Download Indicate whether the statement is true or false.

Name:
Class:
Date:
Practice Final Exam
Indicate whether the statement is true or false.
1. If coplanar lines r and s are cut by transversal t so that alternate exterior angles
are congruent, then r || s.
a. True
b. False
2. The simplified form of the ratio “6 inches to 1 foot” is 6:1.
a. True
b. False
3. For every parallelogram, two adjacent sides must be congruent.
a. True
b. False
4. If
in
, then
.
a. True
b. False
5. If M and N are midpoints of sides
a. True
b. False
and
of
, then
.
6. If the diagonals of a quadrilateral are perpendicular, the quadrilateral must be a square.
a. True
b. False
Copyright Cengage Learning. Powered by Cognero.
Page 1
Name:
Class:
Date:
Practice Final Exam
7. If
, then the altitude to base
of
is
.
a. True
b. False
8. Where m
cone.
a. True
b. False
= 90°, the solid of revolution determined by revolving right triangle
about leg
is a right circular
9. For any regular polygon, the center can be determined by the intersection of any two angle-bisectors of the polygon.
a. True
b. False
10. As shown,
a. True
b. False
must be an isosceles triangle.
Indicate the answer choice that best completes the statement or answers the question.
11. In
a. 30°
c. 60°
,
. Also,
. Find
.
b. 45°
d. None of These
Copyright Cengage Learning. Powered by Cognero.
Page 2
Name:
Class:
Date:
Practice Final Exam
12. In
a.
c.
(not shown),
is the base of
13. In
,m
a. 1.5 cm
c. 6 cm
.
. Which statement is not necessarily true?
b.
is the vertex angle of
d.
is the longest side of
= 45° and OA = 6 cm. Find
b. 3 cm
d. 12 cm
.
14. For a triangle whose perimeter measures 36 units, the radius of the inscribed circle is 3.Find the area of the triangle.
a. 27
b. 54
c. 108
d. None of These
15. Where
a. (45,28,53)
c. (10,18,28)
16. If m
a. x = 8
c. x = 8.5
,
, and
b. (47,53,56)
d. (20,21,29)
,m
b. x = 8.25
d. x = 8.75
, determine the Pythagorean Triple generated by
, and m
and
.
, find x.
17. Suppose that lines r and s are both perpendicular to line t. Then r is parallel to s if:
a. r and s are collinear
b. r and s intersect
c. r and s are coplanar
d. None of These
Copyright Cengage Learning. Powered by Cognero.
Page 3
Name:
Class:
Date:
Practice Final Exam
18. A right circular cone has a diameter for its base of length 16 inches and an altitude of 15 inches. Use
exact lateral area of the cone.
a.
b.
c.
d. None of These
to find the
19. Which property justifies this conclusion?
If X is a point on
and R-X-S, then
.
a. Segment-Addition Postulate
b. Line-Addition Postulate
c. Ruler Postulate
d. None of These
20. If the diagonals of a rhombus measure 10 cm and 24 cm, what is the perimeter of the rhombus?
a. 13 cm
b. 34 cm
c. 52 cm
d. 68 cm
Enter the appropriate word(s) to complete the statement.
21. If X is the midpoint of
and X-V-T on
, find the length of .
,
,
,
, and
22. Complete the following implication with hypothesis “2 + 3 = 5.”
If 2 + 3 = 5, then 2a + 3a = ?:
23.
If m
is separated into smaller angles by
,m
, and m
and
24. In the figure, the exterior sides of the adjacent angles (
Copyright Cengage Learning. Powered by Cognero.
, rays in the interior of
, then m
equals:
and
.
) form perpendicular rays. How are these angles
Page 4
Name:
Class:
Date:
Practice Final Exam
related?
25. If R-S-T on line RT and
while
, then ST measures:
26. For trapezoid RSTV,
. Also, M and N are the midpoints of the nonparallel sides as shown. In relation to
trapezoid RSTV, what name is used for
?
27. In the figure,
bisects
of
28. For regular pentagon ABCDE, diagonals
. If
and
,
, and
are drawn to form
, find WZ.
. Find m
.
29. A regular icosahedron has its faces numbered 1, 2, 3, . . . , 20 and is used as a die. What is the probability that a roll
produces a multiple of 5?
30. If two parallel lines are cut by a transversal, then the interior angles on the same side of the transversal are:
31. If
32. In
.
and
are not supplementary, what conclusion can you draw?
, one side of
is diameter
Copyright Cengage Learning. Powered by Cognero.
while a second side is chord
. If AC = PB = 5, find the length of chord
Page 5
Name:
Class:
Date:
Practice Final Exam
33. A storage tank is in the shape of a right circular cylinder. If r = 3 ft and h = 4 ft,find the exact volume of the cylinder.
34. Find the sum of the interior angles for an octagon.
35. Points A, B, C, and D are collinear in the order A-B-C-D. If AC = 10.8 inches,
BD = 7.7 inches, and BC = 3.2 inches, find AD.
36. If the measures of the angles of a quadrilateral ABCD are in the extended ratio
, what type of quadrilateral is ABCD?
37. When two parallel lines are cut by a transversal, what relationship exists between a pair of interior angles on the same
side of the transversal?
38. In trapezoid RSTV,
. If SR = TV = 6 cm, ST = 4 cm, and m
= 60°, find the area of RSTV.
39. In isosceles trapezoid HJKL,
. Suppose that points M, N, P, and Q are the midpoints of the sides
, and
respectively. If the length of diagonal
is 10 cm, find the perimeter of MNPQ.
Copyright Cengage Learning. Powered by Cognero.
,
,
Page 6
Name:
Class:
Date:
Practice Final Exam
40. In the circle, chords
and
intersect at point P. If m
= 63° and m
= 75°, find m
.
41. A pyramid has a square base with sides of length e. If the altitude of the pyramid also measures e, what is the volume
of the pyramid?
42. In a 45°-45°-90° triangle, the length of a leg is the number a. What expression represents the length of the hypotenuse
of this triangle?
43. In
,
44. Given that
.
. If MQ = 10, RP = 4, and MR = 8, find the area of
is congruent to
,
,
,
.
, and
,find the perimeter
45. In a regular polygon, each central angle measures 15°. How many sides does the polygon have?
46.
measure?
is circumscribed about
47. In the figure,
in such a way that
and
Copyright Cengage Learning. Powered by Cognero.
. Which angle of
. What method allows you to conclude that
has the greatest
?
Page 7
Name:
Class:
Date:
Practice Final Exam
48. Let point E lie in the exterior of
. How many tangent lines can be constructed to
49. In the circle, m
= 76°. Where M is the midpoint of
= 102° and m
from point E?
, find m
.
50. Which three-letter symbol is used to indicate that two triangles are congruent because
their three pairs of sides are congruent?
51. A right circular cone has a radius of 6 inches and an altitude of 8 inches. Use
area of the cone.
in order to find the exact total
52. For a right circular cone, the altitude has the same measure as its radius. If the volume of the cone is approximately 56
, find the length of radius to the nearest tenth of a foot.
53. In addition to being congruent, how are the diagonals of a square related?
54. When a quadrilateral is circumscribed about a circle, how are the sides of the quadrilateral related to the circle?
55. The driver of an automobile leaves home and travels due east 8 miles. Turning right at an intersection, he drives due
south a distance of 6 miles. How far is the driver from home?
56.
has a perimeter of 84 cm. If
57. Given trapezoid RSTV with median
Copyright Cengage Learning. Powered by Cognero.
cm and ST = 25 cm, find the angle of greatest measure in
, find the length of
.
if RS = 12.7 inches and VT = 17.5 inches.
Page 8
Name:
Class:
Date:
Practice Final Exam
58. A prism and pyramid have congruent bases and the same length of altitude. If the volume of the pyramid is 24
find the volume of the prism.
,
59. Parallel lines a, b, and c are cut by transversal d at R, S, and T (where R-S-T) and by transversal e at X, Y, and Z (where
X-Y-Z). If
, how are
and
related?
60. When two lines are cut by a transversal, and are one pair of alternate interior angles that are formed.
If
and
, what relationship (in simplified form) exists between x and y? Answer in the
form x = .
61. In
, chord
62. In the figure,
is 12 inches long and m
intersects two sides of
= 90°. Find the exact area of the segment bounded by
as shown. If
and
are drawn to
.
, what relationship must hold true?
63. A square is inscribed in a circle. If the area of the circle is 28.25
and the area of the square is 18
area of the four-part region that lies outside the square but inside the circle.
64. From an external point E, tangents
and
. If m
, find the
= 78°, find the measure of minor arc
.
65. When two planes intersect, their intersection is a(n):
66. Of the upper case letters R, S, and T, which one has point symmetry?
Copyright Cengage Learning. Powered by Cognero.
Page 9
Name:
Class:
Date:
Practice Final Exam
67. For the triangular prism, the areas of the lateral faces are 18
find the total area of the prism.
, 26
, and 34
. If the area of the base is 16
,
68. For a given polyhedron, there are 24 vertices and 32 faces. How many edges does the polyhedron have? [Hint:
.]
69. What is the name of the geometric figure that is the theoretical limit of an “inscribed” regular polyhedron whose
number of faces increases without limit?
70. For rhombus RSTV, diagonals
rhombus RSTV.
and
intersect at point W. If RT = 16 and VS = 12, find the length of each side of
71. In the figure,
. Also,
,
, and
. Find TX.
72. Given that D-E-F on
, name the property that leads to the conclusion
.
73. Using the symbols ∼ (for negation) and → (for implication), write the contrapositive of
the implication P → Q.
74. In isosceles triangle RST,
. If
, find the length of
.
75. Given line l, how many lines in space can be drawn parallel to l?
76. In
77. Where
,
,
, and
,
. Which angle, if any, measures 90°?
and
. Find
.
78. For the triangle with sides of lengths a, b, and c, find an expression for the semiperimeter s of the triangle.
79. For a regular octagon, the length of the apothem is a = 8.2 cm while the length of each side is s = 6.7 cm. To the
nearest tenth of a square centimeter, find the area of the regular octagon.
Copyright Cengage Learning. Powered by Cognero.
Page 10
Name:
Class:
Date:
Practice Final Exam
80. In this order, points R, S, T, U, and V are equally spaced on a circle in such a way
. What type of figure is formed by the chords
, and
,
,
,
?
81. What does the question mark represent in this extended proportion?
82. For the circle shown, chords
length of
.
and
intersect at point X. With RX = 6, XS = 8, TX = x + 8 and XV = x, find the
83. A right prism has height h and perimeter P for its base. Write the formula for the lateral area L of the prism.
84.
. If AB = 4, XY = 10, and the area of
is 7.8
85. The number of diagonals D for a polygon of n sides is given by
of sides for a polygon that has 14 diagonals.
, find the area of
. Find the number
86. Supply missing reasons for this proof.
Given: m || n
Prove:
Copyright Cengage Learning. Powered by Cognero.
Page 11
Name:
Class:
Date:
Practice Final Exam
S1. m || n R1.
S2.
R2.
S3.
R3.
S4.
R4.
87. Explain (prove) the following property of proportions.
“If
(where
and
), then
.”
88. Supply missing statements and missing reasons for the following proof.
Given:
and
Prove:
S1.
S2.
;
bisects
is an isosceles triangle
;
bisects
R1.
R2. If a ray bisects one
of a
, it divides the opposite
side into segments whose lengths are proportional to
the lengths of the two sides that form the bisected .
S3. R3. Given
S4.
R4.
S5.
S6.
S7. R7.
, so
R5.
R6.
Copyright Cengage Learning. Powered by Cognero.
Page 12
Name:
Class:
Date:
Practice Final Exam
89. Supply missing statements and missing reasons for the following proof.
Given: Chords
and
Prove:
intersect at point N in
)
S1. R1.
S2. Draw
S3.
R2.
R3. The measure of an ext. of a
of measures of the two nonadjacent int.
is
. S4.
the sum
and
R4.
S5. R5. Substitution Property of Equality
S6. R6. Substitution Property of Equality
90. Use an indirect proof to complete the following problem.
Given:
Prove:
and
and
are supplementary (no drawing)
are not both obtuse angles.
91. Use the drawing provided to explain the 45
-45
-90
Theorem.
“In a triangle whose angles measure 45
, 45
, and 90
, the hypotenuse has a length equal to the product of
Given:
,
, and
Prove:
and the length of either leg.”
with
and
Copyright Cengage Learning. Powered by Cognero.
Page 13
Name:
Class:
Date:
Practice Final Exam
92. Supply all statements and all reasons for the proof that follows.
Given:
;
Prove:
93. Using the drawing provided, explain the following statement.
The sum of the interior angles of a quadrilateral is 360.
94.
Consider a circle with diameter length d, radius length r, and circumference C. Given that
formula for the circumference of a circle is given by
.
95. Where and are natural numbers and
Verify that
is a Pythagorean Triple.
, let
,
, and
, explain why the
.
96. Supply missing statements and missing reasons in the following proof.
Given:
Prove:
in the figure shown
Copyright Cengage Learning. Powered by Cognero.
Page 14
Name:
Class:
Date:
Practice Final Exam
S1. R1.
S2.
R2.
S3. R3. Vertical angles are congruent.
S4. R4.
97. Supply missing statements and missing reasons for for the following proof.
Given:
Prove:
;
and
are right angles
S1. R1.
S2.
R2.
S3. R3.Opposite angles of a parallelogram.
S4.
R4.
S5.
R5.
S6. R6. In a proportion, the product of the means equals the
product of the extremes.
98. Explain the following statement.
The measure of each interior angle of an equiangular triangle is 60.
99. Supply missing statements and missing reasons for the proof of this theorem.
“The altitude drawn to the hypotenuse of a right triangle separates the right triangle into two
right triangles that are similar to each other.”
Given: Right triangle ABC with rt.
Prove:
S1. R1.
S2.
S3. and
;
R2.
are comp. R3. The acute angles of a rt.
Copyright Cengage Learning. Powered by Cognero.
are comp.
Page 15
Name:
Class:
Date:
Practice Final Exam
S4. and
are comp. R4.
S5. R5. If 2 s are comp. to the same , these are .
S6. R6.
100. Explain why the following must be true.
Given: Points A, B, and C lie on
in such a way that
also, chords
,
, and
(no drawing provided)
Prove:
must be an isosceles triangle.
;
101. Supply missing statements and missing reasons for the following proof.
Given: In the circle,
Prove:
S1. R1.
S2. Draw R2.
S3.
R3.
S4. R4. Congruent angles have equal measures.
S5. ? and ? R5. The measure of an inscribed angle equals one-half
the measure of its intercepted arc.
S6.
S7.
S8. R8.
R6.
R7.
Copyright Cengage Learning. Powered by Cognero.
Page 16
Name:
Class:
Date:
Practice Final Exam
102. Use the drawing(s) to explain the 30
-60
-90
Theorem.
“In a triangle whose angles measure 30
, 60
, and 90
, the hypotenuse has a length equal to twice the length of the shorter leg, and the length of the longer leg is the product of
and
the length of the shorter leg.”
Given: Right
,
,
and
; also,
Prove:
with
and
103. Supply missing statements and missing reasons for the following proof.
Given:
Prove:
; chords
and
intersect at point V
S1. R1.
S2. Draw
and
. R2.
S3. R3. Vertical angles are congruent.
S4.
R4.
S5. R5. AA
S6.
R6.
S7. R7. Means-Extremes Property of a Proportion
Copyright Cengage Learning. Powered by Cognero.
Page 17
Name:
Class:
Date:
Practice Final Exam
104. Provide the missing statements and missing reasons for the following proof.
Given:
Prove:
and
;
and
S1. R1. Given
S2.
R2.
S3.
S4. R4. CASTC
R3.
105. Supply missing statements and missing reasons for the following proof.
Given:
Prove:
and transversal p;
is a right angle
is a right angle
S1.
and transversal p R1.
S2.
R2.
S3. R3. Congruent measures have equal measures.
S4.
R4.
S5. R5. Substitution Property of Equality
S6. R6. Definition of a right angle
Copyright Cengage Learning. Powered by Cognero.
Page 18
Name:
Class:
Date:
Practice Final Exam
Answer Key
1. True
2. False
3. False
4. True
5. True
6. False
7. False
8. True
9. False
10. False
11. a
12. d
13. a
14. b
15. a
16. c
17. c
18. c
19. a
20. c
21. 29
22. 5a
23.
24. They are complementary.
25. 5x 8
Copyright Cengage Learning. Powered by Cognero.
Page 19
Name:
Class:
Date:
Practice Final Exam
26. median
27.
28. 36°
29.
30. supplementary
31. a is not parallel to b
32.
33.
34. 1080°
35. 15.3 inches
36. a trapezoid
37. They are supplementary.
38.
39. 20 cm
40. 51°
41.
42.
43. 36
44. 27
45. 24 sides
46.
47. SAS
48. 2
49. 140°
Copyright Cengage Learning. Powered by Cognero.
Page 20
Name:
Class:
Date:
Practice Final Exam
50. SSS
51.
52. 3.8 ft
53. perpendicular-bisectors of each other
54. They are tangent to the circle.
55. 10 miles
56.
57. MN = 15.1 inches
58. 72
59. congruent
60. x = 3y
61. (18 - 36)
62.
63. 10.25
64. 102°
65. line
66. S
67. 110
68. 54 edges
69. sphere
70. 10
71.
72. Segment-Addition Postulate
73. ∼ Q → ∼P
74.
Copyright Cengage Learning. Powered by Cognero.
Page 21
Name:
Class:
Date:
Practice Final Exam
75. an infinite number
76.
77. 58°
78. s = (
)
79. 219.8
80. a regular pentagram
81. 6 eggs
82. 4
83.
84. 48.75
85. 7
86. R1. Given
R2. If 2 parallel line are cut by a transversal, then corresponding angles are congruent.
R3. If two lines intersect, the vertical angles formed are congruent.
R4. Transitive Property of Congruence
87. Given that
and
, we add 1 to each side of this equation. By the Addition Property of Equality,
, so that
. But
by the Substitution
Property of Equality. In turn,
.
88. R1. Given
S3.
R4. Definition of congruent line segments
R5. Substitution Property of Equality
R6. Multiplication Property of Equality
S7.
is an isosceles triangle
R7. Definition of isosceles triangle
89. S1. Chords
and
intersect at point N in
R1. Given
R2. Through 2 points, there is exactly one line.
R4. In a circle, the measure of an inscribed angle is one-half that of its intercepted arc.
S5.
S6.
)
Copyright Cengage Learning. Powered by Cognero.
Page 22
Name:
Class:
Date:
Practice Final Exam
90. Suppose that and are both obtuse angles. Then
and
.
It follows that
. But it is given that and are supplementary, so that
With a contradiction of the known fact, it follows that the supposition must be false; thus, and
angles.
91. In
Then
.
are not both obtuse
,
. Thus, the sides opposite these angles are congruent. If
, then
With the right angle at C, we apply the Pythagorean Theorem to obtain
.
, so
. Applying the Square Roots Property, we have
or
. Then
.
92. S1.
;
R1. Given
S2.
and
R2. The measure of a central angle of a circle is equal to the measure of its intercepted arc.
S3.
R3. Substitution Property of Equality
93. In
,
By the Addition Property of Equality,
That is,
. Similarly,
.
.
.
94. Given that
, we use the Multiplication Property of Equality to obtain
Because the length of the diameter of a circle is twice that of a radius,
. By
substitution,
or
.
95. We need to show that
. Where
or
, so that
or
, so that
or
Now
value of
. That is,
,
, and
.
, it follws that
,
, and
, so that
or
for all choices of and .
.
, which is the
96. S1.
in the figure shown
R1. Given
R2. If 2 parallel lines are cut by a trans, the alternate interior angles are congruent.
S3.
S4.
R4. AA
97. S1.
;
and
R1. Given
R2. All right angles are congruent
S3.
R4. AA
R5. CSSTP
S6.
are right angles
Copyright Cengage Learning. Powered by Cognero.
Page 23
Name:
Class:
Date:
Practice Final Exam
98. The sum of the three angles of a triangle is 180. Let x represent the measure of each angle of the equiangular triangle.
Then
, so
. Dividing by 3,
. That is, the measure of each interior angle of an
equiangular triangle is 60.
99. S1. Right triangle ABC with rt.
;
R1. Given
R2. Perpendicular lines meet to form right angles.
R4. The acute angles of a rt. are comp.
S5.
S6.
R6. AA
100. Given that
in
, it follows that
. Then
must be an isosceles triangle.
. But congruent arcs have congruent chords so that
101. S1. In the circle,
R1. Given
R2. Through 2 points, there is exactly one line.
R3. If 2 parallel lines are cut ny a transversal, the alternate interior angles are congruent.
S4.
S5.
and
R6. Substitution Property of Equality
R7. Multiplication Property of Equality
S8.
R8. If 2 arcs of a circle are equal in measure, these arcs are congruent.
102. We reflect
The reflection of
across
to create an equiangular (and equilateral) triangle (
).
, namely
) is conruent to
. Then
and by the Sement-Addition Postulate,
, so
Knowing that
is equilateral, we have
(completing the first part of the
proof).
Using the Pythagorean Theorem in
or
, so
,
or
(completing the final part of the proof)
103. S1.
; chords
and
intersect at point V
R1. Given
R2. Through 2 points, there is exactly one line.
S3.
R4. If 2 inscribed angles of a circle intersect the same arc, these angles are congruent.
S5.
R6. CSSTP
S7.
Copyright Cengage Learning. Powered by Cognero.
Page 24
Name:
Class:
Date:
Practice Final Exam
104. S1.
and
;
R2. Substitution Property of Equality
R3. SSS
S4.
and
105. R1. Given
R2. If 2 parallel lines are cut by a trans, corresponding angles are congruent.
S3.
R4. Given
S5.
S6. is a right angle
Copyright Cengage Learning. Powered by Cognero.
Page 25