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Chapter 5
b.
Chapter 5 Opener
Try It Yourself (p. 183)
2.
30°
60°
90°
120°
150°
y
150°
120°
90°
60°
30°
5
7
=
x
14
70 = 7 x
When x = 60°, y = 180 − 60 = 120°.
10 = x
When x = 90°, y = 180 − 90 = 90°.
So, x is 10 inches.
When x = 120°, y = 180 − 120 = 60°.
When x = 30°, y = 180 − 30 = 150°.
When x = 150°, y = 180 − 150 = 30°.
2
25
=
x
1
2 x = 25
Angle measure (degrees)
1.
x
x = 12.5
So, x is 12.5 millimeters.
Section 5.1
5.1 Activity (pp. 184 –185)
1. a.
y
160
120
80
40
0
x
15°
30°
45°
60°
75°
y
75°
60°
45°
30°
15°
When x = 15°, y = 90 − 15 = 75°.
When x = 30°, y = 90 − 30 = 60°.
When x = 45°, y = 90 − 45 = 45°.
When x = 60°, y = 90 − 60 = 30°.
Angle measure (degrees)
When x = 75°, y = 90 − 75 = 15°.
0
40
80
120
160
x
Angle measure (degrees)
Because the graph is a straight line, the function is
linear. Because x + y = 180°, the equation of the
function is y = 180 − x. The input values of the
function are between 0 and 180, so the domain is
0 < x < 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
y
sometimes acute.
80
c. If x is a right angle, then x is never acute.
60
3. a. ∠ BDE and ∠ DBE , ∠ BDE and ∠ BDC , ∠ DBE
40
and ∠ CBD, and ∠ CBD and ∠ BDC are all
20
complementary angles.
0
0
20
40
60
80
x
Angle measure (degrees)
Because the graph is a straight line, the function is
linear. Because x + y = 90°, the equation of the
function is y = 90 − x. The input values of the
function are between 0 and 90, so the domain is
0 < x < 90.
b. ∠ A and ∠ ABE , ∠ A and ∠ BEF , ∠ A and
∠ F , ∠ A and ∠ CBE , ∠ A and ∠ C , ∠ A and
∠ CDE , ∠ A and ∠ BED, ∠ ABE and ∠ BEF ,
∠ ABE and ∠ F , ∠ ABE and ∠ CBE , ∠ ABE and
∠ C , ∠ ABE and ∠ CDE , ∠ ABE and ∠ BED,
∠ BEF and ∠ F , ∠ BEF and ∠ CBE , ∠ BEF and
∠ C , ∠ BEF and ∠ CDE , ∠ BEF and ∠ BED,
∠ F and ∠ CBE , ∠ F and ∠ C , ∠ F and ∠ CDE ,
∠ F and ∠ BED, ∠ CBE and ∠ C , ∠ CBE and
∠ CDE , ∠ CBE and ∠ BED, ∠ C and ∠ CDE ,
∠ C and ∠ BED, ∠ CDE and ∠ BED are all
supplementary angles.
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Chapter 5
4. Two angles are complementary if the sum of their
measures is 90°. Two angles are supplementary if the
sum of their measures is 180°. Sample answer: 32°
and 58° are complementary angles and 25° and 155°
are supplementary angles.
5. Sample answer should include, but is not limited to:
a real life example of complementary and supplementary
angles with a drawing of each object and approximate
angle measures.
5.1 On Your Own (pp. 186 –187)
1. Because 26° + 64° = 90°, the angles are
complementary.
2. Because 44° + 136° = 180°, the angles are
supplementary.
3. Because 70° + 19° = 89°, the angles are neither
complementary nor supplementary.
4. The angles are supplementary. So, the sum of their
measures is 180°.
x + 85 = 180
x = 95
So, x is 95.
5. The angles are vertical angles. Because vertical angles
are congruent, the angles have the same measure. So, x
is 90.
6. The angles are complementary. So, the sum of their
measures is 90°.
x + 69 = 90
x = 21
So, x is 21.
5.1 Exercises (pp. 188–189)
Vocabulary and Concept Check
1. Two angles are complementary if the sum of their
measures is 90°. Two angles are supplementary if the
sum of their measures is 180°.
2. When two lines intersect, four angles are formed.
Because there are two pairs of opposite angles formed,
two pairs of vertical angles are formed.
Practice and Problem Solving
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.
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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.
6. Because 122° + 68° = 190°, the angles are neither
complementary nor supplementary.
7. Because 42° + 48° = 90°, the angles are
complementary.
8. Because 59° + 31° = 90°, the angles are
complementary.
9. Because 115° + 65° = 180°, the angles are
supplementary.
10. Because 24° + 156° = 180°, the angles are
supplementary.
11. Because 45° + 55° = 100°, the angles are neither
complementary nor supplementary.
12. The angles are complementary. So, the sum of their
measures is 90°.
x + 35 = 90
x = 55
So, x is 55.
13. The angles are vertical angles. Because vertical angles
are congruent, the angles have the same measure. So, x
is 128.
14. The angles are supplementary. So, the sum of their
measures is 180°.
x + 117 = 180
x = 63
So, x is 63.
15. Vertical angles are congruent, not complementary. So,
the angles have the same measure, and the value of x is
35.
16. The angles are supplementary. So, the sum of their
measures is 180°.
x + 127 = 180
x = 53
So, x is 53.
17. The angles are vertical angles. Because vertical angles
are congruent, the angles have the same measure.
2 x + 1 = 75
2 x = 74
x = 37
So, x is 37.
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Chapter 5
18. The angles are complementary. So, the sum of their
measures is 90°.
4 x + 2 x = 90
6 x = 90
x = 15
So, x is 15.
19. The angles are supplementary. So, the sum of their
measures is 180°.
7 x + x + 20 = 180
8 x + 20 = 180
8 x = 160
x = 20
So, x is 20.
20. Sample answer: 100° is the measure of an angle that can
be supplementary but not complementary. Because
complementary angles are all 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.
21. a. Because ∠ CBE is a right angle, ∠ CBD and ∠ DBE
are complementary angles. Because ∠ ABE is a right
angle, ∠ ABF and ∠ FBE are complementary
angles.
b. Because AC is a straight line, the following pairs of
angles are supplementary angles: ∠ ABF and ∠ FBC ,
∠ ABE and ∠ EBC , and ∠ ABD and ∠ DBC.
22. Because the 50° angle and ∠1 are supplementary, the
sum of their measures is 180°.
x + 50 = 180
x = 130
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°.
23. 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°.
3x + 2 x = 90
5 x = 90
x = 18
So, the measure of the larger angle is 3x = 3(18) = 54°.
24. Because vertical angles are congruent, the angles have
the same measure. So, let x represent the measure of one
complementary angle.
x + x = 90
2 x = 90
x = 45
So, the measures of two vertical angles that are
complementary angles are 45°.
Let y represent the measure of one supplementary angle.
y + y = 180
2 y = 180
y = 90
So, the measures of two vertical angles that are
supplementary angles are 90°.
25. Because the angles with measures 7x° and y° are
supplementary to the right angle, the sum of all three
measures is 180°.
7 x + 90 + y = 180
7 x + y = 90
Because the angle with the sum of the angle measures
5x° and 2 y° is a vertical angle to the right angle, the sum
of their measures is 90°.
5 x + 2 y = 90
A system of equations is 7 x + y = 90 and
5 x + 2 y = 90.
7 x + y = 90
y = 90 − 7 x
5 x + 2 y = 90
2 y = 90 − 5 x
y = 45 −
5
x
2
5
x
2
9
90 = 45 + x
2
9
45 = x
2
90
= x
9
10 = x
90 − 7 x = 45 −
y = 90 − 7 x = 90 − 7(10) = 90 − 70 = 20
So, x = 10 and y = 20.
Fair Game Review
26. x + 60 + 45 = 180
x + 105 = 180
x = 75
The value of x is 75.
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Chapter 5
27. x + 58.5 + 92.2 = 180
x + 150.7 = 180
x = 29.3
The value of x is 29.3.
28. x + x + 110 = 180
2 x + 110 = 180
2 x = 70
x = 35
The value of x is 35.
1
2
29. B; m = − ; (6, 4)
5
4
(2, 6)
−2 1
3
(6, 4)
−2
2
student will design an abstract art painting that
includes triangles and count the different types of
triangles used.
4. You can classify them as acute, obtuse, right, or
equiangular depending on the measures of their angles.
5. Answer should include, but is not limited to: The student
will find and name real-life triangles in architecture.
5.2 On Your Own (pp. 192–193)
1. x + 78 + 27 = 180
The value of x is 75. The triangle has all acute angles.
So, it is an acute triangle.
2. x + 44 + 45 = 180
1
O
b. Answer should include, but is not limited to: The
x = 75
(4, 5)
−2 1
right; yellow triangle: obtuse; blue triangle: obtuse
x + 105 = 180
y (0, 7)
1
3. a. Sample answer: red triangle: acute; pink triangle:
1 2 3 4 5 6 7x
Because the line crosses the y-axis at (0, 7), the
1
y-intercept is 7. So, the equation is y = − x + 7.
2
Section 5.2
5.2 Activity (pp. 190 –191)
1. a–f. Answer should include, but is not limited to:
x + 89 = 180
x = 91
The value of x is 91. The triangle has an obtuse angle.
So, it is an obtuse triangle.
3. x + x + 120 = 180
2 x + 120 = 180
2 x = 60
x = 30
Students will follow directions for steps (a)–(d)
for four different triangles. Their conclusion should
be that the sum of the angle measures of any
triangle is 180°.
4. x + x + x = 180
2. a. A right triangle has one right angle. Because the green
3 x = 180
triangle has one right angle, it is a right triangle.
b. An acute triangle has all acute angles. Because the
angle measures in the blue, red, and yellow triangles
are all acute, they are all acute triangles.
c. An obtuse triangle has one obtuse angle. Because the
purple triangle has an angle measure of 100°, which is
obtuse, it is an obtuse triangle.
d. An equiangular triangle has three congruent angles.
Because the yellow triangle has three congruent
angles, it is an equiangular triangle.
e. An equilateral triangle has three congruent sides.
Because the yellow triangle has three congruent sides,
it is an equilateral triangle.
f. An isosceles triangle has at least two sides that are
congruent. Because the blue triangle has two sides that
are congruent, it is an isosceles triangle. Because the
yellow triangle has two sides that are congruent, it is
also an isosceles triangle.
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The value of x is 30. Two of the sides are congruent and
one angle is obtuse. So, it is an obtuse isosceles triangle.
x = 60
The value of x is 60. All three angles are congruent and
acute and all three sides are congruent. So, it is an acute,
isosceles, equilateral, and equiangular triangle.
5. x + 63.9 + 61.8 = 180
x + 125.7 = 180
x = 54.3
The value of x is 54.3.
5.2 Exercises (pp. 194 –195)
Vocabulary and Concept Check
1. An equilateral triangle has three congruent sides and an
isosceles triangle has at least two congruent sides. So, an
equilateral triangle is a specific type of isosceles triangle.
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Chapter 5
2. To find the missing angle of the triangle, write an
equation such that the sum of the angles of the triangle is
180°. Next find the sum of the two known angles. Then
subtract the sum of the angles from 180.
x + 45 + 102 = 180
x + 147 = 180
x = 33
Because the value of x is 33, the measure of the missing
angle is 33°.
Practice and Problem Solving
3. Two of the sides are congruent and one angle is a right
angle. So, it is a right isosceles triangle.
4. All three angles are acute. So, it is an acute triangle.
11. x + x + 132 = 180
2 x + 132 = 180
2 x = 48
x = 24
The value of x is 24. Two of the sides are congruent and
one angle is obtuse. So, it is an obtuse isosceles 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 triangle because
it has one obtuse angle.
13. a. x + x + 40 = 180
2 x + 40 = 180
2 x = 140
x = 70
5. Two of the sides are congruent and one angle is an obtuse
angle. So, it is an obtuse isosceles triangle.
6. x + 53 + 37 = 180
x + 90 = 180
x = 90
The value of x is 90. One of the angles is a right angle.
So, it is a right triangle.
7. x + 73 + 13 = 180
x + 86 = 180
x = 94
The value of x is 94. One angle is obtuse. So, it is an
obtuse triangle.
8. x + 48 + 84 = 180
x + 132 = 180
x = 48
The value of x is 48. Two of the sides are congruent and
all of the angles are acute. So, it is an acute isosceles
triangle.
9. x + x + 45 = 180
2 x + 45 = 180
2 x = 135
x = 67.5
The value of x is 67.5. Two of the sides are congruent and
all of the angles are acute. So, it is an acute isosceles
triangle.
10. x + x + 60 = 180
2 x + 60 = 180
2 x = 120
The value of x is 70.
b. The triangle has two congruent sides, so it is an
isosceles triangle. The triangle has all acute angles,
so it is an acute triangle.
14. yes; Because 76.2 + 81.7 + 22.1 = 180, a triangle can
have the given measures.
15. no; Because 115.1 + 47.5 + 93 = 255.6 and not 180,
a triangle cannot have the given measures.
x + 47.5 + 93 = 180
x + 140.5 = 180
x = 39.5
So, the angle measures 39.5°, 47.5°, and 93° can form
a triangle.
2
1
+ 64 + 87 = 157 and not 180,
3
3
a triangle cannot have the given measures.
16. no; Because 5
1
x + 64 + 87 = 180
3
1
x + 151 = 180
3
2
x = 28
3
So, the angle measures 28
2°
1°
, 64 , and 87° can form
3
3
a triangle.
3
1
3
+ 53 + 94 = 180, a triangle can
4
4
4
have the given measures.
17. yes; Because 31
x = 60
The value of x is 60. All three sides and angles are
congruent and all of the angles are acute. So, it is an acute
isosceles, equilateral, and equiangular triangle.
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Chapter 5
22. x + x + x + 3 + 9 + 5 + 2 = 28
18. a. Green triangle:
x + 65 + 50 = 180
3x + 19 = 28
x + 115 = 180
3x = 9
x = 65
x = 3
The value of x is 65.
The value of x is 3.
Purple triangle:
x + 25 + 130 = 180
x + 155 = 180
23.
1 ⎛ circumference ⎞
⎜
⎟ + three sides of rectangle = P
2 ⎝ of circle
⎠
1
(2π r ) + x + (2 x − 4) + x = P
2
x = 25
The value of x is 25.
Red triangle:
x + 45 + 90 = 180
1
( 2 • 3.14 • 3) + x + 2 x − 4 + x = 25.42
2
4 x + 5.42 = 25.42
x + 135 = 180
4 x = 20
x = 45
The value of x is 45.
b. The angles opposite the congruent sides of the triangle
are congruent.
c. An isosceles triangle has at least two congruent
angles.
19. 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.
20. a. x + x + 36 = 180
x = 5
The value of x is 5.
24. A; Because you are starting out with $10, the y-intercept
is a positive value. Each time you send a text message,
your money decreases. So, the slope is negative. The
equation is y = −0.25 x + 10.
Section 5.3
5.3 Activity (pp. 196 –197)
1. b. Pentagon: n = 5
Sample answer:
2 x + 36 = 180
2 x = 144
x = 72
The value of x is 72.
b. Sample answer: 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 x = 60, 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 > 60. If x = 90, then
the two upright cards would be vertical, which is not
possible. The card structure would not be stable. So,
x < 90.
Fair Game Review
21. 2 x + x + 2 x + 8 + 5 = 48
5 x + 13 = 48
5 x = 35
Draw two lines that divide the pentagon into three
triangles. Because the sum of the angle measures of
each triangle is 180°, the sum of the angle measures
of the pentagon is 3(180°) = 540°.
c. Hexagon: n = 6
Sample answer:
Draw three lines that divide the hexagon into four
triangles. Because the sum of the angle measures of
each triangle is 180°, the sum of the angle measures of
the hexagon is 4(180°) = 720°.
x = 7
The value of x is 7.
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d. Heptagon: n = 7
e. S = 180n − 360
= 180(10) − 360
Sample answer:
= 1800 − 360
= 1440
The sum of the angle measures of a polygon with
10 sides is 1440°.
3. yes; Because the equation from Activity 2 was derived
Draw four lines that divide the heptagon into five
triangles. Because the sum of the angle measures of
each triangle is 180°, the sum of the angle measures
of the heptagon is 5(180°) = 900°.
e. Octagon: n = 8
using triangles, the equation can be used on a concave
polygon because it can also be divided into triangles. The
triangles must be formed such that each line segment
connecting two vertices lies inside the polygon. At least
one of the angle measures in a concave polygon will be
greater than 180°, so you need to expand your definition
of angles to include this type of angle.
4. Sample answer: Create examples of polygons and then
Sample answer:
draw lines to divide the polygon into triangles to find the
sum of the angle measures. Then plot the number of sides
and the sum of the angle measures and look for a pattern.
Finally, write an equation to represent the data.
5.3 On Your Own (pp. 198 –200)
1. The spider web is in the shape of a heptagon. It has
Draw five lines that divide the octagon into six
triangles. Because the sum of the angle measures of
each triangle is 180°, the sum of the angle measures
of the octagon is 6(180°) = 1080°.
2. a.
7 sides.
S = ( n − 2) • 180°
= (7 − 2) • 180°
= 5 • 180°
Sides, n
3
4
5
6
7
8
Angle
Sum, S
180
360
540
720
900
1080
= 900°
The sum of the angle measures is 900°.
2. The honeycomb is in the shape of a hexagon. It has
6 sides.
b.
1080
900
720
540
360
180
−180
−360
S
(8, 1080)
(7, 900)
(6, 720)
(5, 540)
(4, 360)
(3, 180)
1 2 3 4 5 6 7 8n
c. A polygon with n sides can be divided into n − 2
triangles. Because the sum of the angle measures
of each triangle is 180°, the sum of the measures
of a polygon is ( n − 2) • 180°. So, the equation
is S = ( n − 2) • 180 = 180n − 360.
d. Because a polygon can have no fewer than 3 sides, the
domain of the function is all integers greater than or
equal to 3.
S = ( n − 2) • 180°
= (6 − 2) • 180°
= 4 • 180°
= 720°
The sum of the angle measures is 720°.
3. The polygon has 6 sides.
S = ( n − 2) • 180°
= (6 − 2) • 180°
= 4 • 180°
= 720°
The sum of the angle measures is 720°.
125 + 120 + 125 + 110 + 135 + x = 720
615 + x = 720
x = 105
The value of x is 105.
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4. The polygon has 4 sides.
8. An 18-gon has 18 sides.
S = ( n • 2) • 180°
S = ( n − 2) • 180°
= ( 4 − 2) • 180°
= (18 − 2) • 180°
= 2 • 180°
= 16 • 180°
= 360°
= 2880°
The sum of the angle measures is 360°.
The sum of the angle measures is 2880°.
115 + 80 + 90 + x = 360
2880° ÷ 18 = 160°
285 + x = 360
The measure of each angle is 160°.
x = 75
9.
The value of x is 75.
5. The polygon has 5 sides.
S = ( n − 2) • 180°
= (5 − 2) • 180°
= 3 • 180°
= 540°
No line segment connecting two vertices lies outside the
polygon. So, the polygon is convex.
The sum of the angle measures is 540°.
2 x + 145 + 145 + 2 x + 110 = 540
4 x + 400 = 540
10.
4 x = 140
x = 35
The value of x is 35.
6. An octagon has 8 sides.
S = ( n − 2) • 180°
A line segment connecting two vertices lies outside the
polygon. So, the polygon is concave.
= (8 − 2) • 180°
= 6 • 180°
11.
= 1080°
The sum of the angle measures is 1080°.
1080° ÷ 8 = 135°
The measure of each angle is 135°.
7. A decagon has 10 sides.
S = ( n − 2) • 180°
= (10 − 2) • 180°
= 8 • 180°
= 1440°
The sum of the angle measures is 1440°.
1440° ÷ 10 = 144°
The measure of each angle is 144°.
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A line segment connecting two vertices lies outside the
polygon. So, the polygon is concave.
5.3 Exercises (pp. 201–203)
Vocabulary and Concept Check
1. A three-sided polygon is a triangle. Because it is a regular
polygon, all sides are congruent. So, the polygon is an
equilateral triangle.
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Chapter 5
2. Because the second figure is not made up entirely of line
6.
segments and the other three are, the second figure does
not belong.
3. Because the first question asks for one angle measure and
the other three ask for the sum of the angle measures, the
first question is different.
A regular pentagon has n = 5 sides.
S = ( n − 2) • 180°
= (5 − 2) • 180°
= 3 • 180°
= 540°
540° ÷ 5 = 108°
The measure of one angle of a regular pentagon is 108°.
S = ( n − 2) • 180°
= (5 − 2) • 180°
= 3 • 180°
= 540°
The sum of the measures of a regular, convex, or concave
pentagon is 540°.
Practice and Problem Solving
4.
Draw two lines that divide the heptagon into 5 triangles.
Because the sum of the measures of each triangle
is 180°, the sum of the angle measures of a heptagon
is 5(180°) = 900°.
7. The shape has 6 sides.
S = ( n − 2) • 180°
= (6 − 2) • 180°
= 4 • 180°
= 720°
The sum of the angle measures is 720°.
8. The shape has 12 sides.
S = ( n − 2) • 180°
= (12 − 2) • 180°
= 10 • 180°
= 1800°
The sum of the angle measures is 1800°.
9. The shape has 8 sides.
S = ( n − 2) • 180°
= (8 − 2) • 180°
= 6 • 180°
Draw one line that divides the quadrilateral into two
triangles. Because the sum of the measures of each
triangle is 180°, the sum of the angle measures of a
quadrilateral is 2(180°) = 360°.
= 1080°
The sum of the angle measures is 1080°.
10. The formula is incorrect. The correct formula has the
product of two less than the number of sides and 180°.
S = ( n − 2) • 180°
5.
= (13 − 2) • 180°
= 11 • 180°
= 1980°
Draw six lines that divide the 9-gon into 7 triangles.
Because the sum of the measures of each triangle
is 180°, the sum of the angle measures of a 9-gon
is 7(180°) = 1260°.
11. no; S = ( n − 2) • 180°
= (5 − 2) • 180°
= 3 • 180°
= 540°
Because the sum of the given angle measures is
120° + 105° + 65° + 150° + 95° = 535° and not 540°,
a pentagon cannot have the given angle measures.
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Chapter 5
12. The polygon has 4 sides.
17. The polygon has 9 sides.
S = ( n − 2) • 180°
S = ( n − 2) • 180°
= ( 4 − 2) • 180°
= (9 − 2) • 180°
= 2 • 180°
= 7 • 180°
= 360°
= 1260°
155 + 25 + 137 + x = 360
1260° ÷ 9 = 140°
317 + x = 360
The measure of each angle is 140°.
x = 43
18. The polygon has 12 sides.
The value of x is 43.
S = ( n − 2) • 180°
13. The polygon has 6 sides.
= (12 − 2) • 180°
S = ( n − 2) • 180°
= 10 • 180°
= (6 − 2) • 180°
= 1800°
= 4 • 180°
1800° ÷ 12 = 150°
= 720°
The measure of each angle is 150°.
90 + 90 + x + x + x + x = 720
180 + 4 x = 720
4 x = 540
x = 135
The value of x is 135.
19. The sum should have been divided by the number of
angles, which is 20, not 18.
3240° ÷ 20 = 162°
The measure of each angle is 162°.
20. a. The bolt has 5 sides. Find the sum of the angle
14. The polygon has 6 sides.
measures.
S = ( n − 2) • 180°
S = ( n − 2) • 180°
= (6 − 2) • 180°
= (5 − 2) • 180°
= 4 • 180°
= 3 • 180°
= 720°
3x + 45 + 135 + x + 135 + 45 = 720
4 x + 360 = 720
4 x = 360
x = 90
The value of x is 90.
15. Find the number of sides.
S = ( n − 2) • 180
1260 = ( n − 2) • 180°
7 = n−2
9 = n
Because the regular polygon has 9 sides, the measure of
each angle is 1260° ÷ 9 = 140°.
16. A triangle has 3 sides.
S = ( n − 2) • 180° = (3 − 2) • 180° = 180°
= 540°
Divide the sum by the number of angles, 5.
540° ÷ 5 = 108°
The measure of each angle is 108°.
b. Sample answer: Since the standard shape of a bolt is a
hexagon, most people have tools to remove a
hexagonal bolt and not a pentagonal bolt. So, a special
tool is needed to remove the bolt from the fire hydrant.
21. The sum of the angles of the polygon is n • 165, where n
is the number of sides.
S = ( n − 2) • 180
n • 165 = ( n − 2) • 180
165n = 180n − 360
−15n = −360
n = 24
The polygon has 24 sides.
180° ÷ 3 = 60°
The measure of each angle is 60°.
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27. The shape is a regular octagon.
22.
S = ( n − 2) • 180°
= (8 − 2) • 180°
= 6 • 180°
= 1080°
A line segment connecting two vertices lies outside the
polygon. So, the polygon is concave.
1080° ÷ 8 = 135°
The measure of each angle is 135°.
28. Sample answer:
23.
45°
45°
270°
29. Find the sum of the angle measures of the heptagon.
No line segment connecting two vertices lies outside the
polygon. So, the polygon is convex.
S = ( n − 2) • 180°
= (7 − 2) • 180°
= 5 • 180°
24.
= 900°
Let x represent the value of each of the three remaining
angle measures, in degrees.
4 • 135 + 3 • x = 900
540 + 3x = 900
A line segment connecting two vertices lies outside the
polygon. So, the polygon is concave.
25. no; In a regular polygon, all the angles are congruent.
Because a concave polygon has at least one angle that
is greater than 180°, all of the angles are not congruent.
So, a concave polygon cannot be regular.
26. Sample answer:
3x = 360
x = 120
The measure of each remaining angle is 120°.
30. a. The polygon has 11 sides.
b. S = ( n − 2) • 180°
= (11 − 2) • 180°
= 9 • 180°
= 1620°
1620° ÷ 11 ≈ 147°
The measure of each angle is about 147°.
Because not all of the angles are congruent, the polygon
is not regular.
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31. a =
(n − 2) • 180 = 180n − 360 = 180 − 360
S
=
n
n
n
n
Sides of a
Regular
Polygon, n
180 −
360
n
Measure of
One Angle
(degrees), a
3
180 −
360
3
60
4
180 −
360
4
90
5
180 −
360
5
108
6
180 −
360
6
120
c. Sample answer: The tessellation is formed using
equilateral triangles and squares.
Fair Game Review
33.
7
180 −
360
7
128.6
8
180 −
360
8
135
9
180 −
360
9
140
10
180 −
360
10
144
As the number of sides n increases by one, the measure
of one angle a increases by 30°, 18°, 12°, 9°, 6°, 5°,
and 4°, respectively. The rate of change is not constant.
So, the table does not represent a linear function.
32. a. Sample answer:
3
x
=
12
4
x • 4 = 12 • 3
34.
4 x = 36
42 = 21x
x = 9
2 = x
The value of x is 9.
35.
x
14
=
21
3
14 • 3 = 21 • x
x
2
=
9
6
x•6 = 9•2
The value of x is 2.
36.
4
x
=
10
15
4 • 15 = 10 • x
6 x = 18
60 = 10 x
x = 3
6 = x
The value of x is 3.
The value of x is 6.
37. D; Because the ratio of tulips to daisies is 3 : 5, the total
number is a multiple of 3 + 5 = 8. The multiples of
8 are 8, 16, 24, 32, … . So, 16 is the only choice that
could be the total number of tulips and daisies.
Study Help (p. 204)
Available at BigIdeasMath.com.
b. Sample answer:
Squares:
Regular hexagons:
Quiz 5.1–5.3 (p. 205)
1. Because 125° + 65° = 190°, the angles are neither
complementary nor supplementary.
2. Because 63° + 27° = 90°, the angles are
complementary.
3. Because 106° + 74° = 180°, the angles are
supplementary.
4. The angles are supplementary. So, the sum of their
measures is 180°.
x + 34 = 180
x = 146
So, x is 146.
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Chapter 5
5. The angles are complementary. So, the sum of their
12. The polygon has 4 sides.
measures is 90°.
S = ( n − 2) • 180°
x + 74 = 90
= ( 4 − 2) • 180°
x = 16
= 2 • 180°
So, x is 16.
= 360°
x + 122 + 134 + 46 = 360
6. The angles are vertical angles. Because vertical angles
are congruent, the angles have the same measure. So, x
is 59.
x + 302 = 360
x = 58
7. x + 60 + 60 = 180
x + 120 = 180
x = 60
The value of x is 58.
13. The polygon has 7 sides.
S = ( n − 2) • 180°
The value of x is 60. All three angles are congruent and
acute, and all three sides are congruent. So, it is an acute,
isosceles, equilateral, and equiangular triangle.
= (7 − 2) • 180°
= 5 • 180°
= 900°
8. x + 25 + 40 = 180
x + 130 + 140 + 120 + 115 + 154 + 115 = 900
x + 65 = 180
x + 774 = 900
x = 115
x = 126
The value of x is 115. The triangle has an obtuse angle.
So, it is an obtuse triangle.
9. x + x + 90 = 180
The value of x is 126.
14. The polygon has 5 sides.
S = ( n − 2) • 180°
2 x + 90 = 180
2 x = 90
= (5 − 2) • 180°
x = 45
= 3 • 180°
The value of x is 45. The triangle has a right angle and
two angles are congruent. So, it is a right isosceles
triangle.
= 540°
x + 40 + 4 x + 40 + 110 = 540
5 x + 190 = 540
10. The polygon has 8 sides.
5 x = 350
S = ( n − 2) • 180°
x = 70
= (8 − 2) • 180°
= 6 • 180°
= 1080°
The value of x is 70.
15. Because the 115° angle and ∠1 are supplementary, the
sum of their measures is 180°.
The sum of the angle measures is 1080°.
x + 115 = 180
11.
x = 65
So, the measure of ∠1 is 65°.
Because the 115° angle and ∠ 2 are vertical angles, the
angles have the same measure. So, the measure of ∠ 2
is 115°. Because ∠1 and ∠ 3 are vertical angles, the
A line segment connecting two vertices lies outside the
polygon. So, the polygon is concave.
measure of ∠ 3 is 65°.
16.
S = ( n − 2) • 180°
4140 = ( n − 2) • 180°
23 = n − 2
25 = n
The polygon has 25 sides.
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17. x + x + 67.4 = 180
2. yes; 66 + 90 + x = 180
y + 90 + 24 = 180
2 x + 67.4 = 180
156 + x = 180
y + 114 = 180
2 x = 112.6
The triangles have the same angle measures, 66°, 24°,
and 90°. So, they are similar.
Two sides are congruent and all three angles are acute.
So, it is an acute isosceles triangle.
3.
Section 5.4
5.4 Activity (pp. 206 –207)
1. Answer should include, but is not limited to: Students will
describe how to extend two sides of
XYZ and then
draw the third side of the new triangle parallel to the third
side of XYZ . Students will measure the side lengths
and calculate ratios of corresponding side lengths. Ratios
may not be exactly equal due to rounding during
measuring. Students should conclude that if two triangles
have the same angle measures, then the triangles are
similar.
40
x
=
55
50
40
x
55 •
= 55 •
55
50
x = 44
The distance across the river is 44 feet.
5.4 Exercises (pp. 210 –211)
Vocabulary and Concept Check
1. Because the ratio of the corresponding side lengths in
similar triangles are equal, a proportion can be used to
find a missing measurement.
ABC , DEF and GHI are
35°, 82°, and 63°, and the angle measures of JKL
are 32°, 85°, and 63°. So, JKL does not belong with
2. a. true; By definition of similar.
2. The angle measures of
true; By definition of similar.
b. true; By definition of similar.
false; A square and a rhombus with the same side
lengths are not similar because their corresponding
angles are not congruent.
c. true; Shown in Activity 1.
the other three.
Practice and Problem Solving
3– 4. Answer should include, but is not limited to:
true; The similar quadrilaterals will have the same
shape.
d. true; Shown in Activity 1
false; A square and a rectangle have congruent
corresponding angles but the ratios of their
corresponding side lengths are not equal.
e. true; By definition of similar.
false; A square and a rhombus with the same side
lengths do not have identical shapes.
3. Sample answer: If corresponding side lengths of two
triangles are proportional, then the triangles are similar.
If corresponding angles of two triangles are congruent,
then the triangles are similar.
Sample answer: Construction and architecture use
triangles to form buildings.
5.4 On Your Own (p. 209)
1. no; x + 28 + 80 = 180
y + 28 + 71 = 180
x + 108 = 180
y + 99 = 180
x = 72
y = 66
x = 24
x = 56.3
y = 81
Students should draw a triangle with the same angle
measures as those in the textbook. The ratios of
corresponding side lengths,
student’s triangle length
,
book’s triangle length
should be greater than 1, and should all be
approximately equal. (Ratios may differ slightly due
to rounding.)
5. yes; x + 34 + 39 = 180
y + 107 + 39 = 180
x + 73 = 180
y + 146 = 180
x = 107
y = 34
The triangles have the same angle measures, 34°, 39°,
and 107°. So, they are similar.
6. no; x + 36 + 72 = 180
y + 72 + 75 = 180
x + 108 = 180
y + 147 = 180
x = 72
y = 33
The triangles do not have the same angle measures.
So, they are not similar.
The triangles do not have the same angle measures.
So, they are not similar.
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7. no; x + 64 + 85 = 180
y + 26 + 85 = 180
x + 149 = 180
y + 111 = 180
14. Sample answer: 10 ft
y = 69
x = 31
The triangles do not have the same angle measures.
So, they are not similar.
8. yes; x + 48 + 81 = 180
y + 48 + 51 = 180
x + 129 = 180
y + 99 = 180
y = 81
x = 51
The triangles have the same angle measures 48°, 51°,
and 81°. So, the triangles are similar.
9. The proportion was set up incorrectly. If the first ratio is
a length from the smaller triangle over a length from the
larger triangle, the second ratio must be the same.
16
8
=
x
18
16 x = 144
x
5
3
6
15. a. Because AG, GF, and FE are equal and
AE = 9.48 feet, the length of segment
AG is 9.48 ÷ 3 = 3.16 feet.
AG
BG
=
AE
DE
x
3.16
=
9.48
6
18.96 = 9.48 x
2 = x
So, x is 2 feet.
x = 9
b. The length of segment AF is 3.16 • 2 = 6.32 feet.
10. Because the corresponding angle measure is 50°, the
value of x is 50.
11. Because the corresponding angle measure is 65°, the
value of x is 65.
12. Find the missing dimension using indirect measurement.
x
80
=
300
240
240 x = 24,000
Let y = CF .
CF
DE
=
AF
AE
y
6
=
6.32
9.48
9.48 y = 37.92
y = 4
So, CF is 4 feet.
Fair Game Review
x = 100
You take 100 steps from the pyramids to the treasure.
13. no; Consider the two similar triangles below.
16. Because the equation cannot be rewritten in
slope-intercept form, it is nonlinear.
17. Because the equation is of the form y = mx + b,
it is linear.
18. Because the equation is of the form y = mx + b,
it is linear.
1.5y
y
19. Because the equation cannot be rewritten in
slope-intercept form, it is nonlinear.
x
1.5x
1
A = bh
2
1
A = xy
2
1
A = bh
2
1
A = (1.5 x)(1.5 y )
2
1
A = ( 2.25) xy
2
So, the area of the larger triangle is 2.25 times larger than
the original triangle, which is a 125% increase.
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20. C; Find the slope of each line.
Blue: m =
rise
5
=
run
2
Red: m =
rise
4
=
= 2
run
2
Green: m =
rise
5
=
run
2
Because the slopes of the blue line and the green line
5
are , the slopes are equal.
2
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Chapter 5
Section 5.5
2. ∠1 and ∠ 2 are supplementary.
∠1 + ∠ 2 = 180°
5.5 Activity (pp. 212–213)
1. Two lines are parallel if they do not intersect.
Sample answer: Draw one line. Then, draw two points
that are the same distance from the line. Use these two
points to draw a parallel line.
Angles 1, 3, 5, and 7 are congruent. Angles 1 and 3 and
angles 5 and 7 are vertical angles, which are congruent.
Angles 3 and 7 are congruent because the corresponding
angles the parallel lines form with the transversal are
the same.
Angles 2, 4, 6, and 8 are congruent using the same
reasoning.
2. a. Sample answer: Measure the vertical angles and
corresponding angles and make sure they are
congruent.
b. The studs are parallel lines and the diagonal support
beam is a transversal.
3. a. Because the Sun’s rays are parallel, ∠ C ≅ ∠ F .
Because ∠ A and ∠ D are both right angles,
ABC are
DEF , ∠ B ≅ ∠ E.
∠ A ≅ ∠ D. Because two angles of
congruent to two angles of
Therefore,
b. Because
ABC and DEF are similar triangles.
ABC and DEF are similar triangles,
the ratios of the corresponding side lengths are equal.
So, write and solve a proportion to find the height of
the flagpole.
Height of flagpole
Length of flagpole’s shadow
=
Height of boy
Length of boy’s shadow
x
36
=
5
3
3 x = 180
x = 60
The height of the flagpole is 60 feet.
4. Sample answer: A banister on a staircase has spindles
that are parallel to each other. The base of the banister is
a transversal to the spindles. Each angle created by a
spindle and the base has the same measurement.
5. a. Because the flagpole is not being measured directly,
the process is called “indirect” measurement.
63° + ∠ 2 = 180°
∠ 2 = 117°
So, the measure of ∠ 2 is 117°.
3. The 59° angle is supplementary to both ∠1 and ∠ 3.
∠1 + 59° = 180°
∠1 = 121°
So, the measures of ∠1 and ∠ 3 are 121°.
∠ 2 and the 59° angle are vertical angles. They are
congruent. So, the measure of ∠ 2 is 59°.
Find the remaining angle measures:
Using corresponding angles, the measures of ∠ 4 and
∠ 6 are 121°, and the measures of ∠ 5 and ∠ 7 are 59°.
4. Because all of the letters are slanted at a 65° angle, the
dashed lines are parallel. The piece of tape is the
transversal. Using corresponding angles, the 59° angle is
congruent to the angle that is supplementary to ∠1, as
shown. So, the measure of ∠1 is 180° − 65° = 115°.
5. ∠ 3 and ∠ 4 are supplementary angles.
∠ 3 + ∠ 4 = 180°
∠ 3 + 84° = 180°
∠ 3 = 96°
So, the measure of ∠ 3 is 96°.
6. ∠ 4 and ∠ 5 are alternate interior angles. Because the
angles are congruent, the measure of ∠ 5 is 84°.
7. ∠ 4 and ∠ 5 are alternate interior angles and ∠ 5 and
∠ 6 are supplementary. So, ∠ 4 and ∠ 6 are
supplementary.
∠ 4 + ∠ 6 = 180°
84° + ∠ 6 = 180°
∠ 6 = 96°
So, the measure of ∠ 6 is 96°.
b–c. Answer should include, but is not limited to: The
student will use indirect measurement to measure
the height of something outside. The student will
include a diagram of the process used with all
measurements and calculated lengths labeled.
5.5 On Your Own (pp. 214 –216)
1. ∠1 and the 63° angle are corresponding angles. They are
congruent. So, the measure of ∠1 is 63°.
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11. Because ∠1 and ∠ 2 are corresponding angles, the
5.5 Exercises (pp. 217–219)
measure of ∠ 2 is 60°.
Vocabulary and Concept Check
1. Sample answer:
p
q
t
2. Because ∠ 2 and ∠ 6 are corresponding angles and
∠ 6 and ∠ 8 are vertical angles, ∠ 2 ≅ ∠ 6 ≅ ∠ 8.
Because ∠ 5 is supplementary to ∠ 2, ∠ 6, and ∠ 8, the
statement “The measure of ∠ 5 ” does not belong with
the other three.
Practice and Problem Solving
12. Sample answer: The yard lines on a football field are
parallel. The lampposts on a road are parallel.
13. ∠1, ∠ 3, ∠ 5, and ∠ 7 are congruent. ∠ 2, ∠ 4, ∠ 6
and ∠ 8 are congruent.
14. You need to know at least one angle measure. If you
know the measure of ∠1, then ∠ 3, ∠ 5, and ∠ 7 can be
found because they are congruent to ∠1. ∠ 2, ∠ 4, ∠ 6,
and ∠ 8 can also be found because they are
supplementary to ∠1.
15. ∠1 and the 61° angle are corresponding angles. They are
3. Lines m and n are parallel.
congruent. So, the measure of ∠1 is 61°.
4. Line t is the transversal.
∠1 is supplementary to both ∠ 2 and ∠ 4.
5. 8 angles are formed by the transversal.
6. ∠1, ∠ 3, ∠ 5, and ∠ 7 are all congruent. ∠ 2, ∠ 4, ∠ 6,
and ∠ 8 are all congruent.
7. ∠1 and the 107° angle are corresponding angles. They
∠1 + ∠ 2 = 180°
61° + ∠ 2 = 180°
∠ 2 = 119°
So, the measures of ∠ 2 and ∠ 4 are 119°.
∠1 and ∠ 3 are vertical angles. They are congruent.
are congruent. So, the measure of ∠1 is 107°.
So, the measure of ∠ 3 is 61°.
∠1 and ∠ 2 are supplementary.
Using corresponding angles, the measures of ∠ 5
∠1 + ∠ 2 = 180°
107° + ∠ 2 = 180°
∠ 2 = 73°
So, the measure of ∠ 2 is 73°.
8. ∠ 3 and the 95° angle are corresponding angles. They are
congruent. So, the measure of ∠ 3 is 95°.
and ∠ 7 are 119°, and the measure of ∠ 6 is 61°.
16. The 99° angle is supplementary to both ∠1 and ∠ 3.
∠1 + 99° = 180°
∠1 = 81°
So, the measures of ∠1 and ∠ 3 are 81°.
∠ 2 and the 99° angle are vertical angles. They are
∠ 3 and ∠ 4 are supplementary.
congruent. So, the measure of ∠ 2 is 99°.
∠ 3 + ∠ 4 = 180°
Using corresponding angles, the measures of ∠ 4 and
95° + ∠ 4 = 180°
∠ 4 = 85°
So, the measure of ∠ 4 is 85°.
9. ∠ 5 and the 49° angle are corresponding angles. They are
congruent. So, the measure of ∠ 5 is 49°.
∠ 5 and ∠ 6 are supplementary.
∠ 5 + ∠ 6 = 180°
49° + ∠ 6 = 180°
∠ 6 = 131°
So, the measure of ∠ 6 is 131°.
∠ 6 are 99°, and the measures of ∠ 5 and ∠ 7 are 81°.
17. The right angle is supplementary to both ∠1 and ∠ 3.
90° + ∠1 = 180°
∠1 = 90°
So, the measures of ∠1 and ∠ 3 are 90°.
∠ 2 and the right angle are vertical angles. They are
congruent. So, the measure of ∠ 2 is 90°.
Using corresponding angles, the measures of ∠ 4, ∠ 5,
∠ 6, and ∠ 7 are 90°.
10. The lines are not parallel, so corresponding angles
∠ 5 and ∠ 6 are not congruent.
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18. Using corresponding angles, ∠1 is congruent to
∠ 8, which is supplementary to ∠ 4.
∠1 + ∠ 4 = 180°
124° + ∠ 4 = 180°
∠ 4 = 56°
So, if the measure of ∠1 = 124°, then the measure of
∠ 4 = 56°.
26. Using vertical angles, ∠1 is congruent to ∠ 3, and ∠ 3
and ∠ 7 are congruent because they are alternate exterior
angles. So, ∠1 is congruent to ∠ 7. Using corresponding
angles, ∠1 is congruent to ∠ 5, and ∠ 5 and ∠ 7 are
congruent because they are vertical angles. So, ∠1 is
congruent to ∠ 7.
27. The 50° angle is congruent to the alternate interior angle
∠ 2 + ∠ 3 = 180°
formed by the intersection of line a and line c. This angle
is congruent to the corresponding angle formed by the
intersection of line a and line d. This angle is
supplementary to the x° angle. So, the 50° angle is
supplementary to the x° angle.
48° + ∠ 3 = 180°
50 + x = 180
∠ 3 = 132°
x = 130
19. Using corresponding angles, ∠ 2 is congruent to ∠ 7,
which is supplementary to ∠ 3.
So, if the measure of ∠ 2 = 48°, then the measure of
∠ 3 = 132°.
20. Because ∠ 4 and ∠ 2 are alternate interior angles, ∠ 4 is
congruent to ∠ 2. So, if the measure of ∠ 4 = 55°, then
the measure of ∠ 2 = 55°.
21. Because ∠ 6 and ∠ 8 are alternate exterior angles, ∠ 6 is
congruent to ∠ 8. So, if the measure of ∠ 6 = 120°, then
the measure of ∠ 8 = 120°.
22. Using alternate exterior angles, ∠ 7 is congruent to ∠ 5,
which is supplementary to ∠ 6.
So, the value of x is 130.
28. The 115° angle is congruent to the corresponding angle
formed by the intersection of line b and line d. This angle
is congruent to the x° angle because they are alternate
exterior angles. Because the 115° angle is congruent to
the x° angle, the value of x is 115.
29. a. no; The lines look like they will intersect somewhere
to the left of the illustration.
b. Answer should include, but is not limited to: The
student will draw an optical illusion using parallel
lines.
30. a. m + 64° + m = 180°
∠ 7 + ∠ 6 = 180°
2m + 64° = 180°
50.5° + ∠ 6 = 180°
2m = 116°
m = 58°
∠ 6 = 129.5°
So, if the measure of ∠ 7 = 50.5°, then the measure of
So, the value of m is 58.
∠ 6 = 129.5°.
23. Using alternate interior angles, ∠ 3 is congruent to ∠1,
which is supplementary to ∠ 2.
m°
n°
∠ 3 + ∠ 2 = 180°
118.7° + ∠ 2 = 180°
∠ 2 = 61.3°
So, if the measure of ∠ 3 = 118.7°, then the measure of
∠ 2 = 61.3°.
24. Because the two rays of sunlight are parallel, ∠1 and
∠ 2 are alternate interior angles. Because the angles are
congruent, the measure of ∠1 is 40°.
Because the sides of the table are parallel and ∠ m
and ∠ n are alternate interior angles, ∠ m is congruent
to ∠ n. The measure of ∠ n is 58°.
58° + x° + n° = 180°
58° + x° + 58° = 180°
x° + 116° = 180°
x° = 64°
So, the value of x is 64.
25. Because the lines are perpendicular, the lines intersect at
right angles. So, all of the angles formed are right angles.
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b. The goal is slightly wider than the hockey puck. So,
there is some leeway allowed for the measure of x.
By studying the diagram, you can see that x cannot be
much greater. However, x can be a little less and still
have the hockey puck go into the goal.
Fair Game Review
31. 4 + 32 = 4 + 9 = 13
32. 5( 2) − 6 = 5 • 4 − 6 = 20 − 6 = 14
2
33. 11 + ( −7) − 9 = 11 + 49 − 9 = 60 − 9 = 51
2
6. Because the 82° angle and ∠ 6 are vertical angles, the
angles are congruent. So, the measure of ∠ 6 is 82°.
7. Because the 82° angle and ∠ 4 are corresponding angles,
the angles are congruent. So, the measure of ∠ 4 is 82°.
8. Using corresponding angles, the 82° angle is congruent
to ∠ 4, which is supplementary to ∠1. Because the
82° angle and ∠1 are supplementary, the measure of ∠1
is 180° − 82° = 98°.
9. Using alternate exterior angles, ∠1 is congruent to ∠ 7.
So, if the measure of ∠1 = 132°, then the measure of
34. 8 ÷ 2 2 + 1 = 8 ÷ 4 + 1 = 2 + 1 = 3
35. B;
V = π r 2h
20π = π r 2 • 5
4 = r2
∠ 7 = 123°.
10. Using corresponding angles, ∠ 2 is congruent to ∠ 6,
which is supplementary to ∠ 5. Because ∠ 2 and ∠ 5
are supplementary, the measure of ∠ 5 is
180° − 58° = 122°. So, if the measure of ∠ 2 = 58°,
2 = r
So, the radius of the base is 2 inches.
Quiz 5.4–5.5 (p. 220)
1. x + 46 + 95 = 180
x + 141 = 180
x = 39
y + 39 + 46 = 180
y + 85 = 180
y = 95
The triangles have the same angle measures, 39°, 46°,
and 95°. So, they are similar.
then the measure of ∠ 5 = 122°.
11. Because ∠ 5 and ∠ 3 are alternate interior angles, ∠ 5 is
congruent to ∠ 3. So, if the measure of ∠ 5 = 119°, then
the measure of ∠ 3 = 119°.
12. Because ∠ 4 and ∠ 6 are alternate exterior angles, ∠ 4 is
congruent to ∠ 6. So, if the measure of ∠ 4 = 60°, then
the measure of ∠ 6 = 60°.
13. Using corresponding angles, the 72° angle is congruent
to the angle that is supplementary to ∠1 and ∠ 2. So,
the 72° angle is supplementary to both ∠1 and ∠ 2.
2. x + 40 + 51 = 180
∠1 + 72° = 180°
x + 91 = 180
∠1 = 108°
x = 89
y + 40 + 79 = 180
y + 119 = 180
y = 61
The triangles do not have the same angle measures.
So, they are not similar.
3. Because the corresponding angle measure is 95°, the
value of x is 95.
4. Because the corresponding angle measure is 26°, the
value of x is 26.
5. Because the 82° angle and ∠ 2 are alternate exterior
angles, the angles are congruent. So, the measure of ∠ 2
is 82°.
So, the measures of ∠1 and ∠ 2 are 108.
ABC have side lengths a, b, and c. The
perimeter of ABC is a + b + c. Let A′B′C ′ have
side lengths 2a, 2b, and 2c. The perimeter of A′B′C ′ is
14. yes; Let
2a + 2b + 2c = 2( a + b + c), which is 2 times the
perimeter of
ABC.
Chapter 5 Review (pp. 221–223)
1. The angles are complementary angles. So, the sum of
their measures is 90°.
x + 69 = 90
x = 21
So, x is 21.
2. Because the angles are vertical angles, they are
congruent. So, x is 84.
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Chapter 5
3. x + 49 + 90 = 180
8.
x + 139 = 180
x = 41
The value of x is 41. The triangle has one right angle.
So, it is a right triangle.
No line segment connecting two vertices lies outside the
polygon. So, the polygon is convex.
4. x + 35 + 110 = 180
x + 145 = 180
9.
x = 35
The value of x is 35. The triangle has two congruent
sides, so it is an isosceles triangle. The triangle has one
obtuse angle. So, it is an obtuse isosceles triangle.
A line segment connecting two vertices lies outside the
polygon. So, the polygon is concave.
5. The polygon has 4 sides.
S = ( n − 2) • 180°
10.
= ( 4 − 2) • 180°
= 2 • 180°
= 360°
60 + 128 + 95 + x = 360
283 + x = 360
A line segment connecting two vertices lies outside the
polygon. So, the polygon is concave.
x = 77
The value of x is 77.
6. The polygon has 7 sides.
11. yes; x + 68 + 90 = 180
y + 22 + 90 = 180
x + 158 = 180
y + 112 = 180
S = ( n − 2) • 180°
x = 22
= (7 − 2) • 180°
y = 68
The triangles have the same angle measures, 22°, 68°,
and 90°. So, they are similar.
= 5 • 180°
12. Because the corresponding angle measure is 50°, the
= 900
135 + 125 + 135 + 105 + 150 + 140 + x = 900
790 + x = 900
x = 110
value of x is 50.
13. The 140° angle and ∠ 8 are alternate exterior angles.
They are congruent. So, the measure of ∠ 8 is 140°.
The value of x is 110.
14. The 140° angle and ∠ 5 are corresponding angles.
7. The polygon has 6 sides.
They are congruent. So, the measure of ∠ 5 is 140°.
S = ( n − 2) • 180°
15. The 140° angle and ∠ 3 are supplementary. So, the
= (6 − 2) • 180°
measure of ∠ 3 is 180° − 140° = 40°. ∠ 3 and ∠ 7 are
= 4 • 180°
= 720°
100 + 120 + 60 + 2 x + 65 + x = 720
3 x + 345 = 720
3 x = 375
x = 125
The value of x is 125.
corresponding angles. They are congruent. So, the
measure of ∠ 7 is 40°.
16. The 140° angle and ∠ 2 are supplementary. So, the
measure of ∠ 2 is 180° − 140° = 40°.
Chapter 5 Test (p. 224)
1. Because the angles are vertical angles, they are
congruent. So, the value of x is 113.
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2. The angles are complementary angles. So, the sum of
their measures is 90°.
9. no; x + 61 + 70 = 180
y + 39 + 70 = 180
x + 131 = 180
y + 109 = 180
x + 56 = 90
y = 71
x = 49
x = 34
The triangles do not have the same angle measures.
So, they are not similar.
So, the value of x is 34.
3. The angles are supplementary angles. So, the sum of their
measures is 180°.
10. Because the corresponding angle measure is 55°,
the value of x is 55.
x + 74 = 180
11. ∠1 and the 47° angle are supplementary. So, the measure
x = 106
of ∠1 is 180° − 47° = 133°.
So, the value of x is 106.
12. ∠1 and the 47° angle are supplementary. So, the measure
4. x + 23 + 129 = 180
of ∠1 is 180° − 47° = 133°. ∠1 and ∠ 8 are alternate
x + 152 = 180
x = 28
The value of x is 28. The triangle has one obtuse angle.
So, it is an obtuse triangle.
exterior angles. They are congruent. So, the measure
of ∠ 8 is 133°.
13. ∠ 4 and the 47° angle are supplementary. So, the
measure of ∠ 4 is 180° − 47° = 133°.
5. x + 44 + 68 = 180
x + 112 = 180
14. ∠1 and the 47° angle are supplementary. So, the measure
x = 68
of ∠1 is 180° − 47° = 133°. ∠1 and ∠ 5 are
The value of x is 68. The triangle has two congruent
angles and three acute angles. So, it is an acute isosceles
triangle.
corresponding angles. They are congruent. So, the
measure of ∠ 5 is 133°.
15. The triangles are similar, so the ratios of the
6. x + x + x = 180
corresponding side lengths are equal.
3 x = 180
x = 60
The value of x is 60. The triangle has three congruent and
acute angles and three congruent sides. So, it is an acute,
isosceles, equilateral, and equiangular triangle.
7.
d
80
=
105
140
140d = 8400
d = 60
The distance across the pond is 60 meters.
Chapter 5 Standardized Test Practice (pp. 225 –227)
1. 147; Find the sum of the angle measures.
S = ( n − 2) • 180° = (11 − 2) • 180° = 1620°
No line segment connecting two vertices lies outside the
polygon. So, the polygon is convex.
8. The polygon has 5 sides.
Divide the sum by the number of angles, 11.
1620° ÷ 11 ≈ 147°
The measure of each angle is about 147°.
2. B;
S = ( n − 2) • 180°
C = 11 + 1.6t
C − 11 = 1.6t
= (5 − 2) • 180°
C − 11
= t
1.6
= 3 • 180°
= 540°
2 x + 2 x + 125 + 125 + 90 = 540
4 x + 340 = 540
The formula in terms of t is t =
C − 11
.
1.6
4 x = 200
x = 50
The value of x is 50.
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Chapter 5
3. F; Find the slope using the points (0, −5) and (5, 0).
slope =
rise
5
=
=1
run
5
The line crosses the y-axis at (0, −5), so the y-intercept
is −5. So, the equation of the line is y = x − 5.
4. 152; The angles are supplementary. So, the sum of the
angle measures is 180°.
9. G; Because the line crosses the y-axis at (0, 5), the
y-intercept is 5. So, an equation of the line
2
is y = x + 5.
5
10. D; Because the temperature rises 98° − 95° = 3°F,
the Heat Index is 3 • 4 + 122 = 12 + 122 = 134°F.
11. G;
∠1 + ∠ 2 = 180°
5 x − 3 = 11
28° + ∠ 2 = 180°
5 x = 14
∠ 2 = 152°
5. D; Because the equation x + y = 1 is the only equation
of the form Ax + By = C , the equation is linear.
6. H; Because there are 2000 computers, let A + d = 2000
represent one equation. Because each laptop weighs
8 pounds, 8A, and each desktop computer weighs
20 pounds, 20d, and the whole shipment is
34,000 pounds, let 8A + 2d = 34,000 represent the
second equation.
7. A; The function is represented by the points ( −5, 2),
x = 2.8
12. B; The ratios of the corresponding side lengths are equal.
x
5
=
6
10
10 x = 30
x = 3
The length is 3 cm.
13. H; The lines intersect at the point ( 4, 2), so the solution
of the system is ( 4, 2).
(0, 0), and (5, −2). The domain is the set of all x-values,
and the x-values are −5, 0, and 5. So, the domain is
−5, 0, 5.
8. Part A:
S = ( n − 2) • 180°
Part B:
S = ( n − 2) • 180°
= ( 4 − 2) • 180°
= 2 • 180°
= 360°
x + 100 + 90 + 90 = 360
x + 280 = 360
x = 80
The measure of the fourth angle is 80°.
Part C:
Sample answer:
Divide the pentagon into 3 triangles. Because the sum of
the angles of a triangle is 180°, the sum of the angles in a
pentagon is 3 • 180° = 540°.
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