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Answers

Copyright © Big Ideas Learning, LLC Geometry All rights reserved. Answers

A35

91. a. $7.80

b. $9.70

c. 7 lb

92. 2 3y x= − 93. 3 27y x= − −

94. 1 15 2

5y x= + 95. 15

4y x= +

96. 14 2y x= − + 97. 8 93y x= +

98. 4 29y x= − + 99. 2 15 5

y x= − +

100. 13

11y x= − 101. 12

3y x= − +

102. 19 10y x= − 103. 2 2y x= − +

104. 3 22y x= + 105. 7y = −

106. 13

3y x= − + 107. 46x =

108. 136x = 109. 28x =

110. 19x = 111. 35x = 112. 21x =

Chapter 4 4.1 Start Thinking

Translate the original triangle 2 units down; Each ordered pair for A B C′ ′ ′Δ contains y-coordinates that are two less than those of ;ABCΔ When identifying a translation, you can compare the x- and y-values to determine what happens if the figure is plotted.

4.1 Warm Up

1. 2.

( 2, 2)P′ − (0, 1)P′

3. 4.

(4, 0)P′ ( 4, 5)P′ −

5. 6.

(6, 0)P′ (4, 4)P′

4.1 Cumulative Review Warm Up

1.

1 2

p q

∠ ≅ ∠Given

Prove

x

y4

2

−2

−4

42−4 −2

P

P′

x

y4

2

−2

−4

42−4 −2

P

P′

x

y4

2

−2

−4

42−4 −2

P

P′

x

y

4

2

−2

42−4 −2

PP′

x

y4

2

−2

−4

4 6−2 PP′

x

y4

2

−2

−4

42 6−2

P

P′

1

2

3q

p

t

STATEMENTS REASONS

1. p q 1. Given

2. 1 3∠ ≅ ∠ 2. Corresponding Angles Theorem (Thm. 3.1)

3. 3 2∠ ≅ ∠ 3. Vertical Angles Congruence Theorem (Thm. 2.6)

4. 1 2∠ ≅ ∠ 4. Transitive Property of Angle Congruence (Thm. 2.2)

x

y4

2

−2

42−4

A

C

B

B′C′

A′

Answers

Geometry Copyright © Big Ideas Learning, LLC Answers All rights reserved. A36

2.

1 2

p q

∠ ≅ ∠Given

Prove

4.1 Practice A

1. ; 3, 2JK −

2. (3, 1), (0, 2), (1, 3)A B C′ ′ ′− − −

3. 4, 6−

4. ( , ) ( 6, 4)x y x y→ + +

5. (6, 6)Q′ 6. ( 2, 11)M ′ − −

7. 8.

9. no; Sample answer: The translation from A to A′ could be different than the translation from A′ to

.A′′

10. , ) ( 2, 3);( y x yx → − − 0.25 mi

4.1 Practice B

1. ( 4, 1), (1, 7), ( 1, 3)F G H′ ′ ′− −

2. 11, 17−

3. , ) ( 5, 6)( y x yx → + −

4. ( 6, 7)G′ − 5. ( 14, 8)H ′ −

6.

7. no; Multiplying x by 2 does not simply move or translate the object, it stretches the shape.

8. ( , ) ( 3, 1); nox y x y→ + −

9. Sample answer:

yes; All the vectors are parallel. This makes sense because the vertices are all translated by the same vector, so the segments joining the vertices to their images all have the same slope. Because they have the same slope, they are parallel.

1

2

3

q

p

t

STATEMENTS REASONS

1. p q 1. Given

2. 1 3∠ ≅ ∠ 2. Corresponding Angles Theorem (Thm. 3.1)

3. 3 2∠ ≅ ∠ 3. Vertical Angles Congruence Theorem (Thm. 2.6)

4. 1 2∠ ≅ ∠ 4. Transitive Property of Angle Congruence (Thm. 2.2)

x

y4

−4

2−4 −2

A

C

B

B′

C′

A′

x

y4

−4

−2

2 4−4 −2

E

C

DD′

C′

E′

x

y

8

−8

−4

4 8 12−4

EC

D

D′

C′

C″

D″

E″

E′

x

y8

−8

84−4−8

G

FH

F′ H′

G′

x

y

4

2

−2

4−2

A

BC

A′

B′C′

6

6

L″

K″

x

y

−8

−4

−4

K

J

LL′

J′

K′

12

−12

J″

Answers


A37

10. 3, 12, (7, 1)a b P′= − =

4.1 Enrichment and Extension

1. a. 34

b. 3

c. 13

2. ,a c b d+ +

3. a. 4, 1−

b. 6, 9−

c. 8, 7−

d. 2, 2− −

4. x-y plane 5. x-z plane 6. y-z plane

7. x-y and y-z planes

8. a. 17 b. 38

4.1 Puzzle Time

AN UMBRELLA

4.2 Start Thinking

yes; Distances are preserved relative to the surface of reflection; no; It appears as though your image in the reflection is holding up its left hand because your image is being reflected as is, so it appears to be backward.

4.2 Warm Up

1.

(5, 3)P′

2. 3.

(4, 3)P′ − (1, 5)P′ −

4. 5.

( 1, 1)P′ − − (4, 6)P′ −

6.

(5, 1)P′ −


1. acute 2. right 3. obtuse

4. obtuse 5. acute 6. straight

4.2 Practice A

1. 2.

3. 4.

5. 6.

x

y4

2

−2

−4

42−4 −2

P P′

x

y4

2

−2

−4

42−4 −2

P P′

x

y

2

−2

−4

42−4 −2

P P′

x

y8

4

−4

−8

84−8 −4

P

P′

x

y4

2

−2

−4

42−4 −2

P

P′

x

y4

−4

42−2−4

A

C

B

B′

C′

A′

x

y

4

−8

8−8A

BB′

A′

CC′

x

y8

4

−8

8−4−8 A

B

B′

A′

−12

C

C′

x

y

3

1

−3

−5

31−1−5

J′

J″

J

K′K″

K

L′

L″

L

x

y

4

−8

8−8

S

P

S′P′

Q′

Q

R

R′

x

y8

−8

8

R

R′

S′S

P

P′

Q′

Q

x

y4

2

−2

−4

42−2

P

P′

Answers


7.

8. x-axis, y-axis, any line joining opposite outer points or opposite inner points of the star

9. ,x y= y-axis, any line joining an outer point to its

opposite inner point of the star

10. (0, 1) 11. 1a =

12. no; Two translations can always be written as a single translation, so the process is the same as a glide reflection.

4.2 Practice B

1.

2.

3. 4.

5. 6.

7. x-axis, y-axis, ,y x= and y x= −

8. (1, 0)P

9. yes; Sample answer: You can reflect it twice in the same line.

10. no; Angle measures and distances remain constant under any rigid motion.

11. yes; Two reflections do not combine into another reflection, so the motion is not a glide reflection.


1. ( , ) and ( , )a c a b− −

rectangle

2. ( , ) and ( , )a c a b− −

3.

isosceles trapezoid

4.

isosceles right triangle

x

y8

−4

4−8

J′J″

K′K″

KL″ J

L′

L

x

y4

2

4−4

CC′

DD′

EE′

x

y8

4

−8

−4

8

C C′

D

E E′

16

D′

x

y8

−8

−4NN′M

M′

L

L′

K

K′x

y8

4

−4

N

N′

L

L′

K

K′

MM′

x

y

2

6

−2

4−2

A″

AB″

BB′

C″

CC′

6

A′

x

y

2

6

−2

4−2

A″A

B″

BB′

C″

CC′

A′

x

y2

−4

2−2

F(a, c)F′(−a, c)

G(a, b)G′(−a, b)

x

y

2

−4

−2

2−2−4

B

B′

A

A′

x

y

2

4

−2

2−2−4

B

A

A′

Answers


A39

5. 6.

4.2 Puzzle Time

A SQUARE DANCE

4.3 Start Thinking

O; H, I, N, O, S, X, and Z; no

4.3 Warm Up

1. 2.

( 2, 4)P′ − (0, 3)P′

3. 4.

( 6, 0)P′ − ( 2, 6)P′ − −

5. 6.

(0, 2)P′ (0, 4)P′ −


1. Reflexive Property of Segment Congruence (Thm. 2.1)

2. Symmetric Property of Angle Congruence (Thm. 2.2)

4.3 Practice A

1. 2.

3.

4.

5. yes; 45 , 90 , 135 , 180° ° ° ° 6. yes; 30 , 120° °

7.

square

8. H, I, N, O, S, X, Z; H: 180 ; I: 180 ; N: 180 ;° ° °

O: all angles; S: 180 ; X: 90 , 180 ; Z: 180° ° ° °

4.3 Practice B

1.

x

y

−2

2−2−4

A′ A″

B′B″

C′ C″

x

y

4

6

8

2

−2−4−6

A′

A″B′

B″ C′

C″

x

y4

2

−2

−4

42−4 −2

P

P′

x

y4

2

−2

−4

42−4 −2

P

P′

x

y8

4

−4

−8

84−8 −4

PP′x

y8

4

−4

−8

84−8 −4

P

P′

x

y4

2

−2

−4

42−4 −2P

P′

x

y4

2

−2

−4

42−4 −2

P

P′

A

A′

B

B′

C

C′

D

D′

Px

y4

−4

4−4

J

J′L

L′

K

K′

MM′

x

y

4

−2

4

R″

RS″

S

S′

T″

T

T′

R′

x

y4

−4

42−2−4

A(2, 0)

B(0, 2)

(−2, 0)

(0, −2)

x

y

−4

4

T T′

U U′ V

V′W

W′

x

y

4

2−4

R″

R

S″

SS′

T″

TT′

R′

Answers


2.

3.

4. yes; 90 , 180 , 270° ° ° 5. yes; 180°

6. no; A 360° rotation just takes the shape back to the original, so it must be the same.

7. yes; A rotation of 180° maps the figure onto itself; 6 times

8. no 9. yes

10. 0, 8; 0:180 ; 8: 180 ;° ° Some students may also

include the number 1, depending on how it is written.

4.3 Enrichment Extension

1. 2 4

3 3y x= − + 2. 8y x= − +

3. 2

33

y x= + 4. 1

52

y x= − +

5. a. 0 0 0( , )x y a x+ −

b. 0 0 0 0( , )x b y a x y− + − +

c. (2, 7)−

d. (9, 12)

6. a.

b.

4.3 Puzzle Time

WHAT A SHAME WE WILL NEVER MEET

4.4 Start Thinking

Answers should include two or more shapes from each of the categories listed (circle, square, triangle, and rectangle) and should include a table for each category. An example of one category is shown.

Sample answer: Circle: penny, jar lid

Circle

Both objects have one continuous curve (360 ),° but no

angles; Both have the same shape; The objects are not the same size.

4.4 Warm Up

1.

triangle

2.

parallelogram

x

y4

−4

4 6 82

C″

C

C′

D″

D

D′

E″

EE′

x

y

4

6

2

42−2−4

F″

F′

G″G′

Angles Shape Size

Penny no angles circle smaller

Jar lid no angles circle larger

x

y4

−4

−2

42−2−4

A

B

C

x

y4

2

−4

42−4

E

GH

F

x

y

4

2

−2

4−2

F″

F′

G″

G′

6

x

y4

4

C″

D″

D

D′

E″

E

E′

C

C′

Answers


A41

3.

square

4.

trapezoid


1. (6, 2.2)P − 2. (5.4, 1)P −

3. (1.25, 2.75)P 4. (1, 1.25)P −

4.4 Practice A

1. Triangle 1, triangle 5, and triangle 8 are all congruent to one another; Square 6 is congruent to square 2; Parallelogram 4 is congruent to parallelogram 7.

2. Triangle 1, triangle 2, and triangle 9 are all congruent to one another; Rectangle 3 is congruent to rectangle 10; Triangle 5 is congruent to triangle 8.

3. translation 8 units right and 2 units up

4. reflection in the x-axis, followed by a translation 2 units right

5. congruent; ABCΔ can be mapped on to STUΔ by a reflection in the y-axis, followed by a translation 1 unit right and 7 units down.

6. not congruent; It is not possible to transform polygon EFGH to polygon WXYZ using only rigid transformations.

7. , ,CD CD DE D E′′ ′′ ′′ and ;CE C E′′ ′′

EE DD CC′′ ′′ ′′≅ ≅

8. 14° 9. 45°

4.4 Practice B

1. Squares 1, 3, and 7 are congruent; Triangles 6 and 8 are congruent; Parallelograms 4 and 5 are congruent; Each can be translated to one of the others by a sequence of rigid motions.

2. not congruent; Although three of the four points retain their relative positions, the fourth point changes the shape, so the quadrilaterals are not congruent.

3. 8 units 4. 160°

5. no; The angle between lines a and b is not 90 .°

6. 180°

7. sometimes; It depends on the shape of the object undergoing the transformation.

8.

9. yes; It will produce a translation of the same image after any even number of reflections. This is not true for odd numbers of reflections, which will result in a reflection of the original image.


1. 7

1

−

2. 1 5 7 1

3 3 1 1

− −

3. 6 1

9 10

− −

4. 5 15 5

0 11 4

−

5. 1 5 3

1 0 1

−

6. 1 1 1

3 3 3

− − −

7. 1 1 1 1 5 3 0 4 2

3 3 3 1 0 1 4 3 2

− − − + = −

8.

The coordinates of triangle A B C′ ′ ′ are the same as the image matrix in Exercise 7.

x

y4

2

−4

−2

42−2−4

JM

L K

A″

A′A

st

x

y

2

−2

62

B

C

C′A

B′A′

x

y2

−4

−6

−2

4 62−2Q

R

P

S

Answers


4.4 Puzzle Time

A POLYGON

4.5 Start Thinking

yes; The ratio of the distance of the flashlight from the wall and the diameter of the circle remains constant. As the flashlight moves closer to the wall, the circle gets smaller.

4.5 Warm Up

1. 8 units 2. 4 units


1. 2.

AB CD≅ AB ≅ CD

3. 4.

AB ≅ CD AB CD≅

4.5 Practice A

1. 3; enlargement 2. 2.5; reduction

3. 4.

5. 6.

7.

8. 0.77

9. the old film-style camera

10. no; Every dimension would dilate by the same scale factor k, so the area would increase by 2 ,k one factor of k for each dimension.

11. no; A scale factor of 1 does not dilate the object at all. The object is neither enlarged nor reduced.

4.5 Practice B

1. 16; reduction 2. 1.5; enlargement

3.

4. 5.

6.

7. It would look like it is 80 millimeters across.

8. A dilation with a scale factor of 0k = would send all the vertices to the center of the dilation, so the object would be reduced to a point.

x

y8

−8

−4

8−4−8

B

C

D

A

x

y8

4

−8

−4

4−4−8 B

C D

A

x

y

8

4

−4

84−4

B D

C

A x

y8

4

−8

84−4−8

B

CD

A

x

y

B,

CP

D

AA′

D′ C′

B′

x

y

B

C

P

D

AA′

D′ C′

B′

x

y

B

C,

P

D

A

A′

D′ C′

B′

x

y

4

−8

8−4

Q

R

PP′ Q′

R′S′

S

P B,

C

D

A

A′

B′

C′

D′

P B

C

D

AA′

B′

C′D′

x

y4

2

−4

−2

2

M

M′ LL′

KK′

J

J′

x

y

−12

8 12−4

X

X′

Y

Y′

Z

Z′

WV

V′

W′

x

y8

−8

84−4−8

AB

DA′

B′

C′

D′

C

Answers


A43

9. yes; The perimeter is additive, so it is scaled by the same factor by which the object is dilated.

10. The scale factor is x; 2x =


1.

2. The length and width double; The perimeter doubles; The area increases by a factor of 4.

3. The perimeter is 4 times as large, and the area is 16 times as large.

4. The perimeter increases by a factor of a, and the area increases by a factor of 2.a

5. ( , ) (2 , 2 )x y x y→ 6. 223.04 mm

4.5 Puzzle Time

THE OUTSIDE

4.6 Start Thinking

Sample answer:

square ABCD: (1, 1), (1, 1), ( 1 1), ( 1, 1),A B C D− − − −

square EFGH: (3, 3), (3, 3), ( 3 3), ( 3, 3);E F G H− − − −

To find square EFGH, use a scale factor of 3, with the origin as the center of the dilation.

4.6 Warm Up

1. 9.5n = 2. 18w =

3. 4.4x = 4. 123.5y =

5. 12c = 6. 34n =


1. inductive reasoning; A pattern is used to reach the conclusion.

2. deductive reasoning; Facts about numbers and the laws of logic are used to reach the conclusion.

3. deductive reasoning; The laws of logic are used to reach the conclusion.

4.6 Practice A

1. 2.

3. reflection in the y-axis, followed by a dilation with a scale factor of 2

4. yes; The triangle is a translation; ( , ) ( 5, 1)x y x y→ + − followed by a dilation of

( )2 23 3

( , ) , ;x y x y→ Points C and F do not follow

these transformations, so it is not a similarity transformation.

5. yes; The quadrilateral can first be rotated 180° about the origin (or, reflected in the y-axis and then the x-axis). Then the figure can be dilated with a scale factor of 0.5k = and translated to its final position.

6. Rotate PQRΔ so that side a is parallel to side b.

Translate GHIΔ so that point G maps to point P. Because translations preserve angle measure, and all of the angles of an equilateral triangle are 60 ,°

GHIΔ lies on .PQRΔ Because, GI

coincides with

PR

and GH

coincides with ,PQ GI

lies on PR

and GH lies on .PQ Finally, dilate PQRΔ about

point P by a scale factor of b

a so that it is the same

size as .GHIΔ Because a similarity transformation maps PQRΔ onto ,GHIΔ the triangles are similar.

x

y

8

4

84 1612

Length Width Perimeter Area

1. Points A, B, C, and D

2 4 12 8

2. Points ,A B′ ′ , and C D′ ′

4 8 24 32

3. Points D, E, F, and G

8 16 48 128

x

y4

2

−4

−2

42−2−4

E

FG

H

D A

BC

x

y

4

2−2−4−6

R

Q

P

R′R″

Q″

P″

Q′P′

x

y

−8

−4

8−4R

Q

P″

P

R′ R″

Q″P′

Q′

Answers


7. no; A square and a rectangle are not similar, so you cannot use a similarity transformation to change the shape of the object.

8. no; For example begin with a unit square centered at the origin. If you perform a dilation centered at the origin with a scale factor 2 and then translate 1 unit right, the result is not the same as if you first translate the square 1 unit right and then perform a dilation centered at the origin with a scale factor of 2.

9. All white triangles are dilations and translations. There are no rotations in the image.

4.6 Practice B

1.

2.

3. a 180° rotation followed by a dilation with a scale factor of 3

4. similar; The transformation is a translation 6 units to the right and 3 units up, followed by a dilation with a scale factor 2.

5. not similar; The transformation is a reflection in the y-axis, followed by a translation, however two vertices were translated 2 units up and two vertices were translated 3 units up.

6. Rotate onto ABE DBCΔ Δ such that ABE∠ coincides with .DBC∠ Because rotations preserve

length and measure, ABE DBC∠ ≅ ∠ and AE is

still parallel to .DC So, AB coincides with BD

and BE coincides with .BC Dilate ABEΔ until

AE coincides with .CD Therefore, ABEΔ is similar to .DBCΔ

7. no; Circles of different size are simply a dilation of each other, so they remain similar.

8. no; The edges are distorted and curved, and are not an exact replica of the original text. So, a magnifying glass does not produce a perfect similarity transformation because the image is distorted.

9. no; Similar triangles do not need to be the same size, so there are more similar triangles than there are congruent triangles.


1. 15 square units 2. 4

3. 4 square units 4. 64 square units

5. The area has increased by a factor of 24 16.=

6. a. 1 23 33, ,k a b= = = −

b. Because k is 3, the radius of Circle A is 3 times the radius of Circle B, so 3 .t r=

4.6 Puzzle Time

THE TEACHER TOLD THEM NOT TO USE TABLES

Cumulative Review

1. 5 4x − 2. 7 11x + 3. 4 15x +

4. 9 99x− − 5. 4 8x − 6. 6 36x− +

7. 3x 8. 9 6x − 9. 3 6x− +

10. 9x− + 11. 5 88x − 12. 20 74x −

13. 32x = 14. 10x = − 15. 12x = −

16. 23x = 17. 30x = 18. 8x = −

19. 96x = − 20. 8x = 21. 3x =

22. 22x = − 23. 4x = − 24. 3x = −

25. 20x = − 26. 2x = 27. 8x = −

28. 12x = −

29. a. $9.35

b. $10.65

30. a. 48 fluid ounces

b. 6 cups

c. 3 pints

d. 1.5 quarts

x

y4

2

−4

−2

4−4

E

E′

D

D′

D″

C

C′

C″

E″

x

y8

4

−8

−4

8 12EE′

D″C″

E″

D

D′

C

C′

Answers


A45

31. yes 32. no 33. yes 34. no

35. a. 1.5 ft

b. $1.60 per ft

c. $2.40

36. a. 12 in. by 18 in.

b. 2216 in.

c. $6.48

d. $13.52

37. 80° 38. 80° 39. 80° 40. 100°

41. 50° 42. 20° 43. 70° 44. 180°

45. 150° 46. 100° 47. 271.5 ft 48. 2363 yd

49. 2717.5 mm 50. 240 cm 51. 254 in.

52. 2735 mi 53. 3101.25 in. 54. 32132230 in.

55. a. $3.75 b. $0.08

56. 3 24y x= − − 57. 4 7y x= −

58. 12 17y x= + 59. 5 11y x= − −

60. 14 11y x= − 61. 2

3 9y x= − −

62. 12 2y x= + 63. 3

8 2y x= −

64. 57 7y x= + 65. 1

4 13y x= − −

66. 2y x= − + 67. 13 3y x= −

68. 3 5y x= − 69. 12 7y x= − +

70. 12 4y x= − + 71. 4 8y x= −

72. 13y x= − 73. 3 20y x= − −

74. 14 3y x= + 75. 2 12y x= − −

76. 4 19y x= − 77. 15 2y x= +

78. 5y = − 79. 12 5y x= − −

80. (5, 0), (2, 3), (4, 7)A B C′ ′ ′ −

81. (3, 6), (0, 9), (2, 1)A B C′ ′ ′ −

82. (1, 4), ( 2, 7), (0, 3)A B C′ ′ ′− −

83. (7, 2), (4, 5), (6, 5)A B C′ ′ ′ −

84. (2, 3), ( 1, 6), (1, 4)A B C′ ′ ′− −

85. (4, 2), (1, 1), (3, 9)A B C′ ′ ′− −

86. (5, 8), (2, 11), (4, 1)A B C′ ′ ′

87. ( 2, 7), ( 5, 10), ( 3, 0)A B C′ ′ ′− − −

Chapter 5 5.1 Start Thinking

If 120 ,m A∠ = ° then 60m B∠ = ° because together

they make a straight angle. If 40 ,m D∠ = ° then

140m E∠ = ° under the same reasoning. If the sum of

, ,m B m C∠ ∠ and m D∠ is 180°, then 80 .m C∠ = °

5.1 Warm Up

1. 1 31m∠ = ° 2. 2 59m∠ = °

3. 3 59m∠ = ° 4. 4 90m∠ = °


1. 69x = 2. 6x =

3. 60x = 4. 30x =

5.1 Practice A

1. scalene; right 2. isosceles; acute

3. scalene; not a right triangle

4. isosceles; right triangle

5. 32x = 6. 29x = 7. 30x = 8. 17x =

9. 19.5 , 70.5° °

10. no; An exterior angle will be acute when it is adjacent to the obtuse angle of an obtuse triangle.

11. 145°

5.1 Practice B

1. scalene; obtuse 2. isosceles; acute

3. 14x = 4. 9x =

5. 1169

x = 6. 7; 19.5x y= =

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