
11 29
F
C
20
20
14
E
K
40
H
36
1. sin B 0.3793
2. cos C 0.3793
3. cos B 0.8621
4. sin D 0.7143
5. sin F 0.7143
6. cos G 0.3500
Find each missing measure. Round to the nearest tenth. 7.
8.
34 ft
x ft
9.
40 m
x°
26°
28 cm
42 m
x° 15 cm
14.9
17.8
10.
11. 5 cm
57.6 74 m
12. 78 in.
x°
78 m
84 in.
x°
52°
x cm
3.1
71.6
13.
68.2
14.
19 mm
15. 7 ft
86 km 23 mm
3 ft
x°
x°
x° 70 km
55.7
©
Glencoe/McGraw-Hill
35.5
25.4
78
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NAME
14–1
DATE
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Skills Practice
Inscribed Angles Determine whether each angle is an inscribed angle. Name the intercepted arc for the angle.
no; XY
1. XMY
2. RST yes; RT
3. MCN no; MN
R
M
M X A
C
T
B
N
S
Y
Find each measure.
5. mDE 84
55
4. mEDA
35
7. mBDE 90
8. mZYW
35
9. mVZ 90
10. mXYV
25
11. mWY 80
6. mBDA
12. mXY 70
13. mXZY
X
D
50°
E
V
C
P
40°
42° M
45°
110°
B 70°
A
Z
Exercises 4–7
35
70°
W
Exercises 8–13
Find the value of x in each circle. 14.
B
A
15.
16.
N (2x 4)°
E
(x 10)°
M (6x 1)°
(7x 8)°
D
M (9x 4)°
R
P
Q
P
N
C
O (8x 2)°
9
28 x°
17.
18.
6 (3x 5)°
R
19.
B
X
O
V
A
E (x 5)°
80°
Y
X
160 ©
(6x 2)°
(8x 4)°
Z W
Glencoe/McGraw-Hill
C R
D
20
3 79
Y
Geometry: Concepts and
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NAME
14–2
DATE
PERIOD
Skills Practice
Tangents to a Circle Find each measure. Round to the nearest tenth if necessary. Assume segments that appear to be tangent are tangent. 1. BC
2. DF
3. SR Q 50 ft
D 11 cm
3 in.
A
R
P
E
B G
S
5 in. 13 cm
C
F
4 in.
24 cm
4. mBAC
50 ft
5. mARC
6. EG
A 48°
X
B
C
28 cm
E
A 65°
35 cm
G C
R
66
25
7. mZAY A
21
8. FE
10°
9. MR A
B 16 cm
7 cm
X
X
F
Z
M C 20 in.
D Y
30 cm
S
N
E
O R
80
14 cm
10. BC
P
15 in.
35 in.
11. HG
12. mXZY H
X
A 16 m
D
Y
B 30 m
4 mi
C G
Z
I J
18 m ©
Glencoe/McGraw-Hill
4 mi
45 80
Geometry: Concepts and Applications
F
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14–3
DATE
PERIOD
Skills Practice
Secant Angles Find each measure. 1. mEFB
2. mABC
E
3. mLPM
A
70° B
L
80°
P
A
F
36°
O
K
B
C
P
71°
D 46° C
22
4. mXYZ
78 6. mWZ
5. mCED
X A
28° B
Y 44°
X
E
60°
T
M
N
58
120°
85°
Y
P
C
G
40°
24°
D
Z
W
Z
30
26
7. mFE
8. mSQ 90°
C
36 9. mAE C
R
D
S
95°
A
M
24°
Q
W
36°
T
B 70° D
G 50°
H
E
A
F E
10
47
142
Find the value of x in each circle. Then find the given measure. 10. mBC
11. mHG A
E
12. mJL
46° F
I J
110°
76°
E D
C
10; 70 ©
B
9x °
Glencoe/McGraw-Hill
K
7x °
7x °
H
7; 52
G
M
(4x 2)°
K
Q L
(6x 2)°
(7x 3)°
5; 32 81
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DATE
PERIOD
Skills Practice
Secant-Tangent Angles Find the measure of each angle. Assume segments that appear to be tangent are tangent. 1. 1 2. 2 3. ABC 110°
2 70°
20°
1
B
A 230°
C
25
50
4. 3
5. 4
125 6. XYZ 4
X 70°
Z Y
124° 3
120° 160°
56
45
7. 5
8. 6
120 9. CDE
5
6
150°
C
46°
D
E
100°
30
27
10. 7
90
11. TUY
12. 8 T
118°
112°
90°
7
U
8
Y
62
©
Glencoe/McGraw-Hill
135
22
82
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14–5
DATE
PERIOD
Skills Practice
Segment Measures Find the value of x in each circle. If necessary, round to the nearest tenth. 1.
2. 6
3
6
9
3.
15
9
x 12
x
7
18
x
14
13.5
4.
11
10
5.
4
6.
6
5
12
18
10
2
x
x x
7
4
7. 12
26
8.
x
9.
6
7 6
8
9
18
24
x
x
4
12
32
Find each measure. If necessary, round to the nearst tenth. 10. DE
11. IH C
12. WX X
F
W
12 10
B
24
D
4
G
Y 5
8 13
A
20 ©
I E
Glencoe/McGraw-Hill
16
Z
H
6 83
Geometry: Concepts and Applications
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NAME
14–6
DATE
PERIOD
Skills Practice
Equations of Circles Write an equation of a circle for each center and radius or diameter measure given. 1. (1, 1), r 2
(x
1)2
2. (3, 2), d 2
(y
1)2
4
(x 3)2 (y 2)2 1
3. (1, 5), r 3
(x
1)2
4. (4, 3), d 4
(y
5)2
9
(x 4)2 (y 3)2 4
5. (0, 2), r 4
x2
(y
6. (5, 0), r 1
2)2
16
(x 5)2 y 2 1
7. (0, 0), r 6
x2
y2
8. (1, 1), d 6
36
(x 1)2 (y 1)2 9
9. (5, 5), r 5
(x
5)2
10. (3, 3), d 20
(y
5)2
25
(x 3)2 (y 3)2 100
11. (6, 1), d 10
(x
6)2
(y
12. (4, 4), d 14
1)2
25
(x 4)2 (y 4)2 49
13. (3, 7), d 22
(x
3)2
14. (5, 2), r 6
(y
7)2
2
(x 5)2 (y 2)2 6
15. (0, 2), r 10
x2
(y
2)2
16. (7, 0), d 25
10
(x 7)2 y 2 5
Find the coordinates of the center and the measure of the radius for each circle whose equation is given. 17. (x 5)2 (y 2)2 49
18. (x 3)2 (y 7)2 100
(3, 7), 10
(5, 2), 7 19. (x 1)2 (y 8)2 121
20. x2 y2 64
(1, 8), 11
(0, 0), 8
21. x2 (y 9)2 81
22. (x 3)2 y2 25
(0, 9), 9
(3, 0), 5
23. (x 6)2 (y 6)2 36
24. x2 y2 5
(6, 6), 6
(0, 0), 5
25. x2 (y 4)2 7
26. (x 1)2 (y 1)2 10
(0, 4), 7 ©
Glencoe/McGraw-Hill
(1, 1), 10 84
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Page 85
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DATE
PERIOD
Skills Practice
Logic and Truth Tables For Exercises 1–16, use conditionals a, b, c and d. a: A triangle has three sides. b: January is a day of the week. c: 5 5 20
d: Parallel lines do not intersect.
Write the statements for each negation. 1. a
A triangle does not have three sides. 2. b
January is not a day of the week. 3. c
5 5 20
4. d
Parallel lines intersect. Write a statement for each conjunction or disjunction. Then find the truth value. 5. a b A triangle has three sides or January is a day of the week; true. 6. a b A triangle has three sides and January is a day of the week; false. 7. a c A triangle has three sides or 5 5 20; true. 8. a c A triangle has three sides and 5 5 20; false. 9. a d A triangle has three sides or parallel lines do not intersect; true. 10. a d A triangle has three sides and parallel lines do not intersect; true. 11. b c January is a day of the week or 5 5 20; false. 12. b c January is a day of the week and 5 5 20; false. 13. b d January is a day of the week or parallel lines do not intersect; true. 14. b d January is a day of the week and parallel lines do not intersect; false. 15. c d 5 5 20 or parallel lines do not intersect; true. 16. c d 5 5 20 and parallel lines do not intersect; false.
©
Glencoe/McGraw-Hill
85
Geometry: Concepts and Applications
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DATE
PERIOD
Skills Practice
Deductive Reasoning Use the Law of Detachment to determine a conclusion that follows from statements (1) and (2). If a valid conclusion does not follow, write no valid conclusion. 1. (1) If a figure is a triangle, then it is a polygon. (2) The figure is a triangle.
The figure is a polygon. 2. (1) If I sell my skis, then I will not be able to go skiing. (2) I did not sell my skis.
no valid conclusion 3. (1) If two angles are complementary, then the sum of their measures is 90. (2) Angle A and B are complementary.
The sum of the measures of angles A and B is 90. 4. (1) If the measures of the lengths of two sides of a triangle are equal, then the triangle is isosceles. (2) Triangle ABC has two sides with lengths of equal measure.
Triangle ABC is isosceles. 5. (1) If it rains, we will not go on a picnic. (2) We do not go on a picnic.
no valid conclusion Use the Law of Syllogism to determine a conclusion that follows from statements (1) and (2). If a valid conclusion does not follow, write no valid conclusion. 6. (1) If my dog does not bark all night, I will give him a treat. (2) If I give my dog a treat, then he will wag his tail.
If my dog does not bark all night, then he will wag his tail. 7. (1) If a polygon has three sides, then the figure is a triangle. (2) If a figure is a triangle, then the sum of the measures of the interior angles is 180.
If a polygon has three sides, then the sum of the measures of the interior angles is 180. 8. (1) If the concert is postponed, then I will be out of town. (2) If the concert is postponed, then it will be held in the gym.
no valid conclusion 9. (1) All whole numbers are rational numbers. (2) All whole numbers are real numbers.
no valid conclusion 10. (1) If the temperature reaches 70°, then the swimming pool will open. (2) If the swimming pool opens, then we will not go to the beach.
If the temperature reaches 70°, then we will not go to the beach. ©
Glencoe/McGraw-Hill
86
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Page 87
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DATE
PERIOD
Skills Practice
Paragraph Proofs Write a paragraph proof for each conjecture.
A
1. If ABD is an isosceles triangle with base BD and C is the midpoint of B D , then ACD ACB. D
B
C
If ABD is an isosceles triangle with base B D , then AD A B . If C is the , then CD C B . AC A C by the Reflexive Property, so midpoint of BD ACD ACB by SSS. 2. If lines a and b are parallel and W X X Y , then WXS YXZ.
S
W
a
X
b
Z
Y
If lines a and b are parallel, then SWX XYZ since they are alternate interior angles. WXS YXZ since they are vertical angles. Then it is X X Y , so WXS YXZ by ASA. given that W 3. If ACDE is an isosceles trapezoid with bases and E D , then AED CDE. AC
A
C B
E
D
If ACDE is an isosceles trapezoid with bases AC and E D , then the legs C D . Also, an isosceles trapezoid has congruent are congruent, so AE D E D by the Reflexive Property, base angles, so AED CDE. Now, E so AED CDE by SAS. 4. If RSTX is a rhombus, then RXT RST.
S
R
X
T
If RSTX is a rhombus, then RS RX and XT ST. RT RT by the Reflexive Property, so RXT RST by SSS. ©
Glencoe/McGraw-Hill
87
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DATE
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Skills Practice
Preparing for Two-Column Proofs Complete each proof. 1. If m1 m2 and m3 m4, then mABC mROD. Given: m1 m2, m3 m4 Prove: mABC mROD Proof: Statements a. m1 m2, m3 m4 b. mABC m1 m3 mROD m2 m4 c. m1 m3 m2 m4 d. mABC mROD 2. If
7x 6
D R
O A 1 3
Reasons
B
C
a. Given b. Angle Addition Postulate c. Addition Property of Equality d. Substitution Property of Equality
14, then x 12.
Given:
7x 6
14
Prove: x 12 Proof: Statements
Reasons
7x a. 14
a. Given
b. 7x 84 c. x 12
b. Multiplication Property of Equality c. Division Property of Equality
6
3. If ABC is a right triangle with C a right angle and mA mB, then mA 45. Given: ABC is a right triangle with C a right angle and mA mB. Prove: mA 45 Proof: Statements Reasons a. ABC is a right triangle with C a right angle and mA mB b. mA mB mC 180 c. mC 90 d. mA mB 90 e. mA mA 90 f. 2mA 90 g. mA 45 ©
2 4
Glencoe/McGraw-Hill
A
C
B
a. Given b. c. d. e. f. g. 88
Angle Sum Theorem Definition of Right Angle Subtraction Property of Equality Substitution Property of Equality Substitution Property of Equality Division Property of Equality Geometry: Concepts and Applications
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Page 89
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DATE
PERIOD
Skills Practice
Two-Column Proofs Write a two-column proof. 1. Given: ITC is an isosceles triangle with base IC , TS bisects ITC C S Prove: IS Proof: Statements Sample Answer: Reasons
I S C
T
a. ITC is an isosceles triangle
a. Given
b. TS bisects ITC
b. Given
c. IT C T
c. Definition of isosceles triangle
d. ITS CTS
d. Definition of angle bisector
e. S T S T
e. Reflexive Property
f. ITS CTS
f. SAS
g. IS C S
g. CPCTC
with base IC .
A
B
D
C
2. Given: ABCD is a square. Prove: AC B D Proof: Statements Sample Answer:
Reasons
a. ABCD is a square.
a. Given
b. A D BC
b. Definition of a square
c. ADC and BCD are right
c. Definition of a square
d. ADC BCD
d. Definition of congruent angles
e. D C DC
e. Reflexive Property
f. ΔADC ΔBCD
f. SAS
g. A C BD
g. CPCTC
angles.
©
Glencoe/McGraw-Hill
89
Geometry: Concepts and Applications
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NAME
15–6
DATE
PERIOD
Skills Practice
Coordinate Proofs Position and label each figure on a coordinate plane. 1–5. Sample answers given. 1. a square with sides m units long y
C (m, m)
D (0, m)
O
B (m, 0) x A (0, 0)
2. a right triangle with legs p and r units long y Z (0, r )
Y (p, 0)
x
O X (0, 0)
3. a rhombus with sides c units long y S (a, d)
R (c a, d)
O P (0, 0) Q (c, 0) x
4. an isosceles triangle with base d units long and heights s units long y C (0, s)
A ( 12d, 0)
B (12 d, 0)
x
O
5. a rectangle with length x units and width y units y D ( x2, y)
C ( x2 , y)
A ( x2, 0)
B ( x2, 0)
O ©
Glencoe/McGraw-Hill
x
90
Geometry: Concepts and Applications
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Page 91
NAME
16–1
DATE
PERIOD
Skills Practice
Solving Systems of Equations by Graphing Solve each system of equations by graphing. 1. y x 2. y 2x 4 y x 1 (1, 2) y x 2 (1, 1) y
y
y
y x 2
yx1
(1, 2) y x 1
(1, 1)
x
O
(2, 3) y x 5
y 2x 4
yx
4. y x 5 y x 1 (3, 2)
5. y x 4 y x 2 (3, 1)
y
6. y x y 3x 4 (2, 2)
y
y x 1 O
x
O
x
O
3. y x 1 y x 5 (2, 3)
y
yx4
y 3x 4
x
(3, 2)
(3, 1) O
y x 2
yx5
7. y 2x 1 y 3x 2 (1, 1)
x
8. y x 6 y 2x 6 (4, 2)
y
y 2x 1
(1, 1)
9. y x 1 y x 5 (3, 2) y
y 2x 6 O
y x (2, 2)
y
y 3x 2
x
10. y x 5 y 3x 1 (1, 4)
O
(4, 2) yx6
y x 5 (3, 2)
x O
yx1
x
11. y x 1 12. y x 1 y 2x 4 (3, 2) y 2x 5 (4, 3)
y
y
y
yx5
x
O
y 2x 5
y 2x 4
(1, 4)
y 3x 1 O
©
Glencoe/McGraw-Hill
x
yx1 O
(3, 2)
91
x
O
x
y x 1 (4, 3) Geometry: Concepts and Applications
Geometry_0-07-869312-8
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NAME
DATE
PERIOD
Skills Practice
Solving Systems of Equations by Using Algebra Use substitution to solve each system of equations. 1. y 7 2. y 3 x 2y 5 xy8
(7, 1)
(5, 3)
3. y x 2x y 15
(5, 5)
4. y x 7 2x 3y 19
5. y x 2 3x 2y 4
6. y x 1 4x 3y 4
7. 3x y 2 2x 2y 8
8. 2x y 0 3x y 4
9. 4x y 13 2x 3y 11
(8, 1)
(1, 5)
(0, 2)
(4, 8)
(1, 0)
(2, 5)
Use elimination to solve each system of equations. 10. x y 7 11. x y 4 xy1 2x y 13
12. x y 3 3x y 5
13. x y 5 2x y 4
15. x y 2 3x y 4
(4, 3)
(3, 2)
©
Glencoe/McGraw-Hill
(3, 7)
14. x y 2 2x y 1
(3, 5)
92
(2, 1)
(1, 1)
Geometry: Concepts and Applications
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Page 93
NAME
16–3
DATE
PERIOD
Skills Practice
Translations Find the coordinates of the vertices of each figure after the given translation. Then graph the translation image. 1. (3, 4) 2. (2, 3) 3. (1, 5) y C'
y
Q'
B'
U'
A'
C
E' y
D'
Q F' E
D' O B
O
x
x U
A'
x
O
D
D
A
A F
A´(0, 2), B´(2, 3), C´(1, 5) 4. (3, 2)
Q´(1, 4), U´(2, 2), A´(2, 0), D´(1, 1) 5. (3, 1)
6. (4, 0) y
R y
T
D´(2, 4), E´(0, 5), F´(1, 1)
T'
R'
B y
G
G'
A P' O
A'
A
A
P A'
B'
H
x
x
O H'
x
O
A' D'
D C
T´(0, 2), R´(2, 2), A´(2, 1), P´(0, 0) 7. (0, 3)
G´(1, 3), H´(0, 0), A´(3, 1)
A´(0, 2), B´(3, 4), C´(3, 4), D´(1, 3)
8. (2, 4)
9. (5, 0) yR
y
M
C'
X' y
S
X
N R' P
T
M' O
S'
OV
N' x
Z'
x
Y' Z O
Y x
P' T' V'
M´(1, 1), N´(3, 0), P´(3, 2) ©
Glencoe/McGraw-Hill
R´(0, 1), S´(2, 0), T´(1, 3), V´(2, 4) 93
X´(0, 4), Y´(0, 1), Z´(4, 1)
Geometry: Concepts and Applications
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NAME
16–4
DATE
PERIOD
Skills Practice
Reflections Find the coordinates of the vertices of each figure after a reflection over the given axis. Then graph the reflection image. 1. x-axis 2. y-axis 3. x-axis y
y
X'
B
y
X
E
D
A C C'
O
A'
x
F
Y x
Z' Z O
Y'
G G' D'
B'
E'
A´(2, 3), B´(3, 4), C´(2, 1) 4. y-axis
D´(2, 3), E´(2, 4), F´(4, 2), G´(3, 1)
X´(3, 4), Y´(4, 1), Z´(1, 1) 5. x-axis
y
J'
x F'
O
6. y-axis y
J
y Q
I
I'
R
R' R P x
O
x
O
X' R'
M
M'
Q'
T' T
K
K'
J´(4, 4), K´(4, 4), M´(2, 2), I´(2, 2) 7. x-axis X
Y x Y'
C
O
D'
E' E O
Glencoe/McGraw-Hill
O
x
X'
B´(–1, 4), C´(–4, 3), D´(–3, 1), E´(0, 2)
94
B
A
A'
W´(–1, –3), X´(3, –4), Y´(4, –1), Z´(–2, 0)
y
D
W'
©
R´(2, 2), X´(4, 1), T´(1, 3) 9. x-axis
B' y B
C'
W
Z Z'
P´(2, 1), Q´(1, 3), R´(4, 2) 8. y-axis
y
x X
O
P'
C C'
x
B'
D´(3, –4), E´(2, 0), F´(–2, –3)
Geometry: Concepts and Applications
Geometry_0-07-869312-8
9/16/08
4:08 PM
Page 95
NAME
16–5
DATE
PERIOD
Skills Practice
Rotations Find the coordinates of the vertices of each figure after the given rotation about the origin. Then graph the rotation image. 1. 90˚ counterclockwise 2. 90˚ clockwise 3. 180˚ counterclockwise y
B'
y
y
R
D
R'
B A' A O
S S' O
x
C C' O
x
x
D'
A´(1, 1), B´(2, 4)
4. 180˚ clockwise
5. 90˚ counterclockwise
6. 90˚ clockwise
y
y
F'
C´(1, 1), D´(4, 3)
S´(1, 1), R´(4, 3)
y
J
H' G H
G' E' O
OI' I
x
E
K
M O M'
x
x J'
F
K'
E´(2, 1), F´(3, 4)
7. 180˚ clockwise
G´(2, 1), H´(1, 3), I´(0, 0)
8. 90˚ counterclockwise
y
J´(4, 2), K´(1, 4), M´(0, 0)
9. 90˚ clockwise y
y
Q N S P
Q' OQ
R O R'
x
T' T O
x
x
X'
P' Q'
X
S'
N´(2, 3), P´(4, 1), Q´(0, 0)
©
Glencoe/McGraw-Hill
W
W'
N'
Q´(4, 2), R´(0, 0), S´(2, 4)
95
T´(0, 0), W´(3, 3), X´(4, 1)
Geometry: Concepts and Applications
Geometry_0-07-869312-8
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Page 96
NAME
16–6
DATE
PERIOD
Skills Practice
Dilations Find the coordinates of the dilation image for the given scale factor k, and graph the dilation image. 1.
1 2
2. 2
3.
y
y
1 3 y
Z'
D
B C
B'
Z Y'
A A'
D'
Y
C'
x
O
O
A´(1, 1), B´(2, 3)
x
Y´(2, 4), Z´(6, 8)
4. 4
6. y
C'
x
C´(1, 2), D´(2, 3)
5. 2
A' y
O
1 4 y
A
T' X
T A'
C
A
I' E' x
E
A´(0, 8), C´(8, 8) E´(8, 0), G´(0, 0)
7.
R O R'
I
G G' O
X' x
T´(2, 6), R´(0, 0), I´(6, 2)
1 2
8.
2 3
A´(2, 2), X´(0, 1), E´(1, 0)
9. 2
y
y
y A F M'
S
A'
C M
T'
S'
R'
x
O F'
E' O x
E
R C'
x
O T
F´(1, 1), M´(3, 2), T´(2, 3) ©
Glencoe/McGraw-Hill
B' O B x
A´(0, 6), B´(0, 0), C´(2, 4) 96
T T'
R´(2, 4), S´(2, 6), T´(4, 2) Geometry: Concepts and Applications
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