mscc7a ws 0700a - pleasantville high school

Chapter 7

Copyright © Big Ideas Learning, LLC Big Ideas Math Red Accelerated All rights reserved. Worked-Out Solutions

211

Chapter 7 Opener Try It Yourself (p. 269)

1. Align the center of the protractor with the angle’s vertex.

The angle measure is 70 .° So, the angle is acute.


The angle measure is 90 .° So, the angle is a right angle.


The angle measure is 115 .° So, the angle is obtuse.

4. Draw a ray. Place the center of the protractor on the endpoint of the ray and align the protractor so the ray passes through the 0° mark. Make a mark at 55 .° Then draw a ray from the endpoint at the center of the protractor through the mark at 55 .°




Section 7.1 7.1 Activity (pp. 270–271)

1. a. Sample answers:

b.

90° 180° 0°

2. a. Right: , ,AFE ABE∠ ∠ , , BEF BAF∠ ∠

, ,BED BCD∠ ∠ , CDE CBE∠ ∠

Acute: , , , CBD EBD BDE BDC∠ ∠ ∠ ∠

Obtuse: ABD∠

b. and ,ABE CBE∠ ∠ and ,FEB DEB∠ ∠

and ,EBD CBD∠ ∠ and ,EDB CDB∠ ∠

and ;ABD CBD∠ ∠ They share a common side

and a common vertex.

9090

8010070

11060120

50130

4014

0

3015

0

2016

0

10 170

0 180

10080

11070 12060 13050 14040 15030

1602017010

1800

70°

90 9080 10

070 11

060 12

050 13

0

40 140

30 150

20160

10170

0180

100

8011

070

120

6013

050

14040

15030

16020

17010

1800

90°

9090

8010070

11060120

50130

4014

0

3015

0

2016

0

10 170

0 180

10080

11070 12060 13050 14040 15030

1602017010

1800

115°

55°

160°

85°

180°

Acute Straight Right Obtuse

Chapter 7

Big Ideas Math Red Accelerated Copyright © Big Ideas Learning, LLC Worked-Out Solutions All rights reserved. 212

3. a. 4;

Sample answer:

b. Sample answer: 1 70 ,∠ = ° 2 110 ,∠ = ° 3 70 ,∠ = °

4 110 ;∠ = °

The measure of 1∠ is equal to the measure of 3.∠ The measure of 2∠ is equal to the measure of 4.∠

4. Four angles are formed with four pairs of adjacent angles. All four angle measures are equal when one of the angle measures is 90°. In the other cases, the nonadjacent angles have equal measures.

5. Sample answer:

7.1 On Your Own (pp. 272–273)

1. Sample answer:

adjacent: and ,XWY ZWY∠ ∠ and XWY XWV∠ ∠

vertical: and ,VWX YWZ∠ ∠ and YWX VWZ∠ ∠

2. Sample answer:

adjacent: and ,LJM LJK∠ ∠ and LJM NJM∠ ∠

vertical: and ,KJL PJN∠ ∠ and PJQ MJL∠ ∠

3. The angles are adjacent angles. The angles make up a straight angle, so the sum of their measures is 180 .°

85 180

95

x

x

+ ==

So, x is 95.

4. The angles are vertical angles. Because vertical angles are congruent, the angles have the same measure. So, x is 90.

5. The angles are adjacent angles. Because the angles make up a right angle, the sum of their measures is 90 .°

( )2 1 69 90

2 68 90

2 22

11

x

x

x

x

− + =

+ ===

So, x is 11.

6.

7.1 Exercises (pp. 274–275)

Vocabulary and Concept Check

1. 2 pairs; 4 pairs

2. 1 and 3;∠ ∠ 2 and 4;∠ ∠

Two angles are vertical angles when they are opposite angles formed by the intersection of two lines. Vertical angles are congruent angles.

Practice and Problem Solving

3.

4. and ABE CBD∠ ∠

5. Sample answer:

adjacent: and ;FGH HGJ∠ ∠ and HGJ JGK∠ ∠

vertical: and ;FGH JGK∠ ∠ and FGK JGH∠ ∠

6. Sample answer:

adjacent: and ;LMN NMP∠ ∠ and PMQ QMR∠ ∠

vertical: and ;LMN QMR∠ ∠ and SMR PMN∠ ∠

7. The angles listed are adjacent angles. A pair of vertical angles is and .ACB ECF∠ ∠


35 90

55

x

x

+ ==

So, x is 55.

9. The angles are vertical angles. Because vertical angles are congruent, the angles have the same measure. So, x is 128.

10. The angles are adjacent angles. Because the angles make up a straight angle, the sum of their measures is 180 .°

117 180

63

x

x

+ ==

So, x is 63.

Oak

mon

t Rd

832

832

1

3

4 2

30°40°

75°

75°

60°60°

A C

B

DE

120°

120°

Chapter 7


213

11. The angles are vertical angles. Because vertical angles are congruent, the angles have the same measure.

4 25 75

4 100

25

x

x

x

− ===

So, x is 25.


4 2 90

6 90

15

x x

x

x

+ ===

So, x is 15.

13. The angles are adjacent angles. Because the angles make up a straight angle, the sum of their measures is 180 .°

( )7 20 180

8 20 180

8 160

20

x x

x

x

x

+ + =

+ ===

So, x is 20.

14.

15.

16.

17.


2 41 127

2 86

43

x

x

x

+ ===

So, x is 43.

19. a. Sample answer:

b. Sample answer:

c. Sample answer:

20. Sample answer: One procedure is to draw one angle with its given measurement and then draw the second angle adjacent to it. A second procedure is to draw an angle whose measure is the sum of the two given angles. Then draw one of the angles inside that angle, sharing one of the sides. The other angle formed is the second given angle.

21. never; 1 and 3∠ ∠ are vertical angles and will always be congruent.

22. always; 1 and 4∠ ∠ are adjacent angles that form a straight angle, and the sum of their angle measures will always be 180 .°

23. sometimes; 2 and 3∠ ∠ are adjacent angles that form a straight angle, and the sum of their angle measures will always be 180 .° They will be congruent only when they each have an angle measure of 90°.

24. always; 1 and 2∠ ∠ are adjacent angles that form a straight angle, and the sum of their angle measures will always be 180 .° 3 and 4∠ ∠ are also adjacent angles that form a straight angle, and the sum of their angle measures will also always be 180 .°

25.

26. no; The ladder makes a 60° angle with the ground. Assuming that the angle between the ground and the wall is 90°, and knowing that the sum of the angles of a triangle add to 180°, the ladder makes a 30° angle with the wall.

25°25°

85°

85°

110°

110°

135°

135°

35°50°

115°40°

60°75°

13

24

Chapter 7


Fair Game Review

27. 6 54

9

n

n

− >< −

28. 1

172

34

x

x

− ≤

≥ −

29. 1.62.5

4

m

m

− <−

>

30. B; 2 1

2 1

8 3 5

6 2 4

y ym

x x

− −= = =− −


1. a.

b. When the sum of their angle measures is 90°. c.

d. When the sum of their angle measures is 180°. 2. a. If x and y are complementary angles, then both x and

y are always acute.

b. If x and y are supplementary angles, then x is sometimes acute.

c. If x is a right angle, then x is never acute.

d. If x and y are complementary angles, then x and y are sometimes adjacent.

e. If x and y are supplementary angles, then x and y are sometimes vertical.

3. a. supplementary; The angles are 60° and 120°. b. complementary; The angles are 45° and 45°.

c. neither; The angles are 180° and 90°.

d. supplementary; The angles are 90° and 90°.

4. a. Sample answers:

complementary: 1 and 2,∠ ∠ 3 and 4,∠ ∠

5 and 6,∠ ∠ and 7 and 8∠ ∠

supplementary: 9 and 10,∠ ∠ 10 and 11,∠ ∠

11 and 12,∠ ∠ and 9 and 12∠ ∠

b. 9 and 11,∠ ∠ 10 and 12∠ ∠

5. Two angles are complementary when the sum of their measures is 90 .° Two angles are supplementary when the sum of their measures is 180 .° Sample answer: 32° and 58° are complementary angles and 25° and 155° are supplementary angles.

7.2 On Your Own (pp. 278–279)

1. Because 26 64 90 ,° + ° = ° the angles are

complementary.


supplementary.

3. Because 70 19 89 ,° + ° = ° the angles are neither

complementary nor supplementary.

4. The two angles make up a straight angle. So, the angles are supplementary, and the sum of their measures is 180 .°

( )4 5 41 180

4 46 180

4 134

33.5

x

x

x

x

+ + =

+ ===

So, x is 33.5.

5. The angles make up a right angle. So, the angles are complementary angles, and the sum of their measures is 90°.

( )2 3 90

3 3 90

3 93

31

x x

x

x

x

+ − =

− ===

So, x is 31.

6.



1. Two angles are complementary when the sum of their measures is 90 .° Two angles are supplementary when the sum of their measures is 180 .°

−10−11−12 −8−9 −7 −6

−35−36−37 −33−34 −32 −31

3 541 2 6 7

x 10° 20° 25° 30° 45° 50° 75°

y 80° 70° 65° 60° 45° 40° 15°

x 20° 30° 60° 90° 130° 140° 150°

y 160° 150° 120° 90° 50° 40° 30°

15°

Chapter 7


215

2. Adjacent angles are not defined by their measure, so they can be supplementary, complementary, or neither.

Examples:

Supplementary:

Complementary:

Neither:


3. Because the sum of the angle measures of x and y is 180 ,° x can be acute, obtuse, or 90 .° So, the statement

is sometimes true.

4. Because the sum of the angle measures of x and y is 90 90 180 ,° + ° = ° the angles are supplementary.

So, the statement is always true.

5. Because the sum of the angle measures of x and y is 90 ,° the measure of angle y is less than 90 .° So,

the statement is never true.




complementary.


complementary.


supplementary.


supplementary.



12. The two angles make up a right angle. So, the angles are complementary angles, and the sum of their measures is 90°.

3 45 90

3 45

15

x

x

x

+ ===

So, x is 15.

13. The two angles make up a right angle. So, the angles are complementary angles, and the sum of their measures is 90°.

( )20 90

2 20 90

2 110

55

x x

x

x

x

+ − =

− ===

So, x is 55.


( )3 25 2 180

5 25 180

5 155

31

x x

x

x

x

+ + =

+ ===

So, x is 31.

15. Because the 50° angle and 1∠ are supplementary, the

sum of their measures is 180 .°

50 180

130

x

x

+ ==

So, the measure of 1∠ is 130 .°

Because the 50° angle and 2∠ are vertical angles, the

angles have the same measure. So, the measure of 2∠

is 50 .° Because 1∠ and 3∠ are vertical angles, the

angles have the same measure. So, the measure of 3∠

is 130 .°


( )2 21 180

3 21 180

3 159

53

x x

x

x

x

+ + =

+ ===

So, x is 53.

17.

18.

19.

60° 120°

30°60°

30°20°

20°

35°

80°

Chapter 7


20.

21. Sample answer: One procedure is to draw one angle, then draw the other using a side of the first angle. Another procedure is to draw a right angle, then draw one of the angles inside the right angle, sharing one of the sides.

22. Sample answer: 100° is the measure of an angle that can be supplementary, but not complementary. Because complementary angles are less than 90 ,° any angle

that is greater than 90° and less than 180° cannot be complementary. So, an obtuse angle can be a supplementary angle but cannot be a complementary angle.

23. a. The two angles make up a straight angle. So, the angles are supplementary, and the sum of their measures is 180 .°

Let be the measure of .x CBD∠

6.2 180

7.2 180

25

x x

x

x

+ ===

So, the measure of CBD∠ is 25°.

b. Angles and EBD CBD∠ ∠ make up a right angle. So, the angles are complementary, and the sum of their measures is 90 .°

Let be the measure of .y EBD∠

25 90

65

y

y

+ ==

So, the measure of EBD∠ is 65°. Because FBE∠ and EBD∠ are congruent, the measure of FBE∠ is 65°.

24. yes; Sample answer 1: LMQ∠ is a straight angle. By

removing ,NMP∠ the remaining two angles

( ) and LMN PMQ∠ ∠ have a sum of 90°.

Sample answer 2: SML∠ and LMN∠ are complementary angles. SML∠ and PMQ∠ are vertical

angles and have the same measure (congruent). So, LMN∠ and PMQ∠ are also complementary.

25. Let 3x represent the measure of the larger angle and 2x represent the measure of the smaller angle. Because the angles are complementary, the sum of their measures is 90 .°

3 2 90

5 90

18

x x

x

x

+ ===

So, the measure of the larger angle is ( )3 3 18 54 .x = = °

26. Because vertical angles are congruent, the angles have the same measure. So, let x represent the measure of one complementary angle.

90

2 90

45

x x

x

x

+ ===

So, the measures of two vertical angles that are complementary angles are 45 .°

Let y represent the measure of one supplementary angle.

180

2 180

90

y y

y

y

+ ===

So, the measures of two vertical angles that are supplementary angles are 90 .°

27. Because the angles with measures 7x° and 20° are supplementary to the right angle, the sum of all three measures is 180 .°

7 90 20 180

7 110 180

7 70

10

x

x

x

x

+ + =+ =

==

Because the angle with the sum of the angle measures 5x° and 2y° is a vertical angle to the right angle, the sum

of their measures is 90 .° Substitute 10 for x and solve for y.

( )5 2 90

5 10 2 90

50 2 90

2 40

20

x y

y

y

y

y

+ =

+ =

+ ===

So, 10x = and 20.y =

130°

30°

Chapter 7


217

Fair Game Review

28. 7 8

15

x

x

+ = −= −

The value of x is 15.−

Check: ?

7 8

15 7 8

8 8

x + = −

− + = −

− = −

29. 1 3

3 41 3

3 44 9

12 125

12

n

n

n

n

= +

− =

− =

− =

The value of n is 5

.12

−

Check:

?

?

?

1 3

3 41 5 3

3 12 41 5 9

3 12 121 4

3 121 1

3 3

n= +

= − +

= − +

=

=

30. 12.7 3.4

9.3

y

y

− = −− =

The value of y is 9.3.−

Check: ?

12.7 3.4

12.7 9.3 3.4

12.7 12.7

y− = −

− = − −

− = −

31. B; 3.7% 003.7% 0.037= =


1. a.

b.

c. not possible;

d. not possible;


b. Sample answer:

c. Sample answer:

4 cm6 cm

7 cm

2 cm6 cm

7 cm

2 cm4 cm

7 cm

B

A

4

7

7

C

B

A

3

5

6

C

B

A

2

8

7

C

2 cm4 cm

6 cm

Chapter 7


d. Sample answer:

3. a.

The measure of the third angle is 70°.

b.


c.


d.


4. a.


b.


c.


5. Knowing the side lengths or angle measures, you can use different colored straws, a ruler and protractor, or geometry software to see if the triangle is possible to construct.

6. Sample answers:

Side lengths in Activity 2a:

Side lengths in Activity 2b:

Side lengths in Activity 2c:

Side lengths in Activity 2d:

The sum of any two side lengths of a triangle must be greater than the remaining side length.

Side Length 4 cm 7 cm 7 cm

Sum of Other Two Side Lengths

14 cm 11 cm 11 cm



11 cm 9 cm 8 cm

Side Length 2 in. 8 in. 7 in.


15 in. 9 in. 10 in.



2 cm 2 cm 2 cm

B

A

1

1

1 C

70°

40° 70°

45°

60° 75°

60°

30° 90°

40°

40° 100°

B

A

C45° 55°

80°

B

A

C

40°90°

50°

B

A C35°35°

110°

Chapter 7


219

7. Angle measures for each triangle formed in Activity 3:

The sum of the angles of a triangle is 180°.

7.3 On Your Own (pp. 284–285)

1. The triangle has a right angle and two congruent sides. So, the triangle is a right isosceles triangle.

2. The triangle has all congruent angles and all congruent sides. So, the triangle is an equiangular and equilateral triangle.

3. Sample answer:

The triangle has two congruent angles and one right angle. So, it is a right isosceles triangle.

4. Sample answer:

The triangle has no congruent angles and one right angle. So, it is a right scalene triangle.



1. Triangles can be classified by the size of their angles and by the number of congruent angles. An acute triangle has all acute angles. An obtuse triangle has 1 obtuse angle. A right triangle has 1 right angle. An equiangular triangle has 3 congruent angles.

Triangles can also be classified by the number of congruent sides they have. Scalene triangles have no congruent sides. Isosceles triangles have at least 2 congruent sides. Equilateral triangles have 3 congruent sides.

2. “Construct a triangle with no congruent sides” is different; Constructed triangle that is equilateral, equiangular and with 3 congruent sides:

Constructed triangle with no congruent sides:


3. Sample answer:

4. Sample answer:

3a. 3b. 3c. 3d.

1st angle 40° 60° 90° 100°

2nd angle 70° 75° 30° 40°

3rd angle 70° 45° 60° 40°

Sum 180° 180° 180° 180°

45°45°

2 in.

1 in.

60°4 cm

4 cm6 cm

9 cm

12 cm

5 cm

60°

60°60°

Chapter 7


5. Sample answer:

6. The triangle has two are congruent sides and one right angle. So, the triangle is a right isosceles triangle.

7. The triangle has all congruent sides and all congruent angles. So, the triangle is an equilateral and equiangular triangle.

8. The triangle has two congruent sides and one obtuse angle. So, it is an obtuse isosceles triangle.

9. The triangle has no congruent sides and a right angle. So, it is a right scalene triangle.

10. The triangle has no congruent sides and all acute angles. So, it is an acute scalene triangle.

11. The triangle has no congruent sides and one obtuse angle. So, it is an obtuse scalene triangle.

12. The solution incorrectly concludes that a triangle is acute if it has some acute angles. Because 98 90 ,° > ° this

angle is obtuse. The triangle is an obtuse scalene triangle because it has one obtuse angle and no congruent sides.

13. The triangle has two congruent sides and all acute angles. So, it is an acute isosceles triangle.

14. Sample answer:

The triangle has no congruent sides and one right angle. So, it is a right scalene triangle.

15. Sample answer:

The triangle has no congruent sides and one obtuse angle. So, it is an obtuse scalene triangle.

16. Sample answer:

The triangle has two congruent sides and one obtuse angle. So, it is an obtuse isosceles triangle.

17. Sample answer:

18. Sample answer:

19. Sample answer:

20. Sample answer: Your friend is not correct because without any more information about the lengths of the sides or other angles, you could be constructing a right triangle or an obtuse triangle.

21. no; The sum of the angle measures of a triangle must be 180 .° Because 50 70 100 220,+ + = no triangle can

be constructed.

22. many; You can change the length of the second side used to create the 60° angle to construct many triangles.

23. many; You can change the angle formed by the two given sides to construct many triangles.

24. one; Only one line segment can be drawn between the end points of the two given sides.

25. no; The sum of any two side lengths must be greater than the remaining length. Because the two 2-inch sides have a sum less than 5, no triangle can be constructed.

65°

55°

60°

75°

15° 90°

60°

20° 100°

120°

30° 30°

3 in.

2 in.

40°

8 cm

45°

60°

Chapter 7


221

26. no; An equilateral triangle cannot have a right angle because it is equiangular with all three angles being 60 .°

27. a. Green triangle:

65 50 180

115 180

65

x

x

x

+ + =+ =

=

The value of x is 65.

Purple triangle:

25 130 180

155 180

25

x

x

x

+ + =+ =

=


Red triangle:

45 90 180

135 180

45

x

x

x

+ + =+ =

=


b. The angles opposite the congruent sides of the triangle are congruent.

c. An isosceles triangle has at least two congruent angles.

Fair Game Review

28. 2

1

2

x y

x y

=

=

The equation can be written as .y kx= So, x and y show

direct variation. The constant of proportionality is 1

.2

k =

29. 6

6

y x

y x

− == +

The equation cannot be written as .y kx= So, x and y do

not show direct variation.

30. 5

5

xy

yx

=

=

The equation cannot be written as .y kx= So, x and y do

not show direct variation.

31. B; ( ) 18800 0.06 72

12I Prt

= = =

The balance is $800 $72 $872+ = after 18 months.

7.3 Extension (pp. 288–289)

Practice

1. 44 45 180

89 180

91

x

x

x

+ + =+ =

=

The value of x is 91. The triangle has one obtuse angle and no congruent sides. So, it is an obtuse scalene triangle.

2. 78 27 180

105 180

75

x

x

x

+ + =+ =

=

The value of x is 75. The triangle has all acute angles and no congruent sides. So, it is an acute scalene triangle.

3. 53 37 180

90 180

90

x

x

x

+ + =+ =

=

The value of x is 90. The triangle has one right angle and no congruent sides. So, it is a right scalene triangle.

4. 73 13 180

86 180

94

x

x

x

+ + =+ =

=

The value of x is 94. The triangle has one obtuse angle and no congruent sides. So, it is an obtuse scalene triangle.

5. 84 48 180

132 180

48

x

x

x

+ + =+ =

=

The value of x is 48. The triangle has all acute angles and two congruent sides. So, it is an acute isosceles triangle.

6. 60 60 180

120 180

60

x

x

x

+ + =+ =

=

The value of x is 60. The triangle has all congruent angles and all congruent sides. So, it is an equilateral and equiangular triangle.

7. yes; Because 76.2 81.7 22.1 180,+ + = a triangle can

have the given measures.

8. no; Because 115.1 47.5 93 255.6+ + = and not 180, a triangle cannot have the given measures.

47.5 93 180

140.5 180

39.5

x

x

x

+ + =+ =

=

So, the angle measures 39.5 ,° 47.5 ,° and 93° can form

a triangle.

Chapter 7


9. no; Because 2 1

5 64 87 1573 3

+ + = and not 180,

a triangle cannot have the given measures.

1

64 87 1803

1151 180

32

283

x

x

x

+ + =

+ =

=

So, the angle measures 2

28 ,3

° 164 ,

3° and 87° can form

a triangle.

10. yes; Because 3 1 3

31 53 94 180,4 2 4

+ + = a triangle can

have the given measures.

11. 45 180

2 45 180

2 135

67.5

x x

x

x

x

+ + =+ =

==

The value of x is 67.5. The triangle has two congruent sides and all acute angles. So, the triangle is an acute isosceles triangle.

12. 60 180

2 60 180

2 120

60

x x

x

x

x

+ + =+ =

==

The value of x is 60. All three sides are congruent and all three angles are congruent. So, the triangle is an equilateral and equiangular triangle.

13. 132 180

2 132 180

2 48

24

x x

x

x

x

+ + =+ =

==

The value of x is 24. The triangle has two congruent sides and one obtuse angle. So, the triangle is an obtuse isosceles triangle.

14. ( )2 15 90 180

3 105 180

3 75

25

x x

x

x

x

+ + + =

+ ===

The value of x is 25 and the value of 2 15 2(25) 15 65x + = + = . The triangle has no

congruent sides and one right angle. So, the triangle is a right scalene triangle.

15. ( ) ( )10 3 5 180

5 5 180

5 175

35

x x x

x

x

x

+ + − + =

+ ===

The value of x is 35, the value of 10 35 10 45,x + = + = and the value of

( )3 5 3 35 5 100.x − = − = The triangle has no

congruent sides and one obtuse angle. So, the triangle is an obtuse scalene triangle.

16. If a triangle had two obtuse angles, then the sum of the angle measures would be greater than 180 .° So, a triangle can have at most one obtuse angle, and at least two acute angles.

17. a. 36 180

2 36 180

2 144

72

x x

x

x

x

+ + =+ =

==


b. Because the length of each card is the same, the triangle formed must remain an isosceles triangle. So, the cards can be stacked such that the base of the triangle is shorter when the value of x is greater than 72, and longer when the value of x is less than 72. If

60,x = then the three cards form an equilateral

triangle. This is not possible because the two upright cards would have to be exactly on the edges of the base card. So, x must be greater than 60. If 90,x =then the two upright cards would be vertical, which is not possible. The card structure would not be stable. So, x must be less than 90. This means that the value of x is limited to 60 90.x< <

Study Help Available at BigIdeasMath.com.

Quiz 7.1–7.3 1. Sample answer:

adjacent: and ;PQR RQS∠ ∠ and RQS SQT∠ ∠

vertical: and ;PQR SQT∠ ∠ and PQT SQR∠ ∠

2. Sample answer:

adjacent: and ;YUZ ZUV∠ ∠ and ZUV VUW∠ ∠

vertical: and ;YUX WUV∠ ∠ and XUW YUV∠ ∠

3. The angles are adjacent angles. Because the angles are supplementary, the sum of their measures is 180 .°

34 180

146

x

x

+ ==

So, x is 146.

Chapter 7


223

4. The angles are adjacent angles. Because the angles are complementary, the sum of their measures is 90 .°

74 90

16

x

x

+ ==

So, x is 16.


10 59

49

x

x

+ ==

So, x is 49.

6. The two angles make up a straight angle. So, the angles are supplementary, and their sum is 180°.

( )2 5 75 180

2 80 180

2 100

50

x

x

x

x

+ + =

+ ===

So, x is 50.

7. The two make up a right angle. So, the angles are complementary, and their sum is 90°.

( )3 6 90

4 6 90

4 96

24

x x

x

x

x

+ − =

− ===

So, x is 24.

8.

9.

10.

11. 25 40 180

65 180

115

x

x

x

+ + =+ =

=

The value of x is 115. The triangle has one obtuse angle and no congruent sides. So, the triangle is an obtuse scalene triangle.

12. 90 180

2 90 180

2 90

45

x x

x

x

x

+ + =+ =

==

The value of x is 45. The triangle has a right angle and two angles are congruent. So, the triangle is a right isosceles triangle.

13. ( )20 60 60 180

20 120 180

100 180

80

x

x

x

x

− + + =

− + =+ =

=

The value of x is 80 and the value of 20 80 20 60.x − = − = The triangle has all congruent

angles and all congruent sides. So, the triangle is an equilateral and equiangular triangle.

14. One way is to use vertical angles to find that the measure of 2 is 115 .∠ ° Another way is to use supplementary angles to find that the measure of 3 is 65 .∠ ° Then use supplementary angles to find that the measure of

2 is 115 .∠ °


1. Sample answers are given.

b. c.

65°

35° 80°

70°

5 cm

7 cm

Chapter 7


d. e.

f.

2. a. The quadrilateral is a trapezoid.

b. The quadrilateral is a square.

c. The quadrilateral is a rectangle.

d. The quadrilateral is a rhombus.

e. The quadrilateral is a kite.

f. The quadrilateral is a parallelogram.

3. a.

or

b.


b. Sample answer:

c. Sample answer:

d. Sample answer:

90°3

3

3 3

90°

90°90°

B

A

C

D

90°

5

5

2 290°

90°

90°

BA

CD

6

6

11

BA

C

80°

80°

100°

100°

D

4

4

4

4

B

A

C

D109°

109°71°

71°

Chapter 7


225

5. Answers are estimates and are not limited to:

The sum of the angles of a quadrilateral is equal to 360°.

6. You can classify quadrilaterals using the properties of their sides and angles. A quadrilateral with exactly one pair of parallel sides is a trapezoid. A quadrilateral with two pairs of congruent adjacent sides and opposite sides that are not congruent is a kite. A quadrilateral with opposite sides that are parallel and congruent is a parallelogram. A parallelogram with four right angles is a rectangle, and a parallelogram with four congruent sides is a rhombus. A parallelogram that has both is a square.

7.4 On Your Own (pp. 294–295)

1. The quadrilateral has four congruent sides. So, it is a rhombus.

2. The quadrilateral has exactly one pair of parallel sides. So, it is a trapezoid.

3. The quadrilateral has four right angles. So, it is a rectangle.

4. 80 100 100 360

280 360

80

x

x

x

+ + + =+ =

=


5. 81 90 124 360

295 360

65

x

x

x

+ + + =+ =

=


6.



1. All of the statements are true.

2. Two quadrilaterals with four right angles are a rectangle and a square.

3. Kite does not belong. It is the only type of quadrilateral listed that does not have opposite sides that are parallel and congruent.


4. The quadrilateral has four congruent sides and four right angles. So, it is a square.

5. The quadrilateral has exactly one pair of parallel sides. So, it is a trapezoid.

6. The quadrilateral has four congruent sides. So, it is a rhombus.

7. The quadrilateral has two pairs of congruent adjacent sides and opposite sides that are not congruent. So, it is a kite.

8. The quadrilateral has opposite sides that are parallel and congruent. So, it is a parallelogram.

9. The quadrilateral has four right angles. So, it is a rectangle.

10. 65 115 115 360

295 360

65

x

x

x

+ + + =+ =

=


11. 40 82 128 360

250 360

110

x

x

x

+ + + =+ =

=


12. 52 90 90 360

232 360

128

x

x

x

+ + + =+ =

=


13. 90 90 122 360

302 360

58

x

x

x

+ + + =+ =

=

The measure of the angle at the tail end of the kite is 58 .°

14.

1a. 1b. 1c. 1d. 1e. 1f.

1st angle 53° 90° 90° 53° 90° 63°

2nd angle 127° 90° 90° 127° 117° 117°

3rd angle 90° 90° 90° 53° 117° 63°

4th angle 90° 90° 90° 127° 36° 117°

Sum 360° 360° 360° 360° 360° 360°

5 cm

3 cmNot actual size

Chapter 7


15.

16.

17.

18. A square is always a rectangle.

19. A square is always a rhombus.

20. A rhombus is sometimes a square.

21. A parallelogram is never a trapezoid.

22. A trapezoid is never a kite.

23. A rhombus is sometimes a rectangle.

24. a. The new shape of the door is a trapezoid because it has exactly one pair of parallel sides after you cut it.

b. 90 90 91.5 360

271.5 360

88.5

x

x

x

+ + + =+ =

=

The measure of the new angle at the bottom left side of the door is 88.5 .°

25.

26. a. 125, 55x y= =

b. Opposite angles of a parallelogram are equal.

c. Consecutive interior angles of a parallelogram are supplementary.

Fair Game Review

27. 3 1

12 4=

28. 18 2

27 3=

29. 42 6

35 5=

30. B;

percent of decrease original amount new amount

original amount

40 32

408

401

50.2, or 20%

−=

−=

=

=

=


1. a. length: 4 cm; width: 2.5 cm

b. length: 160 m; width: 100 m

Sample answer: Use a proportion or a ratio table.

4×

4×

2.5×

2.5×

c. drawing length 4 cm 1 cm

actual length 160 m 40 m= =

drawing width 2.5 cm 1 cm

actual width 100 m 40 m= =

They are the same. The measurements in the drawing are proportional to the measurements of the actual food court.

100°

100°3 cm3 cm

3 cm3 cm

135° 45°

75°

4 cm

Quadrilaterals

Kites

TrapezoidsSquares

RectanglesRhombi

Parallelograms

Drawing (cm) 1 4

Actual (m) 40 160

Drawing (cm) 1 2.5

Actual (m) 40 100

Chapter 7


227

2.

a. The size of the drawing increases.

b. length: 8 cm; width: 5 cm; The dimensions doubled.

3. a. length: 6 units; width: 6 units

b. length: 48 in.; width: 48 in.

Sample answer: Use a proportion or a ratio table.

6×

6×

c. sketch length 6 units 1 unit

actual length 48 in. 8 in.= =

sketch width 6 units 1 unit

actual width 48 in. 8 in.= =

They are the same. The measurements in the drawing are proportional to the measurements of the actual painting.

4.

a. The size of the drawing decreases.

b. length: 2 units; width: 2 units; The dimensions are reduced by one third.

5. Sample answer: Increase or decrease each dimension of the drawing by the same factor.

6. Food court:

Painting:

Sample answer: The ratio of the perimeters for the original drawings to the actual objects is equal to the ratios found in Activities 1(c) and 3(c). The ratio of the areas for the original drawings to the actual objects is equal to the square of those ratios.

7. Answer should include, but is not limited to: Students will look at maps in books and on the Internet and make a list of all the different scales used on the maps.

8. When you zoom out, the measured distance stays the same and the actual distance increases. So, the scale decreases. When you zoom in, the measured distance stays the same and the actual distance decreases. So, the scale increases.

7.5 On Your Own (pp. 300–302)

1. The map distance is about 3 centimeters.

1 cm 3 cm

50 mi mi50 3

150

dd

d

=

= •=

The distance between Traverse City and Marquette is about 150 miles.

2. 1 in. in.

500 km 2900 km2900 500

5.8

x

x

x

=

==

The mantle of the model is 5.8 inches thick.

3. 1 mm 1 mmscale factor

20 cm 200 mm= =

The scale factor is 1 : 200.

Drawing (units) 1 6

Actual (in.) 8 48

Actual Object

Original Drawing

Your Drawing

Perimeter 520 m 13 cm 26 cm

Area 16,000 2m 10 2cm 40 2cm

Actual Object

Original Drawing

Your Drawing

Perimeter 192 in. 24 units 8 units

Area 2304 2in. 36 2units 4 2units

Chapter 7


4. a. Because the drawing does not change size, the perimeter and area do not change.

b. 1 cm 4 cm

3 mm mm3 4

12

ss

s

=

= •=

The side length of the actual computer chip is 12 millimeters. So, the actual perimeter of the computer

chip is ( )4 12 48= millimeters and the actual area is

212 144= square millimeters.

c. 1 cm 10 mm 10scale factor

3 mm 3 mm 3= = =

drawing perimeter 16 cm 160 mm 10

actual perimeter 48 mm 48 mm 3= = =

2

2

2

2

2

2

2

drawing area 16 cm

actual area 144 mm

1 cm

9 mm

1 cm

3 mm

10 mm

3 mm

10

3

=

=

=

=

=

The ratios change but the overall relationship with the scale factor does not. The ratio of the perimeters is still the same as the scale factor, and the ratio of the areas is still the same as the square of the scale factor.



1. A scale is a ratio that compares the measurements of the drawing or model with the actual measurements. A scale factor is a scale without any units.

2. The scale drawing is larger than the actual object because 2 centimeters is greater than 1 millimeter.

3. Convert one of the lengths into the same units as the other length. Then, form the scale and simplify.


4. The drawing length is 5 centimeters.

1 cm 5 cm

5 ft ft5 5

25

=

= •=

xx

x

The actual length of the flower garden is 25 feet.

5. The rose bed is a square and its drawing has a side length of 2 centimeters.

1 cm 2 cm

5 ft ft5 2

10

=

= •=

xx

x

The actual dimensions of the rose bed are 10 feet by 10 feet.

6. The top perennial bed has a drawing length of 3 centimeters and width of 2 centimeters.

1 cm 3 cm

5 ft ft5 3

15

=

= •=

1 cm 2 cm

5 ft ft5 2

10

=

= •=

ww

w

( ) ( )2 2 2 15 2 10 30 20 50P w= + = + = + =

The actual perimeter of the top perennial bed is 50 feet.

The bottom perennial bed has a drawing length of 2 centimeters and width of 1.5 centimeters. From the top perennial bed, you know that 2 centimeters corresponds to 10 feet.

1 cm 1.5 cm

5 ft ft5 1.5

7.5

=

= •=

ww

w

( ) ( )2 2 2 10 2 7.5 20 15 35P w= + = + = + =

The actual perimeter of the bottom perennial bed is 35 feet.

7. The length of the tulip bed on the drawing is 3 centimeters and the width is 1.5 centimeters.

1 cm 3 cm

5 ft ft5 3

15

=

= •=

1 cm 1.5 cm

5 ft ft5 1.5

7.5

=

= •=

ww

w

( )15 7.5 112.5A w= = =

The area of the actual tulip bed is 112.5 square feet. From Exercise 5, the side length of the actual rose bed is

10 feet. So, the area of the actual rose bed is 210 100= square feet.

Area of tulip bed percent of area of rosebed

112.5 100

1.125

p

p

== •=

The area of the tulip bed is 112.5% of the area of the rose bed.

Chapter 7


229

8. The map distance between Kalamazoo and Ann Arbor is about 2 centimeters.

1 cm 2 cm

50 mi mi50 2

100

dd

d

=

= •=

The distance between Kalamazoo and Ann Arbor is about 100 miles.

9. The map distance between Lansing and Flint is about 1 centimeter.

1 cm 1 cm

50 mi mi50d

d

=

=

The distance between Lansing and Flint is about 50 miles.

10. The map distance between Grand Rapids and Escabana is about 4 centimeters.

1 cm 4 cm

50 mi mi50 4

200

dd

d

=

= •=

The distance between Grand Rapids and Escabana is about 200 miles.

11. The map distance between Saginaw and Alpena is about 2.2 centimeters.

1 cm 2.2 cm

50 mi mi50 2.2

110

dd

d

=

= •=

The distance between Saginaw and Alpena is about 110 miles.

12. 1 6.25

1212 6.25

75

=

= •=

xx

x

The actual length of the mattress is 75 inches.

13. 1

12 1515

125

41.25

x

x

x

x

=

=

=

=

1.25 ft12 in.

1 ft× 15 in.=

The length of the model Corvette is 15 inches.

14. 1 32

1212 32

384

=

= •=

xx

x

384 cm1 m

100 cm× 3.84 m=

The actual depth of the water tower is 3.84 meters.

15. 1 5.4

1212 5.4

64.8

xx

x

=

= •=

64.8 ft1 yd

3 ft× 21.6 yd=

The actual wingspan is 21.6 yards.

16. 1

12 2121

127

41.75

x

x

x

x

=

=

=

=

1.75 cm10 mm

1 cm× 17.5 mm=

The diameter of the model football helmet is 17.5 millimeters.

17. The numerators should be the drawing distance and the denominators should be the actual distance.

1 cm 5 cm

20 m m20 5

100 m

xx

x

=

= •=

The actual distance is 100 meters.

18. The segment is 4 centimeters long.

Drawing distance 4 cm 1 cm

Actual distance 120 m 30 m= =

The scale of the drawing is 1 cm : 30 m.

19. The segment is 2.4 centimeters long.

Drawing distance 2.4 cm 1 cm

Actual distance 24 mm 10 mm= =

The scale of the drawing is 1 cm : 10 mm.

20. You need to know either the length or width of the actual object.

Chapter 7


21. Answer should include, but is not limited to: a. Students will choose a product they want to advertise.

Students will design a billboard for the product. The billboard should contain words and a picture. The drawing of the billboard will be neat and labeled clearly.

b. Students will use the dimensions of their drawing of the billboard and the dimensions of the actual billboard to determine a scale factor for their picture.

22. a. The park has a drawing length of 12.5 centimeters and width of 2.5 centimeters.

( ) ( )2 2 2 12.5 2 2.5 25 5 30P w= + = + = + =

The perimeter of the park in the scale drawing is 30 centimeters.

( )( )12.5 2.5 31.25A w= = =

The area of the park in the scale drawing is 31.25 square centimeters.

b. 1 cm 12.5 cm

320 m m320 12.5

4000

=

= •=

1 cm 2.5 cm

320 m m320 2.5

800

www

=

= •=

( ) ( )2 2

2 4000 2 800

8000 1600

9600

P l w= +

= +

= +=

The actual perimeter of the park is 9600 meters.

( )( )4000 800 3,200,000A w= = =

The actual area of the park is 3,200,000 square meters.

23. a. The icon has the shape of a square and has a drawing side length of 4 centimeters.

( )4 4 4 16P s= = =

The perimeter of the icon in the scale drawing is 16 centimeters.

2 24 16A s= = =

The area of the icon in the scale drawing is 16 square centimeters.

b. 1 cm 4 cm

2.5 mm mm2.5 4

10

sss

=

= •=

( )4 4 10 40P s= = =

The actual perimeter of the icon is 40 millimeters.

2 210 100A s= = = The actual area of the icon is 100 square millimeters.

24. The ratio of the perimeters is the scale factor and the ratio of the areas is the square of the scale factor.

25. Original drawing

New drawing

Measurements in the new scale drawing will be 2 times the measurements in the original scale drawing.

Original scale drawing Length: 5 cm

Width, and base of triangle: 2.5 cm

Height of triangle: 2.2 cm

New scale drawing Length: 5 2 10 cm• =

Width, and base of triangle: 2.5 2 5 cm• =

Height of triangle: 2.2 2 4.4 cm• =

26. Original drawing New drawing

Measurements in the new scale drawing will be 1

2the

measurements in the original scale drawing.

Original scale drawing Length: 4 cm

Width, and diameter of circle: 2 cm

New scale drawing

Length: 1

4 2 cm2

• =

Width, and diameter of circle: 1

2 1 cm2

• =

8 m1 cm

4 m 4 m1 cm 1 cm

2 m1 cm

2 m 2 m

1 cm

5 cm

5 cm

10 cm

5 cm

Not actual size

1 cm

2 cm

Chapter 7


231

27. The red region is in the shape of a trapezoid. The red region of the scale drawing has bases of 4 centimeters and 6 centimeters, and a height of 3 centimeters.

Let b and B be the actual bases and let h be the actual height.

1 cm 4 cm

1 ft ft1 4

4

bb

b

=

= •=

1 cm 6 cm

1 ft ft1 6

6

BB

B

=

= •=

1 cm 3 cm

1 ft ft1 3

3

hh

h

=

= •=

Use the formula for the area of a trapezoid to find the actual area.

( )( )( )

2

3 4 6 2

3 10 2

30 2

15

A h b B= + ÷

= + ÷

= ÷

= ÷=

The actual area of the red region is 15 square feet.

28. The blue region is in the shape of a trapezoid. The blue region of the scale drawing has bases of 1 centimeter and 2 centimeters, and a height of 3 centimeters.

Let b and B be the actual bases and let h be the actual height.

1 cm 1 cm

1 ft ft1b

b

=

=

1 cm 2 cm

1 ft ft1 2

2

BB

B

=

= •=

1 cm 3 cm

1 ft ft1 3

3

hh

h

=

= •=

Use the formula for the area of a trapezoid to find the actual area.

( )( )( )

2

3 1 2 2

3 3 2

9 2

4.5

A h b B= + ÷

= + ÷

= ÷

= ÷=

The actual area of the blue region is 4.5 square feet.

29. The green region is in the shape of a triangle. The green region of the scale drawing has a base of 2 centimeters and a height of 3 centimeters.

Let b be the actual bases and let h be the actual height.

1 cm 2 cm

1 ft ft1 2

2

bb

b

=

= •=

1 cm 3 cm

1 ft ft1 3

3

hh

h

=

= •=

Use the formula for the area of a triangle to find the actual area.

( )( )

( )( )

1

21

2 32

1 3

3

A bh=

=

=

=

The actual area of the green region is 3 square feet.

30. a. The length of the blueprint of the bathroom is

1 3

34 4 =

inch and the width is 1 1

2 inch.4 2 =

3in.1 in. 4

16 ft ft3

164

12

=

= •

=

1in.1 in. 2

16 ft ft1

162

8

=

= •

=

w

w

w

( )12 8 96A w= = =

The actual area of the bathroom floor is 96 square feet. So, it will cost 96 5 $480• = to tile the bathroom.

Chapter 7


b. The length of the blueprint of the bedroom is

14 1

4 =

inch and the width is 1 5

54 4 =

inches.

Because 1 inch corresponds to 16 feet, the actual length of the bedroom is 16 feet.

5 in.1 in. 4

16 ft ft5

164

20 ft

=

= •

=

w

w

w

Convert the dimensions to yards.

Length: 16 ft1 yd

3 ft× 1

5 yd3

=

Width: 20 ft1 yd

3 ft× 2

6 yd3

=

21 2 16 20 55 6 35 yd

3 3 3 3 9A w= = • = • =

The cost to carpet the bedroom is 5

18 35 $640.9

=

The length of the blueprint of the living room is

14 1 inch

4 =

and the width is 1 7

74 4 =

inches.

Because 1 inch corresponds to 16 feet, the actual length of the living room is 16 feet.

7 in.1 in. 4

16 ft ft7

164

28 ft

=

= •

=

w

w

w

Convert the dimensions to yards. You already know 1

16 ft 5 yards.3

= So, the length is 1

53

yards.

Width: 28 ft1 yd

3 ft× 1

9 yd3

=

21 1 16 28 75 9 49 yd

3 3 3 3 9A w= = • = • =

The cost to carpet the living room is

718 49 $896.

9 =

So, the total cost to carpet the

bedroom and the living room is $640 $896 $1536.+ =

c. Convert square yards to square feet.

2

2 23 ft1 yd 1 yd

1 yd

× =

2

2

9 ft

1 yd× 29 ft=

Carpet unit cost: 2 2 2

$18 $18 $2

1 yd 9 ft 1 ft= =

Tile unit cost: 2

$5

1 ft

Because $5 per square foot is greater than $2 per square foot, the tile has a higher unit cost.

31. Sample answer: The radius of a baseball is about 3 inches. So, if you use a baseball as a scale model of Earth, the scale is:

3 in. 1 in.

.6378 km 2126 km

=

Then the radius of the model for the Sun would be:

1 in. in.

2126 km 695,500 km

695,500 2126

327

=

=≈

x

x

x

Because 327 inches is about 27 feet long, your model for the Sun would be quite large. So, it is not reasonable to choose a baseball as a model of Earth.

Fair Game Review

32–35. y

x−6−4 4 62

−6

−8

2

−4

4A

C

D B

O

Chapter 7


233

36. D;

For A: 7 7 5 35

35%20 20 5 100

32%

0.45 45%

×= = =×

=

Exclude A because 35% is not the least.

For B: 17%

0.21 21%

3 3 4 1212%

25 25 4 100

=•= = =•

Exclude B because 12% is not the greatest.

For C: 0.88 88%

7 7 12.5 87.587.5%

8 8 12.5 10093%

=•= = =•

Exclude C because 88% is not the least.

For D: 57%

11 11 6.25 68.7568.75%

16 16 6.25 1005.7 570%

•= = =•

=

Choose D because the order is from least to greatest.

Quiz 7.4–7.5 1. The quadrilateral has four congruent sides. So, it is

a rhombus.

2. The quadrilateral has two pairs of congruent adjacent sides and opposite sides that are not congruent. So, it is a kite.

3. 135 45 135 360

315 360

45

x

x

x

+ + + =+ =

=


4. 90 35 90 360

215 360

145

x

x

x

+ + + =+ =

=


5.

6.

7. 1

20 9090

204.5

x

x

x

=

=

=

The height of the model of the basketball player is 4.5 inches.

8. 1 3.75

2020 3.75

75

=

= •=

xx

x

The actual length of the dinosaur is 75 feet.

9. 90 90 110 360

290 360

70

x

x

x

+ + + =+ =

=

The measure of the top left angle is 70 .°

10.

13 in.Model length 2

Actual length 12 ft=

To find the scale, use the fact that 1 ft 12 in.= Multiply the denominator of the scale by 12.

13 in.

2scale factor12 ft1

3 in.2

144 in.1

32

1447

2144

7

2 1447

288

=

=

=

=

=•

=


50°50°

2 cm 2 cm

2 cm2 cm

5 cm

65°

Chapter 7


11. a. Drawing distance 6 in. 1 in.

Actual distance 300 ft 50 ft= =

The scale of the drawing is 1 in. : 50 ft.

b. 1 in.

scale factor50 ft

1 in.

50 12 in.

1 in.

=

=•

=600 in.

1

600=


Chapter 7 Review 1. The angles are adjacent angles. Because the angles are

complementary angles, the sum of their measures is 90 .°

69 90

21

x

x

+ ==

So, x is 21.

2. The angles are vertical. Because the angles are vertical angles, they are congruent.

3 84

81

x

x

+ ==

So, x is 81.


( )29 6 1 90

6 30 90

6 60

10

x

x

x

x

+ + =

+ ===

So, x is 10.


( )4 10 180

5 10 180

5 170

34

x x

x

x

x

+ + =

+ ===

So, x is 34.

5.

6.

7. 49 90 180

139 180

41

x

x

x

+ + =+ =

=

The value of x is 41. The triangle has one right angle and no congruent sides. So, the triangle is a right scalene triangle.

8. ( )12 35 110 180

157 180

23

x

x

x

+ + + =

+ ==

The value of x is 23. The triangle has one obtuse angle and two congruent sides. So, the triangle is an obtuse isosceles triangle.

9. 90 90 128 360

308 360

52

x

x

x

+ + + =+ =

=


10. 80 95 38 360

213 360

147

x

x

x

+ + + =+ =

=


50°

40° 90°

4 in.

3 in.

30°

Chapter 7


235

11.

12. The segment is 6 centimeters long.

Drawing length 6 cm 1 cm

Actual length 30 in. 5 in.= =

The scale of the drawing is 1 cm : 5 in.

13. The segment is 2.5 centimeters long.

Drawing length 2.5 cm 1 cm

Actual length 7.5 in. 3 in.= =

The scale of the drawing is 1 cm : 3 in.

Chapter 7 Test 1. The angles are vertical. Because the angles are vertical

angles, they are congruent.

So, x is 113.

2. The angles are adjacent angles. Because the angles are complementary angles, the sum of their measures is 90 .°

( )6 56 90

62 90

28

x

x

x

+ + =

+ ==

So, x is 28.


( )74 8 2 180

8 76 180

8 104

13

x

x

x

x

+ + =

+ ===

So, x is 13.


( )15 4 5 90

4 10 90

4 80

20

x

x

x

x

+ − =

+ ===

So, x is 20.

5.

The triangle has one right angle and no congruent sides. So, the triangle is a right scalene triangle.

6.

The triangle has one obtuse angle and no congruent sides. So, the triangle is an obtuse scalene triangle.

7.

8.

9. 23 129 180

152 180

28

x

x

x

+ + =+ =

=

The value of x is 28. The triangle has one obtuse angle and no congruent sides. So, the triangle is an obtuse scalene triangle.

120° 120°

5 cm5 cm

5 cm5 cm

80°10° 90°

40°

30° 110°

5 in.

6 in.

50°

Not actual size

Chapter 7


10. 68 180

2 68 180

2 112

56

x x

x

x

x

+ + =+ =

==

The value of x is 56. The triangle has all acute angles and two congruent sides. So, the triangle is an acute isosceles triangle.

11. 180

3 180

60

x x x

x

x

+ + ===

The value of x is 60. The triangle has three congruent angles and three congruent sides. So, it is an equilateral and equiangular triangle.

12. 90 90 90 360

270 360

90

x

x

x

+ + + =+ =

=


13. 95 90 95 360

280 360

80

x

x

x

+ + + =+ =

=


14. 84 96 110 360

290 360

70

x

x

x

+ + + =+ =

=


15.

16.

17. The drawing of the fish has a length of 5 centimeters.

Drawing length 5 cm 50 mm 10

Actual length 15 mm 15 mm 3= = =


18. First convert feet to inches.

12 in.

12.5 ft 150 in.1 ft

× =

1 cm cm

60 in. 150 in.150

602.5

x

x

x

=

=

=

The length of the component in the drawing is 2.5 centimeters.

Chapter 7 Standards Assessment 1. D; You burn an additional 9 calories for each minute of

basketball you play.

2. 75%; original price sale price

percent of discountoriginal price

40 10

4030

400.75, or 75%

−=

−=

=

=

The percent of discount is 75%.

3. H; ( ) ( )2 6 9 2 6 9 4 9 5− − − = + − + = − + =

4. 12;

8

12 182

3 1836

312

x

x

x

x

=

=

=

=

5. B;

5 6 23

6 18

3

x

x

x

− − ≤ −− ≤ −

≥

80°

80°

6 cm

6 cm

6 cm

6 cm

160°

20°

Chapter 7


237

6. F; length 20 m= 1 cm

2 m× 10 cm=

width 12 m= 1 cm

2 m× 6 cm=

( )( )Area 10 6 60w= • = =

The area of the model is 60 square centimeters.

7. D;

510

7

7 5 710

5 7 5

14

y kx

k

k

k

=

=

• = •

=

So, 14 .y x=

( )14 4 56y = =

8. H; The two angles make up a straight angle. So, the angles are supplementary, and the sum of their measures is 180 .°

Let x be the measure of 2.∠

28 180

152

x

x

+ ==

So, the measure of 2∠ is 152 .°

9. C; ( )15 35 5

15 35 5

50 5

5 50 55

250

c

c

c

c

c

− − = −

+ = −

= −

• = − •

= −

Brett should multiply both sides of the equation by 5 to get 250.c = −

10. Part A: 90 mi;

1 1in. 4 in.

2 210 mi mi

145

290

x

x

x

=

=

=

The actual distance between her town and the state capital is 90 miles.

Part B: 1

3 in.;4

1 in. in.2

10 mi 65 mi32.5 10

3.25

x

x

x

=

==

The distance on the map is 3.25, or 1

34

inches.

mscc7a ws 0700a - pleasantville high school

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