Indicate whether the statement is true or false. 1. The bearing of the airplane with the path shown is N 30° E. a. True b. False 2. If M and N are the midpoints of nonparallel sides of trapezoid HJKL, then a. True
and
.
b. False
3. For a rectangle with base length of 1 foot and height 6 inches, the area is 6 . a. True b. False
4. Given that quadrilateral a. True b. False
quadrilateral MNPQ,
5. The region bounded by radii a. True b. False
and
and arc
.
is known as a secant of the circle.
6. The lateral area L of any prism whose altitude measures h and whose base has perimeter P is given by L = hP. a. True b. False 7. If ray BD bisects a. True b. False
, then
.
8. In a prism, the two bases are parallel to each other and congruent to each other. a. True b. False
.
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9. If a lateral edge is perpendicular to a base edge, then the prism shown is a regular triangular prism. a. True b. False
10. With the circle inscribed in = BT, and CR = CT. a. True
, it follows that AR = AS, BS
b. False
11. If AB = 3 and BC = 4 in rectangle ABCD, then the length of diagonal a. True
is 5.
b. False
12. Given that a. True b. False
, it follows that
.
13. The formula for the volume of a right circular cylinder of radius r and altitude h is given by a. True b. False
.
14. An octagonal prism has 9 faces. a. True b. False 15. Two circles that are internally tangent have three common tangent lines. a. True b. False 16. If and a. True b. False
are complementary, then each of these angles 1 and 2 is acute.
17. A pentagon has the same number of diagonals as it has sides. a. True b. False 18. Both the kite and the parallelogram have two pairs of congruent angles. a. True b. False .
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19. One of the methods for proving that two triangles are congruent is abbreviated AAA. a. True b. False
20. If , then a. True b. False
.
Indicate the answer choice that best completes the statement or answers the question.
21. Suppose that you have utilized congruent vertical angles to prove that by the method SAS. By what reason can you, in turn, state that a. Identity b. SAS c. CPCTC
d. Definition of midpoint
22. In the circle, m a. 34° b. 37° c. 71°
.
?
= 68° and m
= 74°. Find m
.
d. 142°
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23. In a. 3
,m b.
c. 6
d.
= 60° and BC = 12. Find AB.
24. In isosceles triangle RST, . If perimeter of is 36, find the value of x. a. x = 5.5 b. x = 6 c. x = 6.5 d. None of These
,
, and the
25. Point P lies in the exterior of so that is tangent to the circle. Also, is a secant that intersects at B and C, where P-BC. If PB = 9 and BC = 7, find PA. a. 10 b. 12 c. 16 d. None of These
26. Suppose that . Which must be true? a. a || b b. a ⊥ c c. c || d d. c is not parallel to d 27. Find the area of a regular hexagon whose sides measure 6 inches each. a. 9 b. 9 c. 54
d. 54
28. is an altitude for length of altitude . a. 3 b. 3.5 c. 4 d. None of These
29. In a. c.
.
. If RV = 5, RS = 7, and WT = 4, find the
,m
= 103° and m b. is a semicircle is a minor arc
d.
= 77°. Which statement is false? would be a diameter would be a right angle
Page 4
30. Consider regular pentagon ABCDE. If diagonal possible, what type of quadrilateral is ACDE? a. parallelogram b. kite c. trapezoid d. isosceles trapezoid
is drawn, then a quadrilateral ACDE is formed. Being as specific as
31. 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 32. Which statement would one prove by the indirect method? a. If and are not congruent angles, then and are not vertical angles. b. If r || s, then . c. If l ⊥ m and m ⊥ p, then l || p. d. If two sides of a triangle are congruent, then two angles of that triangle are congruent. 33. Which type of triangle has sides of lengths a = 8, b = 15, and c = 17? a. acute b. right c. obtuse d. No triangle with these lengths of sides exists.
34. For the right triangular prism shown, the base is a right triangle with sides of lengths 3 in, 4 in, and 5 in. If the prism has a height of 6 inches, find its volume. a. 24 b. 36 c. 72 d. 360 35. Where C is the circumference of a circle and r is its length of radius, the approximate value of the ratio C : r is: a. 1.57 b. 3.14 c. 6.28 d. 12.56 36. Find the sum of the interior angles of a hexagon. a. 180° b. 360° c. 540° d. None of These 37. Given lines and m with , it follows that: a. b. c. d. None of These 38. Find the measure of each interior angle of a regular octagon. a. 45° b. 108° c. 120° d. 135°
.
Page 5
39.
has been constructed tangent to at point T and to a. congruent b. congruent and parallel c. parallel d. None of These
at point W. How are radii
and
related?
40. The midpoints of the sides of rhombus ABCD are joined in order to form quadrilateral MNPQ. Being as specific as possible, what type of quadrilateral is MNPQ? a. parallelogram b. rectangle c. square d. rhombus
41. In rectangle ABCD, , . Find the value of x.
, and
42. In the figure, bisects and
and D is the midpoint of , find the value of x.
43. Given that of ?
, what angle of
is congruent to
44. For , radii and and arc bound a sector. If m correct to the nearest tenth of a square inch.
.
. If
= 100° and PX = 5.6 inches, find the area of the sector
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45. In the figure,
and
bisect each other. If
46. In rhombus MNPQ, diagonal
is congruent to side
and
is
. Find m
larger than
, find
.
.
47. When the midpoints of the sides of a square RSTV are joined in order, quadrilateral MNPQ is formed. Being as specific as possible, what type of quadrilateral is MNPQ? 48. In addition to being congruent, how are the diagonals of a square related? 49. Find the measure of the angle formed by the hands of a clock at exactly 4:10 PM.
50. If m = 45° and OA = 4.3 cm, find l centimeter.
correct to the nearest tenth of a
51. Points A, B, and C are collinear. If AB = 12 and BC = 7, find the two possible lengths of .
52. In the figure, S-T-U-V and conclusion regarding and
.
. Draw a .
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53. 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? 54. Quadrilateral ABCD quadrilateral HJKL. If , , , and , find x. 55. Each base edge of a regular pentagonal pyramid measures 10 cm. If the lateral area of this pyramid is 425 , find the length of the slant height of the pyramid. 56. What is the name of this property? If , then . 57. Find the length of base in an isosceles triangle with perimeter 25 inches and length of leg 8 inches.
58. In rectangle ABCD, AB = 5 and BC = 4. As a square root, find the length of diagonal 59. Let point E lie in the exterior of
. How many tangent lines can be constructed to
60. In , chord diameter . With chords possible, what type of quadrilateral is RSTV?
and
61. In isosceles triangle RST with vertex angle S, 62. Use
. from point E?
, quadrilateral RSTV is formed. Being as specific as
. Describe the measure of
.
to find the exact total area of the right circular cone for which r = 3 cm and l = 5 cm.
63. For
, E lies on side
so that
. For
, what name is given to the line segment
?
64. The measures of the three interior angles of a triangle are x, 2x, and 3x. What type of triangle is the triangle described? 65. Find the measure of each exterior angle of a regular dodecagon (polygon with 12 sides). 66. In 67. In
.
,m
= 90°. What fraction represents the part of the circumference that is the length of , diagonals
and
?
intersect at point T. If MN = 12.5, NP = 8.7, and QN = 14.6, find QT.
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68. In this figure, and are complementary. Also, are complementary. Regarding and , what conclusion may you draw? 69. In
, the length of radius is 6 inches. If m
and m ?
= 60°, how much longer is chord
and
= 90°
than chord
70. 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. 71. One property of proportions takes the form, “If
, then
.” Use this property to complete this statement. Because
, it follows that:
72. In the proof that justify by the reason Identity?
, what statement can you
73. The figure shows right triangle ABC with Also, . Where , , and
. ,
, what reason allows you to conclude that
, ?
74. For the segment shown, let the center of the circle be point O. If the area of the related sector bounded by and and arc is 45 while the area of is 28 , find the area of the segment bounded by
and
.
75. You have proved that by the reason ASA and in turn that by CPCTC. What reason allows you to further conclude that “Z is the midpoint of ?”
.
Page 9
76. In isosceles triangle RST,
. If
77. The total area of a right circular cylinder is of the radius. 78. For trapezoid MNPQ, , and
, find the length of
.
. If the height of the cylinder is 4,find the length
. Also, A is the midpoint of , find the value of x.
while B is the midpoint of
. If
,
79. A triangle has a perimeter of 40 and area of 60. Using A = rP, find the length of radius r for the inscribed circle for this triangle. 80. Given a regular square pyramid, with respect to which figure(s) does this pyramid have symmetry . . . a point, a line, or a plane?
81. Provide missing statements and missing reasons for the proof of the theorem, “If two sides of a triangle are congruent, then the angles opposite those sides are also congruent.” Given: Prove:
with
S1. R1. S2. Draw the angle bisector for bisector. S3. R3. S4. R4.
R2. Every angle has exactly one angle-
82. Where is the degree measure for the arc of a sector of a circle, the ratio of the area of the sector to that of the area of the circle is given by the area of the sector is given by
.
. Use this ratio to explain why .
Page 10
[Note: In the figure, the sector with arc measure
is bounded by radii
,
, and
.]
83. Supply missing statements and missing reasons for the following proof. Given: also, Prove:
so that
bisects
;
S1. so that bisects R1. S2. R2. S3. R3. Given S4. R4. If 2 angles of one triangle are congruent to 2 angles of a second triangle, then the third angles of these triangles are also congruent.
84. Provide all statements and all reasons for this proof. Given:
with
;
with Prove:
.
Page 11
85. Supply missing statements and missing reasons for the following proof. Given: Prove:
in is an isosceles triangle
S1. R1. S2.
R2.
S3.
R3.
S4. ? and ? R4. The degree measure of an iscribed angle is equal to one-half the degree measure of its intercepted arc. S5. R5. S6. R6. Definition of congruent angles S7. R7. If two angles of a triangle are congruent, then the two sides that lie opposite those angles are also congruent. S8. R8.
86. 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
87. Supply missing statements and missing reasons for the following proof. .
Page 12
Given: Rectangle MNPQ with diagonals Prove:
and
S1. R1. S2. and are rt. R2. S3. R3. S4. R4. The diagonals of a rectangle are congruent. S5. R5.
88. Provide the missing statements and missing reasons for the following proof. Given: Prove:
and V is the midpoint of
S1. R1. S2. R2. S3. R3. Given S4. R4. S5. R5. Identity S6. S7. R7.
R6.
89. Provide mssing statements and missing reasons for the proof of the theorem, “Corresponding altitudes of congruent triangles are congruent.” Given: Prove:
;
and
S1. R1. S2. and R2. S3. R3. Given S4. and are rt. R4. S5. R5. All right angles are congruent. S6. R6. S7. R7. .
Page 13
90. Supply missing reasons for this proof. Given: m || n Prove: S1. m || n R1. S2. R2. S3. R3. S4. R4.
91. Supply missing statements and reasons for the following proof. Given: Prove:
and
intersect at point X
S1. R1. S2. and are supp. R2. S3. R3. If the exterior sides of two adjacent angles form a straight line, these angles are supplementary. S4. R4. Two angles that are supplementary to the same angle are congruent.
92. Use the drawing(s) to explain the 30 -60 .
Page 14
-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
93. Supply missing statements and missing reasons for the following proof. Given: ; V is the midpoint of and W is the midpoint of . Prove: S1. R1. S2.
and
R2. Definition of midpoint
S3. and R3. S4. R4. Substitution Property of Equality S5. R5. S6. R6. 94. Supply all statements and all reasons for the following proof. Given: ; M is the midpoint of and N is the midpoint of Prove: MNAB is a trapezoid
.
Page 15
95. Provide missing reasons for the proof of the theorem, “A diagonal of a parallelogram separates it into two congruent triangles.” Given: Prove: S1. S2. S3. S4. S5. S6. S7.
with diagonal with diagonal R2. R3. R4. R5. R6. R7.
R1.
96. Explain (prove) the following property of proportions. “If
(where
and
), then
.”
97. 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.
;
98. 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
.
Page 16
99. Provide all statements and all reasons for the following proof. Given:
,
,
, and Prove: Quad. ABCD is a parallelogram 100. Given: Prove:
and
are supplementary
Supply missing statements and missing reasons for this proof. S1. and are supplementary R1. S2. R2. If the exterior sides of 2 adjacent angles form a straight line, the angles are supplementary. S3. R3. Two angles that are supplementray to the same angle are congruent.
.
Page 17
Answer Key 1. True 2. True 3. False 4. True 5. False 6. True 7. True 8. True 9. False 10. True 11. True 12. False 13. True 14. False 15. False 16. True 17. True 18. False 19. False 20. True 21. c 22. c 23. d 24. b 25. b 26. a 27. c .
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28. c 29. c 30. d 31. c 32. a 33. b 34. b 35. c 36. d 37. b 38. d 39. c 40. b 41. 42. 29 43. 44. 27.4 45. 38° 46. 120° 47. a square 48. perpendicular-bisectors of each other 49. 65° 50. 3.4 cm 51. AC = 5 or AC = 19 52. 53.
.
Page 19
54. 55. 17 cm 56. Means-Extremes Property 57. 9 inches 58. 59. 2 60. isosceles trapezoid 61. 62. 63. altitude 64. right 65. 30° 66. 67. 7.3 68. 69.
or m
=m
inches
70. 71. 72. 73. CSSTP 74. 17 75. The definition of midpoint. 76. 77. 3 units 78. x = 4 79. 3 .
Page 20
80. both line and plane 81. S1. with R1. Given R3. The bisector of the vertex angle of an isosceles triangle separates the triangle into two congruent triangles. S4. R4. CPCTC 82. Given that
, it follows that
of the circle is given by
. Where r is the length of radius of the circle, the area
. By substitution, it follows that
. 83. R1. Given S2. R2. Definition of angle-bisector S3. S4. 84. S1. S2. S3. S4.
with R1. Given with R2. Given R3. Identity R4. SSS
85. S1. in R1. Given R2. Definition of congruent arcs R3. Multiplication (or Division) Property of Equality S4.
and
R5. Substitution Property of Equality S6. S7. S8. is an isosceles triangle R8. Definition of isosceles triangle 86. In
Then
, . 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
.
87. S1. Rectangle MNPQ with diagonals and R1. Given R2. All angles of a rectangle are right angles. R3. Identity S4. S5. R5. HL 88. R1. Given .
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R2. Perpendicular lines form congruent adjacent angles. S3. V is the midpoint of R4. Definition of midpoint S5. R6. SAS S7. R7. CPCTC 89. R1. Given R2. CPCTC S3. and R4. Perpendicular lines form right angles. S5. R6. AAS S7. R7. CPCTC 90. 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 91. S1. and intersect at point X R1. Given S3. and are supp. S4. 92. 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) 93. S1. ; V is the midpoint of R1. Given R3. Division Property of Equality
and W is the midpoint of
.
S4. R5. Identity S6. R6. SAS 94. S1. ; M is the midpoint of and N is the midpoint of R1. Given S2. R2. The line segment that joins the midpoints of 2 sides of a triangle is parallel to the third side .
Page 22
of the triangle. S3. MNAB is a trapezoid R3. Definition of trapezoid 95. R1. Given R2. The opposite sides of a parallelogram are parallel (definition). R3. If 2 lines are cut by a trans, the alternate interior angles are congruent. R4. Same as reason 2 R5. Same as reason 3. R6. Identity R7. ASA 96. Given that and
, we add 1 to each side of this equation. By the Addition Property of Equality, , so that
Property of Equality. In turn,
. But
by the Substitution .
97. Given that in , it follows that Then must be an isosceles triangle.
. But congruent arcs have congruent chords so that
.
98. 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. 99. S1. , R1. Given S2. R2. If 2 coplanar lines are to the same line, they are . S3. , and R3. Given S4. R4. Same as reason 2. S5. Quad. ABCD is a parallelogram R5. Definition of parallelogram 100. R1. Given S2. and are supplementary. S3.
.
Page 23
