Name: ________________________ Class: ___________________ Date: __________ ID: A
1
Geometry End Of The Year Final Exam Review
Calculate the distance between each given pair of points. Reduce the radical.
1. (3, 1) and (6, 5)
2. (2, 8) and (4, 3)
3.
(−6,4)
and
(5,−1)
4.
(9,−2)
and
(2,−9)
5.
(−5,−8)
and
(−2,−9)
Determine the midpoint of a line segment with each set of given endpoints.
6. (8, 0) and (4, 6)
7.
(6,−3)
and
(−4,5)
8.
(−10,−1)
and (0, 4)
Vocabulary - Define the term in your own words.
9. angle bisector
Determine whether each pair of lines are parallel, perpendicular, or neither. Explain your reasoning.
10. line n:
y = −2x − 4
line m:
y = −2x + 8
11. line p:
y = 3x + 5
line q:
y =
1
3
x + 5
12. line r:
y = −5x + 12
line s:
y =
1
5
x − 6
13. line n:
y = 6x + 2
line m:
y = −6x − 2
Name: ________________________ ID: A
2
14. line p:
y
−
x
=
4
line q:
2x + y = 8
15. line r:
2y + x = 6
line s:
3x + 6y = 12
Determine an equation for the parallel line described. Write your answer in both point-slope form and
slope-intercept form.
16. What is the equation of a line parallel to
y = 7x − 8
that passes through
(5,−2)
?
Determine an equation for the perpendicular line described. Write your answer in both point-slope
form and slope-intercept form.
17. What is the equation of a line perpendicular to
y = −3x + 4
that passes through
(−1,6)
?
Use the given information to determine the measures of the angles in each pair.
18. The measure of the complement of an angle is three times the measure of the angle. What is the measure of
each angle?
19. The measure of the supplement of an angle is one fourth the measure of the angle. What is the measure of
each angle?
Name each pair of vertical angles.
20.
Identify the property demonstrated in each example.
21.
GH = MN and MN = OP, soGH = OP
22.
m∠1 = 134° and m∠2 = 134°, so m∠1 = m∠2
Name: ________________________ ID: A
3
Write the given proof as the indicated proof.
23. Write the two-column proof of the Congruent Supplement Theorem as a paragraph proof.
Given:
∠1
is supplementary to
∠2
,
∠3
is supplementary to
∠4
, and
∠2 ≅ ∠4
Prove:
∠1 ≅ ∠3
Statements Reasons
1.
∠1
is supplementary to
∠2
1. Given
2.
∠3
is supplementary to
∠4
2. Given
3.
∠2 ≅ ∠4
3. Given
4.
m∠2 = m∠4
4. Definition of congruent angles
5.
m∠1 + m∠2 = 180°
5.
6.
m∠3 + m∠4 = 180°
6. Definition of supplementary angles
7.
m∠1 + m∠2 = m∠3+ m∠4
7. Substitution Property
8.
m∠1 + m∠2 = m∠3+ m∠2
8.
9.
m∠1 = m∠3
9. Subtraction Property of Equality
10.
∠1 ≅ ∠3
10. Definition of congruent angles
Write congruence statements for the pairs of corresponding angles in each figure.
24.
Name: ________________________ ID: A
4
Prove each statement using the indicated type of proof.
25. Use a two-column proof to prove the Alternate Exterior Angles Theorem. In your proof, use the following
information and refer to the diagram.
Given:
r Ä s,
t is a transversal
Prove:
∠4 ≅ ∠5
26. Use a two-column proof to prove the Alternate Interior Angles Converse Theorem. In your proof, use the
following information and refer to the diagram.
Given:
∠2 ≅ ∠7
, k is a transversal
Prove:
m Ä n
Translate the given triangle such that one vertex of the image is located at the origin and label the
vertices of the translated image. Then, determine its perimeter. Round your answer to the nearest
hundredth, if necessary.
27. triangle JKL
Name: ________________________ ID: A
5
Translate the given trapezoid such that one vertex of the image is located at the origin and label the
vertices of the translated image. Then, determine the perimeter or area. Round your answer to the
nearest hundredth, if necessary.
28. perimeter of trapezoid ABCD
29. area of trapezoid WXYZ
Name: ________________________ ID: A
6
Translate the given composite figure such that one vertex of the image is located at the origin and label
the vertices of the translated image. Then, determine the perimeter or area. Round your answer to the
nearest hundredth, if necessary.
30. area of figure ABCDEFGH
Calculate the volume of each cone. Use 3.14 for
π
.
31.
Calculate the volume of each sphere. Use
π
. Round decimals to the nearest tenth, if necessary.
32. r = 6 inches
Name: ________________________ ID: A
7
Calculate the volume of each cylinder. Use
π
.
33.
List the side lengths from shortest to longest for each diagram.
34.
35.
Solve for x in each diagram.
36.
37.
Name: ________________________ ID: A
8
38.
Without measuring the angles, list the angles of each triangle in order from least to greatest measure.
39.
Determine whether it is possible to form a triangle using each set of segments with the given
measurements. Explain your reasoning.
40. 3 inches, 2.9 inches, 5 inches
41. 4 meters, 5.1 meters, 12.5 meters
42. 13.8 kilometers, 6.3 kilometers, 7.5 kilometers
Determine the length of the hypotenuse of each
45° − 45° − 90°
triangle.
Write your answer as a radical in simplest form.
43.
Use the given information to answer each question.
44. Soren is flying a kite on the beach. The string forms a
45°
angle with the ground. If he has let out 16 meters
of line, how high above the ground is the kite?
Name: ________________________ ID: A
9
Given the length of the short leg of a
30
°
−
60
°
−
90
°
triangle, determine the lengths of the long leg and
the hypotenuse. Write your answers as radicals in simplest form.
45.
Given the length of the hypotenuse of a
30° − 60° − 90°
triangle, determine the lengths of the two legs.
Write your answers as radicals in simplest form.
46.
Vocabulary - Define the term in your own words.
47. similar triangles
Determine whether each pair of triangles is similar. Explain your reasoning.
48.
Name: ________________________ ID: A
10
Use the diagram and given information to write a statement that can be justified using the
Proportional Segments Theorem, Triangle Proportionality Theorem, or its Converse. State the
theorem used.
49.
Explain how you know that each pair of triangles are similar.
50.
Use indirect measurement to calculate the missing distance.
51. Minh wanted to measure the height of a statue. She lined herself up with the statue’s shadow so that the tip of
her shadow met the tip of the statue’s shadow. She marked the spot where she was standing. Then, she
measured the distance from where she was standing to the tip of the shadow, and from the statue to the tip of
the shadow.
What is the height of the statue?
Name: ________________________ ID: A
11
52. Dimitri wants to measure the height of a palm tree. He lines himself up with the palm tree’s shadow so that
the tip of his shadow meets the tip of the palm tree’s shadow. Then, he asks a friend to measure the distance
from where he was standing to the tip of his shadow and the distance from the palm tree to the tip of its
shadow.
What is the height of the palm tree?
Determine the coordinates of each translated image without graphing.
53. The vertices of triangle RST are R (0, 3), S (2, 7), and
T (3, − 1)
. Translate the triangle 5 units to the left and 3
units up to form triangle
R
′
S
′
T
′
.
54. The vertices of quadrilateral WXYZ are
W (−10, 8), X(−2, − 1), Y (0, 0)
, and Z (3, 7). Translate the
quadrilateral 5 units to the right and 8 units down to form quadrilateral
W
′
X
′
Y
′
Z
′
.
Determine the coordinates of each rotated image without graphing.
55. The vertices of triangle ABC are A (5, 3), B (2, 8), and
C (−4, 5)
. Rotate the triangle about the origin
90°
counterclockwise to form triangle
A
′
B
′
C
′
.
56. The vertices of parallelogram HJKL are
H(2, − 6), J (3, − 1), K (7, − 1)
, and
L(6, − 6)
. Rotate the
parallelogram about the origin
90°
counterclockwise to form parallelogram
H
′
J
′
K
′
L
′
.
Determine the coordinates of each reflected image without graphing.
57. The vertices of rectangle DEFG are
D (−7, 1), E (−7, 8), F (1, 8)
, and G (1, 1). Reflect the rectangle over the
y-axis to form rectangle
D
′
E
′
F
′
G
′
.
58. The vertices of parallelogram HJKL are
H (2, − 6), J (3, − 1), K (7, − 1)
, and
L (6, − 6)
. Reflect the
parallelogram over the x-axis to form parallelogram
H
′
J
′
K
′
L
′
.
Name: ________________________ ID: A
12
Determine the ratio
opposite
hypotenuse
using
∠A
as the reference angle in each triangle. Write your answers
as fractions in simplest form.
59.
60.
61.
62.
Name: ________________________ ID: A
13
Determine the ratio
adjacent
hypotenuse
using
∠A
as the reference angle in each triangle. Write your answers
as fractions in simplest form.
63.
64.
Determine the ratios
opposite
hypotenuse
,
adjacent
hypotenuse
, and
opposite
adjacent
using
∠A
as the reference angle in each
triangle. Write your answers as fractions in simplest form.
65.
Use a tangent ratio or a cotangent ratio to calculate the missing length of each triangle.
66.
Name: ________________________ ID: A
14
Calculate the measure of angle X for each triangle.
67.
Solve each problem.
68. To calculate the height of a tree, a botanist makes the following diagram. What is the height of the tree?
69. A lifeguard is sitting on an observation chair at a pool. The lifeguard’s eye level is 6.2 feet from the ground.
The chair is 15.4 feet from a swimmer. Calculate the measure of the angle formed when the lifeguard looks
down at the swimmer.
70. A surveyor is looking up at the top of a building that is 140 meters tall. His eye level is 1.4 meters above the
ground, and he is standing 190 meters from the building. Calculate the measure of the angle from his eyes to
the top of the building.
Name: ________________________ ID: A
15
Use a sine ratio or a cosecant ratio to calculate the missing length of each triangle.
71.
Calculate the measure of angle X for each triangle.
72.
Solve the problem.
73. Jerome is flying a kite on the beach. The kite is attached to a 100-foot string and is flying 45 feet above the
ground, as shown in the diagram. Calculate the measure of the angle formed by the string and the ground.
Calculate the measure of angle X for each triangle.
74.
Name: ________________________ ID: A
16
75.
Calculate the sum of the interior angle measures of each polygon.
76. A polygon has 13 sides.
77. A polygon has 20 sides.
78. A polygon has 25 sides.
For each regular polygon, calculate the measure of each of its interior angles.
79.
80.
Calculate the number of sides for the polygon.
81. The measure of each angle of a regular polygon is
108°
.
Given the regular polygon, calculate the measure of each of its exterior angles.
82. What is the measure of each exterior angle of a regular pentagon?
83. What is the measure of each exterior angle of a regular 12-gon?
Name: ________________________ ID: A
17
Determine the measure of the minor arc.
84.
CD
Determine the measure of each central angle.
85.
m∠XYZ
86.
m∠KWS
Name: ________________________ ID: A
18
Determine the measure of each inscribed angle.
87.
m∠XYZ
88.
m∠SGI
Determine the measure of each intercepted arc.
89.
mQW
Name: ________________________ ID: A
19
Calculate the measure of the angle.
90. The measure of
∠EOG
is
128°
. What is the measure of
∠EFG
?
Use the diagram shown to determine the measure of each angle or arc.
91. Determine
m∠KLJ
.
mKM = 120°
mJN = 100°
92. Determine
mRS
.
mUV = 30°
m∠RTS = 80°
Name: ________________________ ID: A
20
93. Determine
m
∠
D
.
mZXC = 150°
mCB = 30°
Vocabulary - Answer each question.
94. How are inscribed polygons and circumscribed polygons different?
Draw a triangle inscribed in the circle through the three points. Then determine if the triangle is a
right triangle.
95.
Name: ________________________ ID: A
21
Draw a quadrilateral inscribed in the circle through the given four points. Then determine the
measure of the indicated angle.
96. In quadrilateral ABCD,
m∠B = 81°
. Determine
m∠D
.
Use the given arc measures to determine the measures of the indicated angles.
97.
mXY = 20°
mYZ = 50°
m∠XVY =
________
m∠YVZ =
________
Name: ________________________ ID: A
22
98.
mBE = 20°
mCD = 70°
m∠A =
_______
99.
mJK = 164°
mIL = 42°
m∠H =
_______
100.
mMQ = 50°
mNP = 12°
m∠O =
_______
ID: A
1
Geometry End Of The Year Final Exam Review
Answer Section
1. ANS:
x
1
= 3, y
1
= 1, x
2
= 6, y
2
= 5
d = (x
2
− x
1
)
2
+ (y
2
− y
1
)
2
d = (6 − 3)
2
+ (5− 1)
2
d = 3
2
+ 4
2
d = 9 + 16
d = 25
d = 5
REF: 1.2
2. ANS:
x
1
= 2, y
1
= 8, x
2
= 4, y
2
= 3
d = (x
2
− x
1
)
2
+ (y
2
− y
1
)
2
d = (4 − 2)
2
+ (3− 8)
2
d = 2
2
+ (−5)
2
d = 4 + 25
d = 29
d ≈ 5.4
REF: 1.2
ID: A
2
3. ANS:
x
1
= −6, y
1
= 4, x
2
= 5, y
2
= −1
d = (x
2
− x
1
)
2
+ (y
2
− y
1
)
2
d = [5 − (−6)]
2
+ [(−1) − 4]
2
d = 11
2
+ (−5)
2
d = 121+ 25
d = 146
d ≈ 12.1
REF: 1.2
4. ANS:
x
1
= 9, y
1
= −2, x
2
= 2, y
2
= −9
d = (x
2
− x
1
)
2
+ (y
2
− y
1
)
2
d = (2 − 9)
2
+ [(−9) − (−2)]
2
d = (−7)
2
+ (−7)
2
d = 49 + 49
d = 98
d ≈ 7 2
REF: 1.2
5. ANS:
x
1
= −5, y
1
= −8, x
2
= −2, y
2
= −9
d = (x
2
− x
1
)
2
+ (y
2
− y
1
)
2
d = [(−2) − (−5)]
2
+ [(−9) − (−8)]
2
d = 3
2
+ (−1)
2
d = 9 + 1
d = 10
d ≈ 3.2
REF: 1.2
ID: A
3
6. ANS:
x
1
= 8, y
1
= 0
x
2
= 4, y
2
= 6
x
1
+ x
2
2
,
y
1
+ y
2
2
Ê
Ë
Á
Á
Á
Á
Á
Á
Á
Á
ˆ
¯
˜
˜
˜
˜
˜
˜
˜
˜
=
8 + 4
2
,
0 + 6
2
Ê
Ë
Á
Á
Á
Á
Á
Á
ˆ
¯
˜
˜
˜
˜
˜
˜
=
12
2
,
6
2
Ê
Ë
Á
Á
Á
Á
Á
Á
ˆ
¯
˜
˜
˜
˜
˜
˜
= (6, 3)
REF: 1.3
7. ANS:
x
1
= 6, y
1
= −3
x
2
= −4, y
2
= 5
x
1
+ x
2
2
,
y
1
+ y
2
2
Ê
Ë
Á
Á
Á
Á
Á
Á
Á
Á
ˆ
¯
˜
˜
˜
˜
˜
˜
˜
˜
=
6+ (−4)
2
,
−3+ 5
2
Ê
Ë
Á
Á
Á
Á
Á
Á
Á
ˆ
¯
˜
˜
˜
˜
˜
˜
˜
=
2
2
,
2
2
Ê
Ë
Á
Á
Á
Á
Á
Á
ˆ
¯
˜
˜
˜
˜
˜
˜
= (1, 1)
REF: 1.3
8. ANS:
x
1
= −10, y
1
= −1
x
2
= 0, y
2
= 4
x
1
+ x
2
2
,
y
1
+ y
2
2
Ê
Ë
Á
Á
Á
Á
Á
Á
Á
Á
ˆ
¯
˜
˜
˜
˜
˜
˜
˜
˜
=
−10+ 0
2
,
−1+ 4
2
Ê
Ë
Á
Á
Á
Á
Á
Á
ˆ
¯
˜
˜
˜
˜
˜
˜
=
−10
2
,
3
2
Ê
Ë
Á
Á
Á
Á
Á
Á
ˆ
¯
˜
˜
˜
˜
˜
˜
= (−5, 1.5)
REF: 1.3
9. ANS:
An angle bisector is a ray drawn through the vertex of an angle that divides the angle into two congruent
angles.
REF: 1.4
10. ANS:
Parallel. The slope of line n is
−2
, which is equal to the slope of line m, so the lines are parallel.
REF: 1.5
ID: A
4
11. ANS:
Neither. The slope of line p is 3 and the slope of line q is
1
3
. The slopes of the lines are not equal, so the lines
are not parallel. The product of the slopes is
3 ×
1
3
= 1 ≠ −1
, so the lines are not perpendicular.
REF: 1.5
12. ANS:
Perpendicular. The slope of line r is
−5
and the slope of line s is
1
5
. The product of the slopes is
−5×
1
5
= −1
,
so the slopes are negative reciprocals and the lines are perpendicular.
REF: 1.5
13. ANS:
Neither. The slope of line n is 6 and the slope of line m is
−6
. The slopes of the lines are not equal, so the
lines are not parallel. The product of the slopes is
6 × −6 = −36 ≠ −1
, so the lines are not perpendicular.
REF: 1.5
14. ANS:
Neither. The equation for line p can be rewritten as
y = x + 4
, and the equation for line q can be rewritten as
y = −2x + 8
. The slope of line p is 1 and the slope of line q is
−2
. The slopes of the lines are not equal, so the
lines are not parallel. The product of the slopes is
1× (−2) = −2 ≠ −1
, so the lines are not perpendicular.
REF: 1.5
15. ANS:
Parallel. The equation for line r can be rewritten as
y = −
1
2
x + 3
, and the equation for line s can be rewritten
as
y = −
1
2
x + 2
. The slope of line r is
−
1
2
, which is equal to the slope of line s, so the lines are parallel.
REF: 1.5
16. ANS:
Point-slope form:
(y + 2) = 7(x − 5)
Slope-intercept form:
y + 2 = 7x − 35
y = 7x − 35 − 2
y = 7x − 37
REF: 1.5
ID: A
5
17. ANS:
The slope of the new line must be
1
3
.
Point-slope form:
(y − 6) =
1
3
(x + 1)
Slope-intercept form:
y − 6 =
1
3
x +
1
3
y =
1
3
x +
1
3
+ 6
y =
1
3
x +
19
3
REF: 1.5
18. ANS:
x + 3x = 90
4x = 90
x = 22.5
The measure of the angle is
22.5°
and the measure of the complement is
67.5°
.
REF: 2.2
19. ANS:
x + 0.25x = 180
1.25x = 180
x = 144
The measure of the angle is
144°
and the measure of the supplement is
36°
.
REF: 2.2
20. ANS:
∠1 and ∠6, ∠2 and ∠5, ∠3 and ∠8, ∠4 and ∠7, ∠9 and ∠11, ∠10 and ∠12
REF: 2.2
21. ANS:
Transitive Property
REF: 2.3
22. ANS:
Substitution Property
REF: 2.3
ID: A
6
23. ANS:
If angles 1 and 2 are supplementary, then
m∠1 + m∠2 = 180°
by the definition of supplementary angles.
Likewise, if angles 3 and 4 are supplementary, then
m∠3 + m∠4 = 180°
by the definition of supplementary
angles. You can use the Substitution Property to write
m∠1 + m∠2 = m∠3+ m∠4
. You are given that
∠2 ≅ ∠4
, so
m∠2 = m∠4
by the definition of congruent angles. Then, you can use the Substitution Property
to substitute
∠2
for
∠4
into the equation
m∠1 + m∠2 = m∠3+ m∠4
to get
m∠1 + m∠2 = m∠3+ m∠2
. By the
Subtraction Property of Equality
m∠1 = m∠3
. So,
∠1 ≅ ∠3
by the definition of congruent angles.
REF: 2.3
24. ANS:
∠1 ≅ ∠5, ∠3 ≅ ∠7, ∠4 ≅ ∠8, ∠2 ≅ ∠6
REF: 2.4
25. ANS:
Statements Reasons
1.
r Ä s
, t is a transversal
1. Given
2.
∠4 ≅ /2
2. Corresponding Angles Postulate
3.
∠2 ≅ /5
3. Vertical Angles Congruence Theorem
4.
∠4 ≅ /5
4. Transitive Property
REF: 2.4
26. ANS:
Statements Reasons
1.
∠2 ≅ ∠7
and line k is a transversal 1. Given
2. Angles 5 and 2 are vertical angles 2. Definition of vertical angles
3.
∠5 ≅ ∠2
3. Vertical Angles Congruence Theorem
4.
∠5 ≅ ∠7
4. Transitive Property
5. Angles 5 and 7 are corresponding angles 5. Definition of corresponding angles
6.
m Ä n
6. Corresponding Angles Converse Postulate
REF: 2.5
ID: A
7
27. ANS:
The perimeter is approximately 35.9 units.
JK = (10− 14)
2
+ (−8− 6)
2
= 16 + 196
= 212
KL = (4− 10)
2
+ (−4− (−8))
2
= 36 + 16
= 52
JL = (4− 14)
2
+ (−4− 6)
2
= 100+ 100
= 200
Perimeter = JK + KL + JL
= 212 + 52 + 200
≈ 35.9
REF: 3.2
ID: A
8
28. ANS:
The perimeter is approximately 19.83 units.
A
′
B
′
= 3, A
′
D
′
= 5, C
′
D
′
= 6
B
′
C
′
= (x
2
− x
1
)
2
+ (y
2
− y
1
)
2
= (6− 3)
2
+ (0− 5)
2
= (3)
2
+ (−5)
2
= 9+ 25
= 34
Perimeter
= A
′
B
′
+ B
′
C
′
+ C
′
D
′
+ A
′
D
′
= 3+ 34 + 6 + 5
≈ 19.83
REF: 3.4
ID: A
9
29. ANS:
The area is 40 square units.
AX
′
= 8, W
′
Z
′
= 6, X
′
Y
′
= 4
Area =
1
2
(b
1
+ b
2
)h
=
1
2
(6 + 4)(8)
=
1
2
(10)(8)
= 40
REF: 3.4
ID: A
10
30. ANS:
The area of figure ABCDEFGH is 22 square units. I used the rectangle method to determine the area of the
figure.
First, I sketched rectangle WXYZ that contains the figure and calculated that the area of rectangle WXYZ is 42
square units.
Area of Rectangle:
area = bh
= (7)(6)
= 42
Area of Triangles:
Next, I calculated that the total area of the 2 corner rectangles is 16 square units.
Area of Corner Rectangles:
area of rectangle
WA
′
H
′
G
′
=
bh
= (2)(4)
= 8
area of rectangle
F
′
E
′
D
′
Z
=
bh
= (2)(4)
= 8
Then, I calculated that the total area of the 2 corner triangles is 4 square units.
Area of Triangles:
ID: A
11
area of
A
′
XB
′
=
1
2
bh
=
1
2
(2)(2)
= 2
area of
C
′
YD
′
=
1
2
bh
=
1
2
(2)(2)
= 2
Lastly, I subtracted the area of the 4 corner figures from the area of the rectangle to determine that the area of
the figure is 22 square units.
Area of Figure:
area of figure = area of rectangle − area of 4 corner figures
= 42 − (16+ 4)
= 42 − 20
= 22
REF: 3.5
31. ANS:
volume =
1
3
Bh
=
1
3
π
r
2
h
=
1
3
π
(2)
2
(7)
≈
28
3
π
cubiccentimeters
REF: 4.4
32. ANS:
Volume =
4
3
π
r
3
=
4
3
π
(6)
3
= 288
π
≈904.3 cubic inches
REF: 4.5
ID: A
12
33. ANS:
V =
π
r
2
h
=
π
(4)
2
(6)
= 96
π
≈ 301.4cubic millimeters
REF: 4.6
34. ANS:
m∠Y = 180° − (84° + 42°) = 54°
The shortest side of a triangle is opposite the smallest angle. So, the side lengths from shortest to longest are
z, y, x.
REF: 5.2
35. ANS:
m∠X = 180° − (67° + 27°) = 86°
m∠Z = 180° − (64° + 79°) = 37°
The shortest side of a triangle is opposite the smallest angle. Side c is the longest side of
WXY
, and the
shortest side of
WYZ
. So, the side lengths from shortest to longest are b, a, c, d, e.
REF: 5.2
36. ANS:
m∠UTV = 180° − 90° = 90°
m∠SVU = m∠UTV + m∠TUV
x + 8° = 90° + 64°
x + 8° = 154°
x = 146°
REF: 5.2
37. ANS:
m∠KJL = 180° − 132° = 48°
m∠KLN = m∠KJL + m∠JKL
112° = 48° + (2x + 4°)
112° = 52° + 2x
60° = 2x
30° = x
REF: 5.2
ID: A
13
38. ANS:
m∠DFE = 180° − 90° = 90°
m∠DFG = m∠DEF + m∠EDF
90° = (2x + 18°) + (3x + 2°)
90° = 5x + 20°
70° = 5x
14° = x
REF: 5.2
39. ANS:
The smallest angle of a triangle is opposite the shortest side. So, the angles from least to greatest are
∠H, ∠F, ∠G
.
REF: 5.3
40. ANS:
Yes. A triangle can be formed because the sum of the two shortest sides is greater than the longest side.
Sum of the Two Shortest Sides:
3+ 2.9 = 5.9
Longest Side: 5
REF: 5.3
41. ANS:
No. A triangle cannot be formed because the sum of the two shortest sides is less than the longest side.
Sum of the Two Shortest Sides:
4+ 5.1 = 9.1
Longest Side: 12.5
REF: 5.3
42. ANS:
No. A triangle cannot be formed because the sum of the two shortest sides is equal to the longest side.
Sum of the Two Shortest Sides:
6.3 + 7.5 = 13.8
Longest Side: 13.8
REF: 5.3
43. ANS:
c = 7 2
The length of the hypotenuse is
7 2
kilometers.
REF: 5.4
ID: A
14
44. ANS:
a 2 = 16
a =
16
2
a =
16 2
2 2
a =
16 2
2
= 8 2 ≈ 11.3
The kite is approximately 11.3 meters above the ground.
REF: 5.4
45. ANS:
a = 3
feet
b = 3 3
feet
c = 2(3) = 6
feet
REF: 5.5
46. ANS:
c = 20
meters
a =
20
2
= 10
meters
b = 10 3
meters
REF: 5.5
47. ANS:
Triangles in which all corresponding angles are congruent and all corresponding sides are proportional.
REF: 6.1
ID: A
15
48. ANS:
The triangles are similar by the Side-Side-Side Similarity Theorem because all of the corresponding sides are
proportional.
DE
TU
=
15
20
=
3
4
DF
SU
=
18
24
=
3
4
EF
ST
=
27
36
=
3
4
REF: 6.2
49. ANS:
GH Ä FI ,
Converse of Triangle Proportionality Theorem
REF: 6.3
50. ANS:
The known corresponding sides of the triangles are proportional:
6
3
=
2
1
and
8
4
=
2
1
.
The angle between the known sides is a right angle for both triangles, so those angles are congruent.
Therefore, by the Side-Angle-Side Similarity Postulate, the triangles are similar.
REF: 6.6
51. ANS:
The height of the statue is 35 feet.
x
5
=
84
12
x = 35
REF: 6.6
52. ANS:
The palm tree is 24 feet tall.
x
6
=
45
11.25
x = 24
REF: 6.6
53. ANS:
The vertices of triangle
R
′
S
′
T
′
are
R
′
(−5, 6)
,
S
′
(−3, 10)
, and
T
′
(−2, 2)
.
REF: 7.1
ID: A
16
54. ANS:
The vertices of quadrilateral
W
′
X
′
Y
′
Z
′
are
W
′
(−5, 0)
,
X
′
(3, − 9)
,
Y
′
(5, − 8)
, and
Z
′
(8, − 1)
.
REF: 7.1
55. ANS:
The vertices of triangle
A
′
B
′
C
′
are
A
′
(−3,5)
,
B
′
(−8,2)
, and
C
′
(−5, − 4)
.
REF: 7.1
56. ANS:
The vertices of parallelogram
H
′
J
′
K
′
L
′
are
H
′
(6,2), J
′
(1,3), K
′
(1,7)
and
L
′
(6,6)
.
REF: 7.1
57. ANS:
The vertices of rectangle
D
′
E
′
F
′
G
′
are
D
′
(7, 1), E
′
(7, 8), F
′
(−1, 8)
, and
G
′
(−1, 1)
.
REF: 7.1
58. ANS:
The vertices of parallelogram
H
′
J
′
K
′
L
′
are
H
′
(2, 6), J
′
(3, 1), K
′
(7, 1)
, and
L
′
(6,6)
.
REF: 7.1
59. ANS:
opposite
hypotenuse
=
24
26
=
12
13
REF: 9.1
60. ANS:
c
2
= a
2
+ b
2
c
2
= 15
2
+ 8
2
c
2
= 225+ 64 = 289
c = 289 = 17
opposite
hypotenuse
=
15
17
REF: 9.1
ID: A
17
61. ANS:
c
2
= a
2
+ b
2
c
2
= 7
2
+ 24
2
c
2
= 49 + 576 = 625
c = 625 = 25
opposite
hypotenuse
=
7
25
REF: 9.1
62. ANS:
c
2
=
a
2
+
b
2
c
2
= 12
2
+ 9
2
c
2
= 144+ 81 = 225
c
= 225 = 15
opposite
hypotenuse
=
12
15
=
4
5
REF: 9.1
63. ANS:
adjacent
hypotenuse
=
16
34
=
8
17
REF: 9.1
64. ANS:
c
2
= a
2
+ b
2
c
2
= 2
2
+ (2 3)
2
c
2
= 4+ 12 = 16
c = 16 = 4
adjacent
hypotenuse
=
2 3
4
=
3
2
REF: 9.1
ID: A
18
65. ANS:
opposite
hypotenuse
=
18
30
=
3
5
adjacent
hypotenuse
=
24
30
=
4
5
opposite
adjacent
=
18
24
=
3
4
REF: 9.1
66. ANS:
tan 40° =
x
2
2 tan 40° = x
x ≈ 1.68 ft
REF: 9.2
67. ANS:
tan X =
5
9
m∠X = tan
−1
5
9
Ê
Ë
Á
Á
Á
Á
Á
Á
ˆ
¯
˜
˜
˜
˜
˜
˜
≈ 29.05°
REF: 9.2
68. ANS:
tan 70° =
h
20
20 tan 70° = h
h ≈ 54.95 ft
REF: 9.2
69. ANS:
tan x =
15.4
6.2
x = tan
−1
15.4
6.2
Ê
Ë
Á
Á
Á
Á
Á
Á
ˆ
¯
˜
˜
˜
˜
˜
˜
≈ 68.07°
The lifeguard is looking down at an angle of approximately
68.07°.
REF: 9.2
ID: A
19
70. ANS:
tan x =
140− 1.4
190
x = tan
−1
138.6
190
Ê
Ë
Á
Á
Á
Á
Á
Á
ˆ
¯
˜
˜
˜
˜
˜
˜
≈ 36.11°
The surveyor is looking up at an angle of approximately
36.11°.
REF: 9.2
71. ANS:
sin 40° =
x
2
2sin 40° = x
x ≈ 1.29ft
REF: 9.3
72. ANS:
sin X =
20
25
m∠X = sin
−1
20
25
Ê
Ë
Á
Á
Á
Á
Á
Á
ˆ
¯
˜
˜
˜
˜
˜
˜
≈ 53.13°
REF: 9.3
73. ANS:
sinx =
45
100
x = sin
−1
45
100
Ê
Ë
Á
Á
Á
Á
Á
Á
ˆ
¯
˜
˜
˜
˜
˜
˜
≈ 26.74°
The angle formed by the string and the ground is approximately
26.74°.
REF: 9.3
74. ANS:
cos X =
9
13
m∠X = cos
−1
9
13
Ê
Ë
Á
Á
Á
Á
Á
Á
ˆ
¯
˜
˜
˜
˜
˜
˜
≈ 46.19°
REF: 9.4
ID: A
20
75. ANS:
XV
2
= 6
2
+ 8
2
XV
2
= 36 + 64
XV
2
= 100
XV = 10
cos X =
6
10
m∠X = cos
−1
6
10
Ê
Ë
Á
Á
Á
Á
Á
Á
ˆ
¯
˜
˜
˜
˜
˜
˜
≈ 53.13°
REF: 9.4
76. ANS:
The sum is equal to
(n − 2) ⋅180°
:
(13 − 2) ⋅ 180° = 11 ⋅ 180° = 1980°
The sum of the interior angles of the polygon is
1980°
.
REF: 10.4
77. ANS:
The sum is equal to
(
n
− 2) ⋅ 180°
.
(20 − 2) ⋅ 180° = 18 ⋅ 180 = 3240°
The sum of the interior angles of the polygon is
3240°
.
REF: 10.4
78. ANS:
The sum is equal to
(
n
− 2) ⋅ 180°
.
(25 − 2) ⋅ 180° = 23 ⋅180 = 4140°
The sum of the interior angles of the polygon is
4140°
.
REF: 10.4
ID: A
21
79. ANS:
(n − 2)180°
n
=
(8− 2)180°
8
=
(6)180°
8
=
1080°
8
= 135°
The measure of each interior angle is
135°
.
REF: 10.4
80. ANS:
(n − 2)180°
n
=
(7− 2)180°
7
=
(5)180°
7
=
900°
7
≈ 128.6°
The measure of each interior angle is approximately
128.6°
.
REF: 10.4
81. ANS:
108° =
(n − 2)180°
n
108°n = (n − 2)(180°)
108°n = 108°n − 360°
72°n = 360°
n = 5
The regular polygon has 5 sides.
REF: 10.4
82. ANS:
360
5
= 72°
REF: 10.5
ID: A
22
83. ANS:
360
12
= 30°
REF: 10.5
84. ANS:
The measure of
CD
is
60°
.
REF: 11.2
85. ANS:
m∠XYZ = 80°
REF: 11.2
86. ANS:
m∠KWS = 70°
REF: 11.2
87. ANS:
m∠XYZ = 75°
REF: 11.2
88. ANS:
m∠SGI = 14°
REF: 11.2
89. ANS:
mQW = 162°
REF: 11.2
90. ANS:
m∠EFG =
1
2
(m∠EOG) =
128°
2
= 64°
REF: 11.2
ID: A
23
91. ANS:
The measure of angle KLJ is 70 degrees.
First, I determined that the sum of the measures of arcs KJ and MN is 140 degrees
mKM + mJN + mKJ + mMN = 360
120 + 100 + mKJ + mMN = 360
220+ mKJ + mMN = 360
mKJ + mMN = 140
Then, I calculated the measure of angle KLJ.
m∠KLJ =
1
2
(mKJ + mMN)
=
1
2
(140)
= 70
REF: 11.3
92. ANS:
The measure of arc RS is 130 degrees.
m∠RTS =
1
2
(mRS − mUV)
80 =
1
2
(mRS + 30)
160 = mRS + 30
mRS = 130
REF: 11.3
ID: A
24
93. ANS:
The measure of angle D is 75 degrees.
To solve the problem, I calculated the measure of arc ZAB first. Then, I calculated the measure of angle D.
mZXC + mCB + mZAB = 360
150 + 30+ mZAB = 360
180+ mZAB = 360
mZAB = 180
m∠D =
1
2
(mZAB − mCB)
=
1
2
(180− 30)
=
1
2
(150)
= 75
REF: 11.3
94. ANS:
Inscribed polygons are drawn inside of a circle with all vertices touching the circle. Circumscribed polygons
are drawn outside of a circle with all sides tangent to the circle.
REF: 12.1
95. ANS:
No. The triangle is not a right triangle. None of the sides of the triangle is a diameter of the circle.
REF: 12.1
ID: A
25
96. ANS:
m∠D = 180° − 81° = 99°
REF: 12.1
97. ANS:
m∠XVY =75°
m∠YVZ =105°
Because arc YZW is a semicircle, its measure is
180°
.
mYZW = m∠YZ + m∠ZW
180° = 50° + mZW
mZW = 130°
m∠XVY =
1
2
(mXY + mZW)
m∠XVY =
1
2
(20° + 130°)
m∠XVY =
1
2
(150°)
m∠XVY = 75°
m∠YVZ = 180° − 75°
= 105°
REF: 12.4
ID: A
26
98. ANS:
m∠A = 25°
m∠A =
1
2
(mCD − mBE)
m∠A =
1
2
(70° − 20°)
m∠A =
1
2
(50°)
m∠A = 25°
REF: 12.4
99. ANS:
m∠H =61°
m∠H =
1
2
(mJK − mIL)
m∠H =
1
2
(164° − 42°)
m∠H =
1
2
(122°)
m∠H = 61°
REF: 12.4
100. ANS:
m∠O = 19°
m∠O =
1
2
(mMQ − mNP)
m∠O =
1
2
(50° − 12°)
m∠O =
1
2
(38°)
m∠O = 19°
REF: 12.4
