Solutions Manual

[Pages:586]Solutions Manual

This booklet is provided in Glencoe Geometry Answer Key Maker (0-07-860264-5). Also provided are solutions for problems in the Prerequisite Skills, Extra Practice, and Mixed Problem Solving sections.

Copyright ? by The McGraw-Hill Companies, Inc. All rights reserved. Printed in the United States of America. Permission is granted to reproduce the material contained herein on the condition that such material be reproduced only for classroom use; be provided to students, teacher, and families without charge; and be used solely in conjunction with Glencoe Geometry. Any other reproduction, for use or sale, is prohibited without prior written permission of the publisher.

Send all inquiries to: Glencoe/McGraw-Hill 8787 Orion Place Columbus, OH 43240-4027

ISBN: 0-07-860204-1 1 2 3 4 5 6 7 8 9 10 009 12 11 10 09 08 07 06 05 04 03

Geometry Solutions Manual

CONTENTS

Chapter 1 Points, Lines, Planes, and Angles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Chapter 2 Reasoning and Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 24 Chapter 3 Parallel and Perpendicular Lines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 Chapter 4 Congruent Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 Chapter 5 Relationships in Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .120 Chapter 6 Proportions and Similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .151 Chapter 7 Right Triangles and Trigonometry. . . . . . . . . . . . . . . . . . . . . . . . . . .. 189 Chapter 8 Quadrilaterals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 Chapter 9 Transformations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 Chapter 10 Circles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 Chapter 11 Areas of Polygons and Circles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383 Chapter 12 Surface Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 411 Chapter 13 Volume..................................................................447 Prerequisite Skills ..................................................................477 Extra Practice........................................................................502 Mixed Problem Solving and Proof................................................567

Chapter 1 Points, Lines, Planes, and Angles

Page 5 Getting Started

1?4.

y

D( 1, 2) O

B(4, 0) x

2. See students' work; sample answer: Two lines intersect at a point.

3. Micha; the points must be noncollinear to determine a plane.

4. Sample answers: line p; plane R

5. Sample answer:

y

A(3, 2) C( 4, 4)

5. 34 38 68 38 98 118

6. 2 156 5 18 ?3176 481 ?3176 8126 ?11169 ?7 176

7. 78 196 ?1146 196 ?156

8. 1112 9 176 ?223 11561 ?11864 11561 ?3136 2 116

9. 2 17 15

10. 23 (14)? 23 14 37

11. [7 (2)]2? (7 2)2 (5)2 25

12. 92 132? 81 169 250

13. P? 4s 4(5) 20

The perimeter is 20 in. 14. P? 2 2w

? 2(6) 2 212

? 12 5 17 The perimeter is 17 ft. 15. P? 2 2w

2(4.8) 2(7.5) 9.6 15 24.6 The perimeter is 24.6 m.

1-1 Points, Lines, and Planes

W x

O

X Z

Y

6.

Q

P

r

s

7. There are six planes: plane ABC, plane AGE,

plane CDE, plane BCD, plane FAB, and plane DEF.

8. A, K, B or B, J, C

9. No; A, C, and J lie in plane ABC, but D does not.

10. line

11. point

12. plane

Pages 9?11 Practice and Apply

13. n

14. F

15. R

16. W

17. Sample answer: PR

18. Yes, it intersects both m and n when all three

lines are extended.

19. (D, 9)

20. Charlotte

21.

B

W

Q

A

22. R

W T 23. Sample answer:

y

Page 8 Geometry Activity

1. no

2. no

3. On CD; see students' work.

4. See students' work.

R

S

Z

O

x

Q

Page 9 Check for Understanding

1. point, line, plane

1

Chapter 1

24. Sample answer: y

C

R

P D

SO

x

25.

a

b

c

26.

a

F

b

c

27.

s

C

r

D

M

36. Sample answer: points E, A, and B are coplanar,

but points E, A, B, and C are not.

37. AC

38. point

39. lines

40. plane

41. plane

42. two planes intersecting in a line

43. point

44. intersecting lines

45. point

46. line

47.

48.

V

49. See students' work. 50. Sample answer: the image is rotated so that the

front or back plane is not angled. 51. Sample answer:

28.

s

A

BC

29. points that seem collinear; sample answer: (0, 2), (1, 3), (2, 4), (3, 5)

y

O

x

30. There are five planes: plane ABC, plane BCE,

plane ABE, plane ADE, and plane CDE.

31. 1; There is exactly one plane through any three noncollinear points.

32. E, F, C

33. Because A and B determine a line, add point G anywhere on AB.

34. E, F

35. A, B, C, D or E, F, C, B

52. See picture.

53. vertical

54. Sample answer: the paths flown by airplanes flying in formation

55. Sample answer: Chairs wobble because all four legs do not touch the floor at the same time. Answers should include the following. ? The ends of the legs represent points. If all points lie in the same plane, the chair will not wobble. ? Because it only takes three points to determine a plane, a chair with three legs will never wobble.

56. C; Three lines intersect in a maximum of three points. The fourth line can cross the other three lines only one time each, adding three points to the figure. For example,

Chapter 1

2

57. B; 2 x 2 x 2x22x2 x x x x x x 2x 0 Thus, x must be 0.

58.

y

O

x

a line

59.

y

O

x

part of the coordinate plane above the line y 2x 1 60. 12 48, so 12 in. 38 in. 61. 14 146, so 146 in. 14 in. 62. 45 180, so 45 in. 160 in. 63. 10 mm 1 cm

64. 2.5 cm 25 mm, so 2.5 cm 28 mm

65. 0.025 cm 0.25 mm, so 0.025 cm 25 mm

Page 12 Reading Mathematics

1. Points P, Q, and R lie on . Point T is not collinear with P, Q, and R.

2. Planes F, G, and H intersect at line j.

3. The intersection of planes W, X, Y, and Z is point P.

4.

A

B

C

1-2 Linear Measure and Precision

Page 16 Check for Understanding

1. Align the 0 point on the ruler with the leftmost endpoint of the segment. Align the edge of the ruler along the segment. Note where the rightmost endpoint falls on the scale and read the closest eighth of an inch measurement.

2. Sample answers: rectangle, square, equilateral triangle

3. Each inch on the ruler is divided into eighths.

Point about

Q168isocrlo1s34e rintcohtehsel1on68g-i.nch

mark.

Thus,

PQ

is

4. The long marks on the ruler are centimeters, and

the shorter marks are millimeters. There are

10 millimeters for each centimeter. Thus, the bee

is 13 millimeters or 1.3 centimeters long.

5. The measurement is precise to within 0.5 meter.

So, a measurement of 14 meters could be 13.5 to

14.5 meters. 6. The measuring tool is divided into 14-inch

increments. Thus, the measurement is precise

timnoecwhasietushraiennmd12e3n14t14coour18ld18bien3c38bhei.tnwTcheheeenrse.3fo14re, th18e 318

7. EG EF FG EG 2.4 1.3 EG 3.7 So, EG is 3.7 centimeters long.

8.

XY YZ? XZ

XY 158? 3

XY 158 So, XY is

138X1iY58n??che31s38lon1g58.

9.

5x

3x

15

N

L

M

NL 5x? 15

55x ? 155 x? 3

LM 3x LM 3(3) LM 9

10.

30

6x 5

2x 3

N

L

M

NM? NL LM

30? 6x 5 2x 3

30? 8x 2

30 2? 8x 2 2

32? 8x 382 ? 88x

3

Chapter 1

4? x LM 2x 3 LM 2(4) 3 8 3 LM 11

11. BC CD because they both have length 10 inches. BE ED because they both have length 8 inches.

BA DA because they both have length 14.4 inches.

Pages 17?19 Practice and Apply

12. Each inch on the ruler is divided into sixteenths.

Point B is about

i1s1c56loisnecrhteostlhoeng1.156-inch

mark.

Thus,

AB

13. The long marks on the ruler are centimeters, and

the shorter marks are millimeters. Point D is

closer to the 45-millimeter mark. Thus, CD is

about 45 millimeters or 4.5 centimeters long.

14. The long marks on the ruler are centimeters, and the shorter marks are millimeters. The right end of the key is closer to the 33-millimeter mark. Thus, the key is about 33 millimeters or 3.3 centimeters long.

15. Each inch on the ruler is divided into sixteenths.

The right tip of the paperclip is closer to the 16. 1oTarh1m461ee-ngo.fth8uis0s,pitnrhecechiepsseapctooeurwclldiitphbeiins7a912b12oinutctoh18.10S46o12,

inches.

17. The measurement is precise to within 0.5 millimeter. So, a measurement of 22 millimeters could be 21.5 to 22.5 millimeters.

18. The measuring tool is divided into 12-inch increments. Thus, the measurement is precise to

wminiecthahseinusra12enmd12e1n6to12rco14u lid14ncbhe.1bT6eh34tewrineeefconhree1s,6.t12h e 14 1614

19. The measurement is precise to within 0.5 centimeter. So, a measurement of 308 centimeters could be between 307.5 and 308.5 centimeters.

20. The measurement is precise to within 0.005 meter or 5 millimeters. So, a measurement of 3.75 meters 3750 millimeters could be between 3745 and 3755 millimeters.

21. The measuring tool is divided into 14-foot increments. Thus, the measurement is precise to

within 12 14 or 18 foot. Therefore, the

measurement could be between 314 18 318 feet and 314 18 338 feet. 22. AC AB BC AC 16.7 12.8 29.5 So, AC is 29.5 millimeters long.

23. XZ XY YZ XZ 12 34 24 34

XZ 54

So, XZ is 54 or 114 inches long.

24.

PR? PQ QR

214? 156 QR

94 156 ? 156 QR 156

3166 156 ? QR

3116? QR

So, QR is 3116 or 11156 inches long.

25.

RT? RS ST

4.0? 1.2 ST

4.0 1.2? 1.2 ST 1.2

2.8? ST

So, ST is 2.8 centimeters long.

26. WY WX XY 4.8 WX WX 4.8 2WX

WX is congruent to XY.

42.8 2W2X 2.4 WX

So, WX is 2.4 centimeters long.

27. AD? AB BC CD

334? BC BC BC AB is congruent to BC and CD is congruent to BC.

334? 3BC

13 145 13(3BC)

1152? BC So, BC is 1132 or 114 inches long.

28.

7a

12a

28

R

S

T

RS 7a? 28 77a? 278 a? 4

ST 12a

ST 12(4)

ST 48

29.

34

12

2x

R

S

T

RT RS ST 34 12 2x 34 12 12 2x 12 22 2x

222 22x 11 x ST 2x ST 2(11) ST 22

Chapter 1

4

30.

25

2x

3x

R

S

T

RT? RS ST 25? 2x 3x 25? 5x 255? 55x

5? x ST? 3x ST? 3(5) ST? 15

31.

5x 10

16

2x

R

ST

RT? RS ST 5x 10? 16 2x 5x 10 10? 16 2x 10

5x? 6 2x 5x 2x? 6 2x 2x

3x? 6 33x 63

x? 2 ST? 2x ST? 2(2) ST? 4

32.

21

3y 1

2y

R

S

T

RT? RS ST 21? 3y 1 2y 21? 5y 1 21 1? 5y 1 1 20? 5y 250? 55y

4? y ST? 2y ST? 2(4) ST? 8

33.

5y

4y 1

2y 1

R

S

T

RT? RS ST

5y? 4y 1 2y 1

5y? 6y 2

5y 2? 6y 2 2

5y 2? 6y

5y 2 5y? 6y 5y

2? y

ST? 2y 1

ST? 2(2) 1

ST? 3

34. yes; AB CD 3 cm

35. no; EF 6 ft and FG 8 ft

36. no; NP 1.75 in. and LM 0.75 in.

37. yes; WX XY 6 m

38. not from the information given

39. yes; TR? 3(a b) 3c 3a 3b 3c SU? 3a 3(b c) 3a 3b 3c

40. The width of a music CD is 12 centimeters.

41. CF DG, AB HI, CE ED EF EG

42. 144 cm3; 343 mL could be actually as much as 343.5 mL and 200 mL as little as 199.5 mL; 343.5 199.5 144.

43. The lengths of the bars are given in tenths of millions, and 0.1 million 100,000. So the graph is precise to within 50,000 visitors.

44. 50,000 0.05 million, so a measurement of 98.5 million could be 98.45 million to 98.55 million visitors.

45. No; the number of visitors to Washington state parks could be as low as 46.35 million or as high as 46.45 million. The visitors to Illinois state parks could be as low as 44.45 million or as high as 44.55 million visitors. The difference in visitors could be as high as 2.0 million.

46. 12.5 cm; Each measurement is accurate within 0.5 cm, so the least perimeter is 2.5 cm 4.5 cm 5.5 cm.

47. 15.5 cm; Each measurement is accurate within 0.5 cm, so the greatest perimeter is 3.5 cm 5.5 cm 6.5 cm.

48. E

EF 4(CD)

F

49.

2(CD)

E

F

3(AB)

50a. 2

50b. 5

50c. 7

51. Sample answer: Units of measure are used to differentiate between size and distance, as well as for accuracy. Answers should include the following. ? When a measurement is stated, you do not know the precision of the instrument used to make the measure. Therefore, the actual measure could be greater or less than that stated. ? You can assume equal measures when segments are shown to be congruent.

52. allomweaabs leureerror 02.75fftt 0.019 or 1.9%

53. allomweaabs leureerror 01.42.55 iinn.. 0.017 or 1.7%

54. allomweaabs leureerror 04.20.53 ccmm 0.001 or 0.1%

55. allomweaabs leureerror 06.30.57 kkmm 0.0008 or 0.08%

56. B; 5(12 in.) 60 in. or 5 ft

5

Chapter 1

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