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Name: ________________________ Class: ___________________ Date: __________
ID: A
ch 8 review
____
1. How many triangles are formed by drawing diagonals from one vertex in the figure? Find the sum of the
measures of the angles in the figure.
a.
b.
c.
d.
5, 900°
5, 1080°
6, 900°
6, 1080°
____
2. The sum of the measures of the interior angles of a convex quadrilateral is _____.
a. 180°
b. 270°
c. 360°
d. 540°
____
3. The measure of each interior angle of a regular hexagon is ________.
a.
b.
c.
d.
30°
120°
15°
60°
1
Name: ________________________
____
ID: A
4. The measure of each exterior angle of a regular octagon is ______.
a.
b.
c.
d.
22.5°
67.5°
45°
135°
Find the value of x. (The figure may not be drawn to scale.)
____
5.
a.
b.
c.
d.
74
108
49
51
____
6. Find the measure of each exterior angle of a regular polygon with 16 sides.
a. 11.25°
b. 360°
c. 22.5°
d. 157.5°
____
7. Find the measure of one of the exterior angles of a regular polygon with nine sides.
a. 140°
b. 40°
c. 160°
d. 20°
2
Name: ________________________
ID: A
8. Find the measure of the missing angle.
9. Find the value of x.
10. Find x and y.
11. Find the sum of the measures of the interior angles in the figure.
12. A regular pentagon has five congruent interior angles. What is the measure of each angle?
13. What is the measure of each interior angle in a regular octagon?
3
Name: ________________________
ID: A
14. Find the number of sides of a convex polygon if the measures of its interior angles have a sum of 2880°.
15. Find the number of sides of a regular polygon with each interior angle equal to 171°.
16. Find the measure of an interior angle and an exterior angle of a regular polygon with 20 sides.
17. What is the measure of each exterior angle in a regular pentagon?
Open-ended:
18. Using the drawing and the triangle sum theorem, explain why the sum of the measures of the interior angles
of a pentagon is 540°.
Find the measure of an interior angle and the measure of an exterior angle for the regular polygon.
19. 16-gon
20. 32-gon
Find each unknown angle measure.
21.
22.
23. Consider an octagonal stop sign.
a. Find the sum of the interior angles of a stop sign.
b. Find the measure of one of the interior angles of a stop sign.
c. Find the measure of an exterior angle of a stop sign.
4
Name: ________________________
ID: A
Writing:
24. Explain how the formula for finding the sum of the interior angles of a polygon can be derived.
Open-ended:
25. Write an indirect proof to show that the measure of each interior angle of a regular hexagon is not 150°.
26. Write an indirect proof to show that the measure of each interior angle of a regular decagon is 144°.
27. SHORT RESPONSE Write your answer on a separate piece of paper.
Figure ABCDE below is a regular pentagon.
What is the measure, in degrees, of ∠AFE? Explain in words how you arrived at your answer.
____ 28. For parallelogram PQLM below, if m∠PML = 83°, then m∠PQL =______ .
a.
b.
c.
d.
m∠PQM
83°
97°
m∠QLM
5
Name: ________________________
ID: A
____ 29. Consecutive angles in a parallelogram are always ________.
a. congruent angles
b. complementary angles
c. supplementary angles
d. vertical angles
____ 30. Choose the statement that is NOT ALWAYS true.
For any parallelogram _______.
a. the diagonals bisect each other
b. opposite angles are congruent
c. the diagonals are perpendicular
d. opposite sides are congruent
____ 31. Find the value of the variables in the parallelogram.
a.
b.
c.
d.
x = 52°, y = 10.5°, z = 159°
x = 21°, y = 55°, z = 104°
x = 55°, y = 21°, z = 104°
x = 10.5°, y = 52°, z = 159°
____ 32. If ON = 7x − 5, LM = 6x + 3, NM = x − 4, and OL = 2y + 5, find the values of x and y given that LMNO is a
parallelogram.
a.
b.
1
13
x= ;y=
2
2
x = 8; y = −2
c.
d.
6
1
2
x = 2; y = −2
x = 8; y = −
Name: ________________________
ID: A
33. Complete the statement for parallelogram ABCD. Then state a definition or theorem as the reason.
AD ≅ ____
34. Find AM in the parallelogram if PN = 10 and MO = 19.
35. Refer to the figure below.
Given: UVWX is a parallelogram, m∠WXV = 17°, m∠WVX = 29°, XW = 41 , UX = 24 , UY = 15
a. Find m∠WVU.
b. Find WV.
c. Find m∠XUV.
d. Find UW.
7
Name: ________________________
ID: A
36. Use the figure below.
Given: FGHJ is a parallelogram, m∠JHG = 68°, JH = 34, GH = 19
a. Find m∠FJH.
b. Find JF.
c. Find m∠GFJ.
d. Find FG.
True or False:
37. If a quadrilateral is a parallelogram, then consecutive angles are complementary.
38. If a quadrilateral is a parallelogram, then opposite angles are complementary.
Use the diagram to find the given length.
39. AC
40. BD
8
Name: ________________________
ID: A
41. Complete the steps of this proof.
Given: parallelogram WXYZ
Prove: ∆XYZ ≅ ZWX
Statements
Reasons
1. parallelogram WXYZ
1.
2. WZ || XY ; WX || YZ
2. ?
3. ∠WZX ≅ ∠YXZ;
∠WXZ ≅ ∠YZX
?
3. ?
4. ZX ≅ ZX
4. ?
5. ∆XYZ ≅ ∆ZWX
5. ?
42. SHORT RESPONSE Write your answer on a separate piece of paper.
ABCD is a parallelogram. Find the value of x and explain your reasoning. (The figure may not be drawn to
scale.)
____ 43. (2, 3) and (3, 1) are opposite vertices in a parallelogram. If (0, 0) is the third vertex, then the fourth vertex is
_____.
1, − 1
a.
b.
5
, 2
2
c.
−1, 2
d.
5, 4
9
Name: ________________________
ID: A
44. Given the following, determine whether quadrilateral XYZW must be a parallelogram. Justify your answer.
XY ≅ WZ and XW ≅ YZ.
45. Given: ∆ABF ≅ ∆DEC and FB EC
Prove: BCEF is a parallelogram.
46. Given: ABCDF is a parallelogram and FB EC
Prove: BCEF is a parallelogram.
10
Name: ________________________
ID: A
47. Given: VU ≅ ST and SV ≅ TU
Prove: VX = XT
48. Given: VU ≅ ST and VU
Prove: VX = XT
ST
49. Given: SV ≅ TU and SV TU
Prove: VX = XT
11
Name: ________________________
ID: A
50. Draw a figure in the coordinate plane and write a two-column coordinate proof.
Given: Quadrilateral ABCD with A(–5, 0), B(4, –3), C(8, –1), D(–1, 2)
Prove: ABCD is a parallelogram.
51. Use the distance formula to determine whether ABCD below is a parallelogram.
52. Use the Distance Formula to determine whether ABCD below is a parallelogram.
12
Name: ________________________
ID: A
53. Given: ST || UV and TU || SV
Prove: VX = XT
Open-ended:
54. Find a fourth point, D, so that a parallelogram is formed using the vertices
A 0, − 4 , B 5, − 3 , C −4, − 3 , and D in any order. Plot your point and draw the parallelogram in the
coordinate plane.
55. SHORT RESPONSE Write your answer on a separate piece of paper.
Is quadrilateral ABCD a parallelogram? Explain your answer briefly. (The figure may not be drawn to scale.)
13
Name: ________________________
ID: A
____ 56. Which statement is true?
a. All quadrilaterals are squares.
b. All rectangles are squares.
c. All parallelograms are quadrilaterals.
d. All quadrilaterals are parallelograms.
____ 57. Choose the statement that is NOT ALWAYS true. For a rhombus ________.
a. each diagonal bisects a pair of opposite angles
b. all four sides are congruent
c. the diagonals are congruent
d. the diagonals are perpendicular
____ 58. The diagonals of a parallelogram always _________.
a. are congruent
b. are parallel
c. bisect each other
d. are perpendicular
____ 59. Which statement is NOT always true of a rhombus?
a. The diagonals are perpendicular to each other.
b. The diagonals bisect each other.
c. Each diagonal is longer than at least one side.
d. The sum of the diagonals is less than the perimeter.
60. Draw a Venn diagram showing the relationship between squares, rectangles, rhombuses, parallelograms, and
quadrilaterals.
61. Consider the statement, "If a parallelogram is a square, then it is a rhombus."
a. Decide whether it is true or false.
b. Write the converse.
c. Decide whether the converse is true or false.
62. If the diagonals of a parallelogram are perpendicular, then the parallelogram is also what type of figure?
63. If the diagonals of a parallelogram are equal in length, then the parallelogram is also what type of figure?
64. True or false: A rectangle is a parallelogram.
65. True or false: A quadrilateral has three diagonals.
66. True or false: A rhombus is a regular polygon.
67. True or false: A quadrilateral is a polygon with four angles.
68. True or false: A square is a rectangle.
69. True or false: In quadrilateral ABCD, AB and CD are adjacent sides.
14
Name: ________________________
ID: A
70. True or false: Opposite angles in a parallelogram are supplementary.
71. True or false: Opposite sides of a parallelogram are congruent.
72. True or false: A rectangle is an equiangular quadrilateral.
73. True or false: The sum of the measures of the angles of a quadrilateral is 180°.
Open-ended: Give the possible coordinates of the vertices of the quadrilateral so that the diagonals lie
on the x- and y-axes.
74. a rhombus with its horizontal diagonal longer than its vertical diagonal
75. a square with diagonals more than 10 units long
76. a rhombus with its horizontal diagonal shorter than its vertical diagonal
77. a square with diagonals more than 8 units long
True or False:
78. All squares are quadrilaterals.
79. If a quadrilateral is a parallelogram, then it is a kite.
80. Quadrilateral DEFG is a rhombus. What is the value of x? You can use the following fact to help you: If two
sides of a triangle are congruent, then the angles opposite them are congruent. (The figure may not be drawn
to scale.)
81. Determine if the statement is a valid definition. If it is not, state a counterexample.
A square is a figure with four right angles and four congruent sides.
Decide if the argument is valid or invalid. Explain your reasoning.
82. If a figure is a rhombus, then it is a parallelogram.
A square is a rhombus.
Therefore, a square is a parallelogram.
15
Name: ________________________
ID: A
83. Performance Task: Does showing that all four pairs of corresponding sides are congruent prove that two
quadrilaterals are congruent? Explain why or why not. If not, what additional information would you need to
show in order to prove the two quadrilaterals congruent?
84. EXTENDED RESPONSE Write your answer on a separate piece of paper.
The coordinates of the vertices of a quadrilateral are A −8, − 3 , B −6, 3 , C −3, 2 , and D −5, − 4 .
Part A How long is each side of the quadrilateral? Show your work.
Part B What are the slopes of each side of the quadrilateral? Show your work.
Part C What type of quadrilateral is it? Explain your reasoning.
85. Given: ABCD is a rhombus.
Prove: ∆ACB ≅ ∆CAD
86. Writing: Explain the difference between a rhombus and a rectangle.
87. Is the biconditional "A quadrilateral is a square if and only if it is a rectangle" True or False? Explain your
reasoning.
88. a. Is the statement "If a quadrilateral is a square, then it is a rectangle" True or False?
b. State the converse, inverse, and contrapositive of the statement in part (a). Which, if any, of these is a true
statement?
89. a. Is the statement "If a quadrilateral is a rectangle, then it is a parallelogram" True or False?
b. Write the inverse of the statement in part (a) and tell if it is True or False.
90. a. Is the statement "If a quadrilateral is a square, then it is a rhombus" True or False?
b. Write the contrapositive of the statement in part (a) and tell if it is True or False.
16
Name: ________________________
ID: A
Decide if the argument is valid or invalid. If the argument is valid, tell which rule of logic is used. If the
argument is invalid, tell why.
91. If a quadrilateral is a rectangle, then it is a parallelogram.
Quadrilateral ABCD is a parallelogram.
Therefore, quadrilateral ABCD is a rectangle.
Performance Task:
92. Graph the line y = 2x − 3. Find the equations of three other lines that when graphed with y = 2x − 3 enclose a
rectangle on the coordinate plane.
____ 93. Isosceles trapezoid JKLM has legs JK and LM , and base KL. If JK = 8x − 9, KL = 7x + 10, and LM = 10x + 2,
find the value of x.
a. −1
c. 19
11
8
b. −
d.
2
3
____ 94. Choose the statement that is NOT always true.
For an isosceles trapezoid _______.
a. the diagonals are congruent
b. the base angles are congruent
c. the diagonals are perpendicular
d. the legs are congruent
17
Name: ________________________
ID: A
____ 95. For the trapezoid shown below, the measure of the midsegment is _______.
a.
b.
c.
d.
29
58
25
30
____ 96. Choose the figure below which satisfies the definition of a kite.
c.
a.
b.
d.
____ 97. Which type of quadrilateral has no parallel sides?
a. rectangle
b. trapezoid
c. rhombus
d. kite
18
Name: ________________________
ID: A
____ 98. Three vertices of an isosceles trapezoid are shown in the figure below.
What are the coordinates of the missing vertex that make the bases parallel to the x-axis?
−2, 1
−2, 0
a.
c.
b.
−3, 0
d.
−3, 1
99. In what type of trapezoid are the base angles congruent?
100. Given: Trapezoid ABCD with midsegment EF . If EF = 23 and DC = 26, find the length of AB.
101. One side of a kite is 5 cm less than 2 times the length of another. If the perimeter is 8 cm, find the length of
each side of the kite.
19
Name: ________________________
ID: A
102. Find m∠T in the diagram, if m∠R = 130° and m∠S = 60°.
103. Writing: Write a paragraph proof.
Given: kite EFGH
Prove: ∠F ≅ ∠H
104. Writing: Write a paragraph proof.
Given: kite EFGH
Prove: ∠FEG ≅ ∠HEG
Open-ended: Give the possible coordinates of the vertices of the quadrilateral so that the diagonals lie
on the
x- and y-axes.
105. a trapezoid
20
Name: ________________________
ID: A
Open-ended:
106. Use the Venn diagram below to compare and contrast the properties of a parallelogram and the properties of
a kite. The intersection represents the properties that they have in common.
Performance Task:
107. If an equilateral triangle with a side length of 1 unit is folded so that the vertex of the triangle touches the
midpoint of the base, find the perimeter of the trapezoid formed.
Open-ended:
108. Draw a kite in the coordinate plane. Show that your quadrilateral is a kite.
109. Show that quadrilateral ABCD with vertices A(0, 0), B(6, 0), C(5, 2), and D(1, 2) is an isosceles trapezoid.
110. Show that quadrilateral JKLM with vertices J 0, 0 , K 3, 0 , L 6, 2 , and M −3, 2 is an isosceles
trapezoid.
21
Name: ________________________
ID: A
111. SHORT RESPONSE Write your answer on a separate piece of paper.
Figure ABCD below is a trapezoid.
Find the value of a, and then describe two ways to find the value of c and give its value.
____ 112. The coordinates of quadrilateral PQRS are P(–3, 0), Q(0, 4), R(4, 1), and S(1, –3). What best describes the
quadrilateral?
a. a rectangle
b. a square
c. a rhombus
d. a parallelogram
____ 113. Use slope or the Distance Formula to determine the most precise name for the figure: A(–1, –4), B(1, –1),
C(4, 1), D(2, –2).
a. kite
b. trapezoid
c. rhombus
d. square
____ 114. If all four sides of a quadrilateral are congruent, the quadrilateral is _______.
a. a kite
b. a nonsquare rectangle
c. a rhombus
d. a trapezoid
What name best describes the quadrilateral?
____ 115.
a.
b.
c.
d.
parallelogram
rhombus
kite
rectangle
22
Name: ________________________
ID: A
____ 116.
a.
b.
c.
d.
kite
rectangle
parallelogram
triangle
____ 117. Which statement is false?
a. All rhombuses are kites.
b. All squares are rhombuses.
c. Every kite is a rectangle.
d. All squares are quadrilaterals.
____ 118. Which statement is false?
a. Every square is a parallelogram.
b. Some rhombuses are rectangles.
c. Every rhombus is a quadrilateral.
d. Every parallelogram is a rhombus.
____ 119. Which statement is false?
a. If a quadrilateral is a square, then it is not a kite.
b. Some parallelograms are rhombuses.
c. All parallelograms are quadrilaterals.
d. If a quadrilateral is a rectangle, then it is a kite.
120. Describe the figure using as many of these words as possible: rectangle, trapezoid, square, quadrilateral,
parallelogram, rhombus.
121. Identify the quadrilateral which has all sides and angles congruent.
23
Name: ________________________
ID: A
122. Open-ended Problem: List all of the important characteristics of each quadrilateral.
a. square
b. rectangle
c. parallelogram
d. rhombus
e. trapezoid
f. kite
123. Quadrilateral ABCD has vertices A −2, − 2 , B 3, − 2 , C 6, 2 , and D 1, 2 . What type of quadrilateral
is ABCD?
124. Quadrilateral ABCD has vertices A −2, − 5 , B 8, − 5 , C 6, − 1 , and D 0, − 1 . What type of
quadrilateral is ABCD? Explain your reasoning.
125. Quadrilateral ABCD has vertices A −6, − 2 , B 0, − 2 , C 4, 2 , and D 2, 6 . What type of quadrilateral
is ABCD? Explain your reasoning.
Performance Task:
126. Draw a Venn diagram showing the relationships among the various types of quadrilaterals.
127. Theorem 8.18 says that if a quadrilateral is a kite, then its diagonals are perpendicular. Is the converse true?
Justify your reasoning.
128. Prove that quadrilateral PQRS is a rhombus by showing that it is a parallelogram with perpendicular
diagonals.
24
Name: ________________________
ID: A
129. Prove quadrilateral HIJK is an isosceles trapezoid.
130. Prove quadrilateral HIJK is an isosceles trapezoid by showing it is a trapezoid with congruent diagonals.
131. Prove quadrilateral CDEF is a non-isosceles trapezoid.
25
Name: ________________________
ID: A
132. Prove quadrilateral BCDE is an isosceles trapezoid.
133. In the diagram, m∠BAC = 30°, m∠DCA = 110°, ∠BCA ≅ ∠DAC , and AC ≅ BD. Is enough information
given to show that quadrilateral ABCD is an isosceles trapezoid? Explain.
134. ∆ABC ≅ ∆CDA. What special type of quadrilateral is ABCD? Write a paragraph proof to support your
conclusion.
26
ID: A
ch 8 review
Answer Section
1. ANS:
TOP:
KEY:
NOT:
2. ANS:
TOP:
KEY:
NOT:
3. ANS:
STA:
KEY:
BLM:
4. ANS:
STA:
KEY:
BLM:
5. ANS:
TOP:
BLM:
6. ANS:
TOP:
KEY:
NOT:
7. ANS:
TOP:
KEY:
NOT:
8. ANS:
114°
A
PTS: 1
DIF: Level B
REF: MOT70179
Lesson 8.1 Find Angle Measures in Polygons
diagonals | sum | interior angle measures of polygons
BLM: Application
978-0-618-65613-4
C
PTS: 1
DIF: Level B
REF: HLGM0440
Lesson 8.1 Find Angle Measures in Polygons
sum | quadrilateral | interior angle measures of polygons BLM: Application
978-0-618-65613-4
B
PTS: 1
DIF: Level B
REF: HLGM0441
MI.MIGLC.MTH.06.9-12.G1.5.2 TOP: Lesson 8.1 Find Angle Measures in Polygons
regular polygon | hexagon | interior angle measures of polygons
Application NOT: 978-0-618-65613-4
C
PTS: 1
DIF: Level B
REF: HLGM0451
MI.MIGLC.MTH.06.9-12.G1.5.2 TOP: Lesson 8.1 Find Angle Measures in Polygons
octagon | regular polygon | exterior angle measures of polygons
Application NOT: 978-0-618-65613-4
D
PTS: 1
DIF: Level B
REF: MLPA0713
Lesson 8.1 Find Angle Measures in Polygons
KEY: exterior angle measures of polygons
Application NOT: 978-0-618-65613-4
C
PTS: 1
DIF: Level B
REF: GMPA0651
Lesson 8.1 Find Angle Measures in Polygons
regular polygon | exterior angle measures of polygons
BLM: Application
978-0-618-65613-4
B
PTS: 1
DIF: Level B
REF: AXGM0234
Lesson 8.1 Find Angle Measures in Polygons
regular polygon | interior angle measures of polygons
BLM: Application
978-0-618-65613-4
PTS:
TOP:
KEY:
NOT:
9. ANS:
128°
1
DIF: Level B
REF: ACG60049
Lesson 8.1 Find Angle Measures in Polygons
quadrilateral | interior angle measures of polygons
978-0-618-65613-4
PTS:
TOP:
KEY:
NOT:
1
DIF: Level B
REF: MPPA1214
Lesson 8.1 Find Angle Measures in Polygons
quadrilateral | exterior angle measures of polygons
978-0-618-65613-4
1
BLM: Application
STA: MI.MIGLC.MTH.06.9-12.G1.5.2
BLM: Application
ID: A
10. ANS:
x = 103, y = 66
PTS:
TOP:
KEY:
BLM:
11. ANS:
540°
1
DIF: Level B
REF: PHGM0246 STA: MI.MIGLC.MTH.06.9-12.G1.5.2
Lesson 8.1 Find Angle Measures in Polygons
quadrilateral | supplementary angles | interior angle measures of polygons
Application NOT: 978-0-618-65613-4
PTS:
TOP:
KEY:
NOT:
12. ANS:
108°
1
DIF: Level B
REF: PHGM0214
Lesson 8.1 Find Angle Measures in Polygons
polygon | sum | interior angle measures of polygons
978-0-618-65613-4
PTS:
TOP:
KEY:
BLM:
13. ANS:
135°
1
DIF: Level B
REF: HLGM0437 STA: MI.MIGLC.MTH.06.9-12.G1.5.2
Lesson 8.1 Find Angle Measures in Polygons
regular polygon | pentagon | interior angle measures of polygons
Application NOT: 978-0-618-65613-4
PTS:
TOP:
KEY:
BLM:
14. ANS:
18
1
DIF: Level B
REF: HLGM0442 STA: MI.MIGLC.MTH.06.9-12.G1.5.2
Lesson 8.1 Find Angle Measures in Polygons
regular polygon | octagon | interior angle measures of polygons
Application NOT: 978-0-618-65613-4
PTS:
TOP:
KEY:
NOT:
15. ANS:
40
1
DIF: Level B
REF: AGEO0311
Lesson 8.1 Find Angle Measures in Polygons
polygon | sum | interior angle measures of polygons
978-0-618-65613-4
PTS:
TOP:
KEY:
NOT:
1
DIF: Level B
REF: XEGS0702
Lesson 8.1 Find Angle Measures in Polygons
regular polygon | interior angle measures of polygons
978-0-618-65613-4
2
BLM: Application
BLM: Application
BLM: Application
ID: A
16. ANS:
interior angle: 162 degrees; exterior angle: 18 degrees
PTS:
TOP:
KEY:
BLM:
17. ANS:
72°
1
DIF: Level B
REF: GGEO1003 STA: MI.MIGLC.MTH.06.9-12.G1.5.2
Lesson 8.1 Find Angle Measures in Polygons
regular polygon | interior angle measures of polygons | exterior angle measures of polygons
Application NOT: 978-0-618-65613-4
PTS: 1
DIF: Level B
REF: HLGM0444 STA: MI.MIGLC.MTH.06.9-12.G1.5.2
TOP: Lesson 8.1 Find Angle Measures in Polygons
KEY: regular pentagon | exterior angle measures of polygons
BLM: Application
NOT: 978-0-618-65613-4
18. ANS:
Sample answer: By drawing all the diagonals possible from one vertex, the pentagon is divided into 3
triangles. Since the sum of the angles of each triangle is 180°, the sum of the angles of the pentagon is 3 •
180°, or 540°.
PTS: 1
DIF: Level B
REF: MIM20457
TOP: Lesson 8.1 Find Angle Measures in Polygons
KEY: sum | interior angle measures of polygons | pentagon
NOT: 978-0-618-65613-4
19. ANS:
157.5°, 22.5°
PTS: 1
DIF: Level B
REF: MLPA0709
TOP: Lesson 8.1 Find Angle Measures in Polygons
KEY: regular polygon | interior angle measures of polygons
NOT: 978-0-618-65613-4
20. ANS:
about 168.8°, about 11.2°
PTS:
TOP:
KEY:
NOT:
1
DIF: Level B
REF: MLPA0710
Lesson 8.1 Find Angle Measures in Polygons
regular polygon | interior angle measures of polygons
978-0-618-65613-4
3
STA: MI.MIGLC.MTH.06.9-12.G1.4.4
BLM: Comprehension
BLM: Application
BLM: Application
ID: A
21. ANS:
55°, 60°, 75°, 80°
PTS: 1
DIF: Level B
REF: BS022047
TOP: Lesson 8.1 Find Angle Measures in Polygons
BLM: Application NOT: 978-0-618-65613-4
22. ANS:
56°, 84°, 92°, 128°
PTS: 1
DIF: Level B
REF: BS022049
TOP: Lesson 8.1 Find Angle Measures in Polygons
BLM: Application NOT: 978-0-618-65613-4
23. ANS:
a. 1080°
b. 135°
c. 45°
KEY: exterior angle measures of polygons
KEY: exterior angle measures of polygons
PTS: 1
DIF: Level A
REF: PA.13.03.SR.05
TOP: Lesson 8.1 Find Angle Measures in Polygons
KEY: exterior angle measures of polygons | interior angle measures of polygons | sum |regular polygon
BLM: Application NOT: 978-0-618-65613-4
24. ANS:
Sample answer: By drawing all the diagonals from one vertex of a polygon, the polygon is divided into two
fewer triangles than there are sides of the polygon (that is, n − 2 triangles if n = the number of sides.) Since
the sum of the measures of the angles of a triangle is 180°, and since the angles of the polygon are formed by
combinations of the angles of these triangles, the formula is S = (n – 2)180°.
PTS: 1
DIF: Level B
REF: MIM30157 STA: MI.MIGLC.MTH.06.9-12.G1.4.4
TOP: Lesson 8.1 Find Angle Measures in Polygons
KEY: polygon | sum | interior angle measures of polygons
BLM: Comprehension
NOT: 978-0-618-65613-4
25. ANS:
Sample answer: Assume that the measure of each interior angle of a regular hexagon is 150°. Then the sum
of the measures is 6(150°) = 900°. So, (n – 2)180° = 900° and thus n − 2 = 5, or n = 7. But this contradicts
the fact that a hexagon has 6 sides. Therefore, the assumption is false and it must be true that the measure of
each interior angle of a regular hexagon is not 150°.
PTS:
NAT:
STA:
TOP:
KEY:
BLM:
1
DIF: Level B
REF: MIM30184
NCTM 9-12.REA.3 | NCTM 9-12.REA.4
MI.MIGLC.MTH.06.9-12.L3.3.2 | MI.MIGLC.MTH.06.9-12.G1.4.4
Lesson 8.1 Find Angle Measures in Polygons
interior angle measures of polygons | hexagon | proof | indirect
Analysis
NOT: 978-0-618-65613-4
4
ID: A
26. ANS:
Sample answer: Assume that the measure of each interior angle of a regular decagon is not 144°. Then, since
a decagon has 10 sides, the sum of the measures of the interior angles is not 10(144°) or 1440°. So (n –
2)180° ↑ 1440° and thus n − 2 ≠ 8, and n ≠ 10. But this contradicts the fact that a decagon does have 10
sides. Therefore, the assumption is false and it is true that the measure of each interior angle of a regular
decagon is 144°.
PTS: 1
DIF: Level B
REF: MIM30185
NAT: NCTM 9-12.REA.3 | NCTM 9-12.REA.4
STA: MI.MIGLC.MTH.06.9-12.L3.3.2 | MI.MIGLC.MTH.06.9-12.G1.4.4
TOP: Lesson 8.1 Find Angle Measures in Polygons
KEY: proof | indirect | interior angle measures of polygons
BLM: Analysis
NOT: 978-0-618-65613-4
27. ANS:
The measure of ∠AFE is 108°. Pentagon sides AE and AB are congruent because ABCDE is a regular
pentagon. Thus, ∆EAB is isosceles. The measure of ∠EAB is 108°, because each interior angle of a regular
pentagon has a measure of 540° ÷ 5, or 108°. That leaves 72° for the sum of the measures of the two
congruent base angles ∠AEF and ∠ABF, so the measure of ∠AEF is half of 72°, or 36°. Triangle EDA is
congruent to triangle EAB, because two sides and the included angle in ∆EDA are congruent to two sides
and the included angle in ∆EAB. That means that the measure of ∠EAF is 36°, because ∠EAF ≅ ∠AEF. In
∆EAF, m∠AFE = 180° – m∠EAF − m∠AEF , so the measure of ∠AFE is 108°:180° – 36° – 36° = 108°.
PTS: 1
DIF: Level C
REF: MC100116
NAT: NCTM 9-12.GEO.1.c | NCTM 9-12.REA.4 | NCTM 9-12.REA.3
STA: MI.MIGLC.MTH.06.9-12.G1.4.2 | MI.MIGLC.MTH.06.9-12.G1.5.2 |
MI.MIGLC.MTH.06.9-12.G2.3.1 | MI.MIGLC.MTH.06.9-12.G2.3.2
TOP: Lesson 8.1 Find Angle Measures in Polygons
KEY: pentagon | regular polygon | explain
BLM: Analysis
NOT: 978-0-618-65613-4
28. ANS: B
PTS: 1
DIF: Level B
REF: HLGM0457
NAT: NCTM 9-12.GEO.1.a
STA: MI.MIGLC.MTH.06.9-12.G1.4.1 | MI.MIGLC.MTH.06.9-12.G1.4.2 |
MI.MIGLC.MTH.06.9-12.G1.4.3
TOP: Lesson 8.2 Use Properties of Parallelograms
KEY: angle measure | parallelogram
BLM: Comprehension
NOT: 978-0-618-65613-4
29. ANS: C
PTS: 1
DIF: Level A
REF: MLGE0285
NAT: NCTM 9-12.GEO.1.a
STA: MI.MIGLC.MTH.06.9-12.G1.4.1 | MI.MIGLC.MTH.06.9-12.G1.4.2 |
MI.MIGLC.MTH.06.9-12.G1.4.3
TOP: Lesson 8.2 Use Properties of Parallelograms
KEY: parallelogram | consecutive interior angles | property
BLM: Comprehension
NOT: 978-0-618-65613-4
30. ANS: C
PTS: 1
DIF: Level B
REF: MHST0010
STA: MI.MIGLC.MTH.06.9-12.G1.4.1 | MI.MIGLC.MTH.06.9-12.G1.4.2 |
MI.MIGLC.MTH.06.9-12.G1.4.3
TOP: Lesson 8.2 Use Properties of Parallelograms
KEY: property | parallelogram
BLM: Comprehension
NOT: 978-0-618-65613-4
5
ID: A
31. ANS: B
PTS: 1
DIF: Level B
REF: MHN90085
NAT: NCTM 9-12.GEO.1.a
STA: MI.MIGLC.MTH.06.9-12.G1.1.2 | MI.MIGLC.MTH.06.9-12.G1.2.1 |
MI.MIGLC.MTH.06.9-12.G1.2.2 | MI.MIGLC.MTH.06.9-12.G1.4.1 | MI.MIGLC.MTH.06.9-12.G1.4.2 |
MI.MIGLC.MTH.06.9-12.G1.4.3
TOP: Lesson 8.2 Use Properties of Parallelograms
KEY: angle measure | parallelogram | diagonals
BLM: Application
NOT: 978-0-618-65613-4
32. ANS: C
PTS: 1
DIF: Level B
REF: MLGE0400
NAT: NCTM 9-12.GEO.1.a
STA: MI.MIGLC.MTH.06.9-12.G1.4.1 | MI.MIGLC.MTH.06.9-12.G1.4.2 |
MI.MIGLC.MTH.06.9-12.G1.4.3 | MI.MIGLC.MTH.06.9-12.G1.5.2
TOP: Lesson 8.2 Use Properties of Parallelograms
KEY: side lengths | parallelogram
BLM: Application NOT: 978-0-618-65613-4
33. ANS: BC , the opposite sides of a parallelogram are congruent
PTS: 1
DIF: Level B
REF: GGEO0601 NAT: NCTM 9-12.GEO.1.a
STA: MI.MIGLC.MTH.06.9-12.G1.4.1 | MI.MIGLC.MTH.06.9-12.G1.4.2 |
MI.MIGLC.MTH.06.9-12.G1.4.3
TOP: Lesson 8.2 Use Properties of Parallelograms
KEY: property | parallelogram
BLM: Comprehension
NOT: 978-0-618-65613-4
34. ANS:
9.5
PTS: 1
DIF: Level B
REF: PHGM0902 NAT: NCTM 9-12.GEO.1.a
STA: MI.MIGLC.MTH.06.9-12.G1.4.1 | MI.MIGLC.MTH.06.9-12.G1.4.2 |
MI.MIGLC.MTH.06.9-12.G1.4.3
TOP: Lesson 8.2 Use Properties of Parallelograms
KEY: parallelogram | diagonal
BLM: Application NOT: 978-0-618-65613-4
35. ANS:
a. 46°
b. 24
c. 134°
d. 30
PTS: 1
DIF: Level B
REF: MLGE0132 NAT: NCTM 9-12.GEO.1.a
STA: MI.MIGLC.MTH.06.9-12.G1.4.1 | MI.MIGLC.MTH.06.9-12.G1.4.2 |
MI.MIGLC.MTH.06.9-12.G1.4.3 | MI.MIGLC.MTH.06.9-12.G1.4.4
TOP: Lesson 8.2 Use Properties of Parallelograms
KEY: angle measure | parallelogram | diagonals | side lengths
BLM: Application
NOT: 978-0-618-65613-4
6
ID: A
36. ANS:
a. 112°
b. 19
c. 68°
d. 34
PTS: 1
DIF: Level B
REF: MLGE0133 NAT: NCTM 9-12.GEO.1.a
STA: MI.MIGLC.MTH.06.9-12.G1.4.1 | MI.MIGLC.MTH.06.9-12.G1.4.2 |
MI.MIGLC.MTH.06.9-12.G1.4.3 | MI.MIGLC.MTH.06.9-12.G1.4.4
TOP: Lesson 8.2 Use Properties of Parallelograms
KEY: angle measure | parallelogram | diagonals | side lengths
BLM: Application
NOT: 978-0-618-65613-4
37. ANS:
False
PTS: 1
DIF: Level A
REF: MIM20428
STA: MI.MIGLC.MTH.06.9-12.G1.4.1 | MI.MIGLC.MTH.06.9-12.G1.4.2 |
MI.MIGLC.MTH.06.9-12.G1.4.3
TOP: Lesson 8.2 Use Properties of Parallelograms
KEY: parallelogram | property
BLM: Comprehension
NOT: 978-0-618-65613-4
38. ANS:
False
PTS: 1
DIF: Level A
REF: MIM20430
STA: MI.MIGLC.MTH.06.9-12.G1.4.1 | MI.MIGLC.MTH.06.9-12.G1.4.2 |
MI.MIGLC.MTH.06.9-12.G1.4.3
TOP: Lesson 8.2 Use Properties of Parallelograms
KEY: parallelogram | property
BLM: Comprehension
NOT: 978-0-618-65613-4
39. ANS:
10
PTS: 1
DIF: Level B
REF: 7f5cc4e2-cdbb-11db-b502-0011258082f7
TOP: Lesson 8.2 Use Properties of Parallelograms
KEY: Parallelogram | bisect | diagonal
BLM: Knowledge NOT: 978-0-618-65613-4
40. ANS:
8
PTS: 1
DIF: Level B
REF: 7f5dd6e7-cdbb-11db-b502-0011258082f7
TOP: Lesson 8.2 Use Properties of Parallelograms
KEY: Parallelogram | bisect | diagonal
BLM: Knowledge NOT: 978-0-618-65613-4
7
ID: A
41. ANS:
Statements
Reasons
1. parallelogram WXYZ
1. Given
2. WZ || XY ; WX || YZ
2. Definition of parallelogram
3. ∠WZX ≅ ∠YXZ;
3. If 2 parallel lines are intersected by
∠WXZ ≅ ∠YZX
a transversal, then alternate interior
angles are congruent.
4. ZX ≅ ZX
4. Reflexive Property
5. ∆XYZ ≅ ∆ZWX
5. ASA Postulate
PTS: 1
DIF: Level B
REF: BS022244
NAT: NCTM 9-12.REA.4
TOP: Lesson 8.2 Use Properties of Parallelograms
KEY: triangle | parallelogram | proof
BLM: Analysis
NOT: 978-0-618-65613-4
42. ANS:
The fact that ABCD is a parallelogram means that AB and CD are parallel. That leads to
m∠DCB = m∠EBC = 132°, because m∠DCB and m∠EBC are congruent alternate interior angles formed
by a transversal through parallel lines. Because of angle addition, m∠DCB = 51° + x°. Using substitution, 51°
+ x° = 132°, so x = 132 − 51 = 81.
PTS: 1
DIF: Level B
REF: MCT90009
NAT: NCTM 9-12.ALG.2.b | NCTM 9-12.GEO.1.c | NCTM 9-12.GEO.1.a | NCTM 9-12.REA.4 | NCTM
9-12.REA.3
STA: MI.MIGLC.MTH.06.9-12.G1.1.2 | MI.MIGLC.MTH.06.9-12.G1.4.1 |
MI.MIGLC.MTH.06.9-12.G1.4.2 | MI.MIGLC.MTH.06.9-12.G1.4.3
TOP: Lesson 8.2 Use Properties of Parallelograms
KEY: angle measure | parallelogram | interior | exterior
BLM: Analysis
NOT: 978-0-618-65613-4
43. ANS: D
PTS: 1
DIF: Level B
REF: MLGE0286
STA: MI.MIGLC.MTH.06.9-12.G1.4.2 | MI.MIGLC.MTH.06.9-12.G2.3.4
TOP: Lesson 8.3 Show that a Quadrilateral is a Parallelogram
KEY: vertices | coordinates | parallelogram
BLM: Application
NOT: 978-0-618-65613-4
44. ANS:
Yes. If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral
is a parallelogram.
PTS: 1
DIF: Level B
REF: AD010115
STA: MI.MIGLC.MTH.06.9-12.G1.4.1 | MI.MIGLC.MTH.06.9-12.G1.4.2 |
MI.MIGLC.MTH.06.9-12.G1.4.3
TOP: Lesson 8.3 Show that a Quadrilateral is a Parallelogram
KEY: parallelogram | quadrilateral | justify | diagonals
BLM: Application
NOT: 978-0-618-65613-4
8
ID: A
45. ANS:
1. ∆ABF ≅ ∆DEC
1. Given
2. BF ≅ EC
2. Corresponding Parts of ≅ ∆ are ≅.
3. FB
3. Given
EC
4. BCEF is a parallelogram.
4. If 1 pair of opposite sides are and ≅ ,
then the quadrilateral is a parallelogram.
PTS: 1
DIF: Level B
REF: MLGE0301A
NAT: NCTM 9-12.REA.4 | NCTM 9-12.GEO.1.c | NCTM 9-12.REA.3
TOP: Lesson 8.3 Show that a Quadrilateral is a Parallelogram
KEY: triangle | parallelogram | parallel lines | congruence | proof | CPCTC
BLM: Analysis
NOT: 978-0-618-65613-4
46. ANS:
1. ACDF is a parallelogram 1. Given
2. FE CB
2. Definition of a parallelogram
3. FB
3. Given
EC
4. BCEF is a parallelogram.
PTS:
NAT:
TOP:
KEY:
BLM:
47. ANS:
4. Definition of a parallelogram
1
DIF: Level B
REF: MLGE0301B
NCTM 9-12.REA.4 | NCTM 9-12.GEO.1.c | NCTM 9-12.REA.3
Lesson 8.3 Show that a Quadrilateral is a Parallelogram
triangle | parallelogram | parallel lines | congruence | proof | CPCTC
Analysis
NOT: 978-0-618-65613-4
1. VU ≅ ST and SV ≅ TU
1. Given
2. STUV is a parallelogram.
2. If both pairs of opp. sides of a quad.
are ≅ , then the quad. is a parallelogram.
3. VX = XT
3. The diagonals of a
parallelogram bisect each other.
PTS:
NAT:
TOP:
KEY:
NOT:
1
DIF: Level C
REF: XEGS0704A
NCTM 9-12.REA.3 | NCTM 9-12.REA.4 | NCTM 9-12.GEO.1.c
Lesson 8.3 Show that a Quadrilateral is a Parallelogram
parallelogram | congruent | proof | diagonals
BLM: Analysis
978-0-618-65613-4
9
ID: A
48. ANS:
1. VU ≅ ST and VU
ST
2. STUV is a parallelogram.
1. Given
2. If one pair of opp. sides of a quad. are both
and ≅ , then the quad. is a parallelogram.
3. VX = XT
3. The diagonals of a
parallelogram bisect each other.
PTS:
NAT:
TOP:
KEY:
NOT:
49. ANS:
1
DIF: Level C
REF: XEGS0704B
NCTM 9-12.REA.3 | NCTM 9-12.REA.4 | NCTM 9-12.GEO.1.c
Lesson 8.3 Show that a Quadrilateral is a Parallelogram
parallelogram | congruent | proof | diagonals
BLM: Analysis
978-0-618-65613-4
1. SV ≅ TU and SV TU
1. Given
2. STUV is a parallelogram.
2. If one pair of opp. sides of a quad. are both
and ≅ , then the quad. is a parallelogram.
3. VX = XT
3. The diagonals of a
parallelogram bisect each other.
PTS:
NAT:
TOP:
KEY:
NOT:
1
DIF: Level C
REF: XEGS0704C
NCTM 9-12.REA.3 | NCTM 9-12.REA.4 | NCTM 9-12.GEO.1.c
Lesson 8.3 Show that a Quadrilateral is a Parallelogram
parallelogram | congruent | proof | diagonals
BLM: Analysis
978-0-618-65613-4
10
ID: A
50. ANS:
1. Quadrilateral ABCD with A −5, 0 ,
1. Given
B 4, − 3 , C 8, − 1 , D −1, 2
2. slope of AB =
−3 − 0
−3
=
4 − −5
9
slope of BC =
−1 − (−3) 2
=
8−4
4
slope of CD =
2 − (−1)
3
=
−1 − 8
−9
slope of AD =
0−2
−2 1
=
=
−5 − (−1) −4 2
3. AB
DC, AD
BC
4. ABCD is a parallelogram.
2. Definition of slope
3. Lines with = slopes are
.
4. Definition of a parallelogram
PTS: 1
DIF: Level B
REF: MLGE0302
NAT: NCTM 9-12.GEO.1.c | NCTM 9-12.GEO.4.a | NCTM 9-12.REA.3 | NCTM 9-12.GEO.2.a | NCTM
9-12.REA.4
STA: MI.MIGLC.MTH.06.9-12.G1.4.2 | MI.MIGLC.MTH.06.9-12.G2.3.2 |
MI.MIGLC.MTH.06.9-12.G2.3.4
TOP: Lesson 8.3 Show that a Quadrilateral is a Parallelogram
KEY: slope | coordinate proof | two-column | coordinate geometry
BLM: Analysis
NOT: 978-0-618-65613-4
11
ID: A
51. ANS:
Since AB = CD =3 10 and BC = AD = 8, ABCD is a parallelogram.
PTS: 1
DIF: Level B
REF: MLGE0287 NAT: NCTM 9-12.GEO.2.a
STA: MI.MIGLC.MTH.06.9-12.A1.2.9 | MI.MIGLC.MTH.06.9-12.G1.1.5 |
MI.MIGLC.MTH.06.9-12.G1.4.1 | MI.MIGLC.MTH.06.9-12.G1.4.2 | MI.MIGLC.MTH.06.9-12.G1.4.3 |
MI.MIGLC.MTH.06.9-12.G2.3.4
TOP: Lesson 8.3 Show that a Quadrilateral is a Parallelogram
KEY: parallelogram | distance formula | coordinate geometry
BLM: Evaluation
NOT: 978-0-618-65613-4
52. ANS:
If ABCD is a parallelogram, then AB = DC. Since AB = 37 and DC = 40 , ABCD is not a parallelogram.
PTS: 1
DIF: Level B
REF: MLGE0288 NAT: NCTM 9-12.GEO.2.a
STA: MI.MIGLC.MTH.06.9-12.A1.2.9 | MI.MIGLC.MTH.06.9-12.G1.1.5 |
MI.MIGLC.MTH.06.9-12.G1.4.1 | MI.MIGLC.MTH.06.9-12.G1.4.2 | MI.MIGLC.MTH.06.9-12.G1.4.3 |
MI.MIGLC.MTH.06.9-12.G2.3.4
TOP: Lesson 8.3 Show that a Quadrilateral is a Parallelogram
KEY: parallelogram | distance formula | coordinate geometry
BLM: Evaluation
NOT: 978-0-618-65613-4
53. ANS:
1. ST
UV and TU || SV
2. STUV is a parallelogram.
3. VX = XT
1. Given
2. Def. of a parallelogram
3. The diagonals of a
parallelogram bisect each other.
PTS: 1
DIF: Level B
REF: MLGM0038
NAT: NCTM 9-12.REA.4 | NCTM 9-12.REA.3 | NCTM 9-12.GEO.1.c
STA: MI.MIGLC.MTH.06.9-12.G1.4.2 | MI.MIGLC.MTH.06.9-12.G2.3.2
TOP: Lesson 8.3 Show that a Quadrilateral is a Parallelogram KEY: parallelogram | parallel lines | proof
BLM: Analysis
NOT: 978-0-618-65613-4
54. ANS:
Three answers are possible: D 1 1, − 2 , D 2 9, − 4 , or D 3 −9, − 4 . Check students' graphs.
PTS:
NAT:
STA:
TOP:
KEY:
NOT:
1
DIF: Level B
REF: MGEO0039
NCTM 9-12.GEO.4.a | NCTM 9-12.GEO.2.a
MI.MIGLC.MTH.06.9-12.G1.4.2 | MI.MIGLC.MTH.06.9-12.G2.3.4
Lesson 8.3 Show that a Quadrilateral is a Parallelogram
parallelogram | plot | vertices | coordinates
BLM: Synthesis
978-0-618-65613-4
12
ID: A
55. ANS:
Quadrilateral ABCD is not a parallelogram:
The angles in any quadrilateral must have a sum of 360°. The angles indicated have a sum of 144° + 34° +
144°, which is 322°. That means that m∠ABC = 360° – 322°, so m∠ABC = 38°. The measures of the
opposite angles of a parallelogram must be equal, but m∠ABC ≠ m∠ADC. The quadrilateral is not a
parallelogram.
56.
57.
58.
59.
60.
PTS: 1
DIF: Level B
REF: MCT90280 NAT: NCTM 9-12.GEO.1.a
STA: MI.MIGLC.MTH.06.9-12.G1.4.1 | MI.MIGLC.MTH.06.9-12.G1.4.2 |
MI.MIGLC.MTH.06.9-12.G1.4.3 | MI.MIGLC.MTH.06.9-12.G1.4.4
TOP: Lesson 8.3 Show that a Quadrilateral is a Parallelogram
KEY: parallelogram | property | angle measures in polygons
BLM: Evaluation
NOT: 978-0-618-65613-4
ANS: C
PTS: 1
DIF: Level A
REF: TASH0019
NAT: NCTM 9-12.GEO.1.a
STA: MI.MIGLC.MTH.06.9-12.G1.4.3
TOP: Lesson 8.4 Properties of Rhombuses, Rectangles, and Squares
KEY: property | quadrilateral | geometric figure
BLM: Knowledge
NOT: 978-0-618-65613-4
ANS: C
PTS: 1
DIF: Level B
REF: HLGM0475
STA: MI.MIGLC.MTH.06.9-12.G1.4.1 | MI.MIGLC.MTH.06.9-12.G1.4.2 |
MI.MIGLC.MTH.06.9-12.G1.4.3
TOP: Lesson 8.4 Properties of Rhombuses, Rectangles, and Squares
KEY: property | rhombus
BLM: Comprehension
NOT: 978-0-618-65613-4
ANS: C
PTS: 1
DIF: Level A
REF: MLGE0291
STA: MI.MIGLC.MTH.06.9-12.G1.4.1 | MI.MIGLC.MTH.06.9-12.G1.4.2 |
MI.MIGLC.MTH.06.9-12.G1.4.3
TOP: Lesson 8.4 Properties of Rhombuses, Rectangles, and Squares
KEY: parallelogram | bisect | diagonal
BLM: Knowledge NOT: 978-0-618-65613-4
ANS: C
PTS: 1
DIF: Level B
REF: MC100124
STA: MI.MIGLC.MTH.06.9-12.G1.4.1 | MI.MIGLC.MTH.06.9-12.G1.4.2 |
MI.MIGLC.MTH.06.9-12.G1.4.3
TOP: Lesson 8.4 Properties of Rhombuses, Rectangles, and Squares
KEY: property | rhombus
BLM: Comprehension
NOT: 978-0-618-65613-4
ANS:
Diagrams vary.
PTS:
TOP:
KEY:
BLM:
1
DIF: Level B
REF: HLGM0470
Lesson 8.4 Properties of Rhombuses, Rectangles, and Squares
square | rectangle | parallelogram | rhombus | quadrilateral | Venn diagram
Comprehension
NOT: 978-0-618-65613-4
13
ID: A
61. ANS:
a. True.
b. If a parallelogram is a rhombus, then it is a square.
c. False.
PTS: 1
DIF: Level B
REF: MLGE0134 NAT: NCTM 9-12.GEO.1.a
STA: MI.MIGLC.MTH.06.9-12.L3.2.1 | MI.MIGLC.MTH.06.9-12.L3.2.4 |
MI.MIGLC.MTH.06.9-12.G1.4.3
TOP: Lesson 8.4 Properties of Rhombuses, Rectangles, and Squares
KEY: property | parallelogram | rhombus | converse
BLM: Application
NOT: 978-0-618-65613-4
62. ANS:
A rhombus
PTS: 1
DIF: Level A
REF: HLGM0471 NAT: NCTM 9-12.GEO.1.a
STA: MI.MIGLC.MTH.06.9-12.G1.4.1 | MI.MIGLC.MTH.06.9-12.G1.4.2 |
MI.MIGLC.MTH.06.9-12.G1.4.3
TOP: Lesson 8.4 Properties of Rhombuses, Rectangles, and Squares
KEY: perpendicular | parallelogram | rhombus | diagonal
BLM: Knowledge
NOT: 978-0-618-65613-4
63. ANS:
A rectangle
PTS: 1
DIF: Level A
REF: MLGE0290
STA: MI.MIGLC.MTH.06.9-12.G1.4.1 | MI.MIGLC.MTH.06.9-12.G1.4.2 |
MI.MIGLC.MTH.06.9-12.G1.4.3
TOP: Lesson 8.4 Properties of Rhombuses, Rectangles, and Squares
KEY: rectangle | parallelogram | diagonal BLM: Knowledge NOT: 978-0-618-65613-4
64. ANS:
True
PTS:
TOP:
KEY:
NOT:
65. ANS:
False
1
DIF: Level A
REF: DJAF1009A NAT: NCTM 9-12.GEO.1.a
Lesson 8.4 Properties of Rhombuses, Rectangles, and Squares
polygon | triangle | quadrilateral | true | false
BLM: Knowledge
978-0-618-65613-4
PTS:
TOP:
KEY:
NOT:
66. ANS:
False
1
DIF: Level A
REF: DJAF1009B NAT: NCTM 9-12.GEO.1.a
Lesson 8.4 Properties of Rhombuses, Rectangles, and Squares
polygon | triangle | quadrilateral | true | false
BLM: Knowledge
978-0-618-65613-4
PTS:
TOP:
KEY:
NOT:
1
DIF: Level A
REF: DJAF1009C NAT: NCTM 9-12.GEO.1.a
Lesson 8.4 Properties of Rhombuses, Rectangles, and Squares
polygon | triangle | quadrilateral | true | false
BLM: Knowledge
978-0-618-65613-4
14
ID: A
67. ANS:
True
PTS:
TOP:
KEY:
NOT:
68. ANS:
True
1
DIF: Level A
REF: DJAF1009D NAT: NCTM 9-12.GEO.1.a
Lesson 8.4 Properties of Rhombuses, Rectangles, and Squares
polygon | triangle | quadrilateral | true | false
BLM: Knowledge
978-0-618-65613-4
PTS:
TOP:
KEY:
NOT:
69. ANS:
False
1
DIF: Level A
REF: DJAF1009F NAT: NCTM 9-12.GEO.1.a
Lesson 8.4 Properties of Rhombuses, Rectangles, and Squares
polygon | triangle | quadrilateral | true | false
BLM: Knowledge
978-0-618-65613-4
PTS:
TOP:
KEY:
NOT:
70. ANS:
False
1
DIF: Level A
REF: DJAF1009H NAT: NCTM 9-12.GEO.1.a
Lesson 8.4 Properties of Rhombuses, Rectangles, and Squares
polygon | triangle | quadrilateral | true | false
BLM: Knowledge
978-0-618-65613-4
PTS:
TOP:
KEY:
NOT:
71. ANS:
True
1
DIF: Level A
REF: DJAF1009I NAT: NCTM 9-12.GEO.1.a
Lesson 8.4 Properties of Rhombuses, Rectangles, and Squares
polygon | triangle | quadrilateral | true | false
BLM: Knowledge
978-0-618-65613-4
PTS:
TOP:
KEY:
NOT:
72. ANS:
True
1
DIF: Level A
REF: DJAF1009J NAT: NCTM 9-12.GEO.1.a
Lesson 8.4 Properties of Rhombuses, Rectangles, and Squares
polygon | triangle | quadrilateral | true | false
BLM: Knowledge
978-0-618-65613-4
PTS:
TOP:
KEY:
NOT:
73. ANS:
False
1
DIF: Level A
REF: DJAF1009K NAT: NCTM 9-12.GEO.1.a
Lesson 8.4 Properties of Rhombuses, Rectangles, and Squares
polygon | triangle | quadrilateral | true | false
BLM: Knowledge
978-0-618-65613-4
PTS:
TOP:
KEY:
NOT:
1
DIF: Level A
REF: DJAF1009L NAT: NCTM 9-12.GEO.1.a
Lesson 8.4 Properties of Rhombuses, Rectangles, and Squares
polygon | triangle | quadrilateral | true | false
BLM: Knowledge
978-0-618-65613-4
15
ID: A
74. ANS:
Sample answer: (–5, 0), (0, 3), (5, 0), (0, –3)
PTS: 1
DIF: Level B
REF: BS022283
NAT: NCTM 9-12.GEO.2.a | NCTM 9-12.GEO.4.a
STA: MI.MIGLC.MTH.06.9-12.G1.4.2 | MI.MIGLC.MTH.06.9-12.G2.3.4
TOP: Lesson 8.4 Properties of Rhombuses, Rectangles, and Squares
KEY: vertices | coordinate | rhombus | quadrilateral | diagonal | TEKSd2A
BLM: Application NOT: 978-0-618-65613-4
75. ANS:
Sample answer: (–6, 0), (0, 6), (6, 0), (0, –6)
PTS: 1
DIF: Level B
REF: BS022285
NAT: NCTM 9-12.GEO.2.a | NCTM 9-12.GEO.4.a
STA: MI.MIGLC.MTH.06.9-12.G1.4.2 | MI.MIGLC.MTH.06.9-12.G2.3.4
TOP: Lesson 8.4 Properties of Rhombuses, Rectangles, and Squares
KEY: TEKSd2A
BLM: Application NOT: 978-0-618-65613-4
76. ANS:
Sample answer: (–3, 0), (0, 5), (3, 0), (0, –5)
PTS: 1
DIF: Level B
REF: BS022287
NAT: NCTM 9-12.GEO.2.a | NCTM 9-12.GEO.4.a
STA: MI.MIGLC.MTH.06.9-12.G1.4.2 | MI.MIGLC.MTH.06.9-12.G2.3.4
TOP: Lesson 8.4 Properties of Rhombuses, Rectangles, and Squares
KEY: vertices | coordinate | rhombus | quadrilateral | TEKSd2A BLM: Application
NOT: 978-0-618-65613-4
77. ANS:
Sample answer: (–6, 0), (0, 6), (6, 0), (0, –6)
PTS:
NAT:
STA:
TOP:
KEY:
NOT:
78. ANS:
True
PTS:
STA:
TOP:
KEY:
NOT:
1
DIF: Level B
REF: BS022288
NCTM 9-12.GEO.2.a | NCTM 9-12.GEO.4.a
MI.MIGLC.MTH.06.9-12.G1.4.2 | MI.MIGLC.MTH.06.9-12.G2.3.4
Lesson 8.4 Properties of Rhombuses, Rectangles, and Squares
square | coordinate | quadrilateral | diagonal | TEKSd2A BLM: Application
978-0-618-65613-4
1
DIF: Level B
REF: MIM20276 NAT: NCTM 9-12.GEO.1.a
MI.MIGLC.MTH.06.9-12.G1.4.3
Lesson 8.4 Properties of Rhombuses, Rectangles, and Squares
square | rectangle | classify | quadrilateral | kite
BLM: Application
978-0-618-65613-4
16
ID: A
79. ANS:
False
PTS:
STA:
KEY:
NOT:
80. ANS:
57
1
DIF: Level A
REF: MIM20278 NAT: NCTM 9-12.GEO.1.a
MI.MIGLC.MTH.06.9-12.G1.4.3 TOP: Lesson 8.5 Use Properties of Trapezoids and Kites
classify | conditional | quadrilateral | logic
BLM: Comprehension
978-0-618-65613-4
PTS: 1
DIF: Level B
REF: MCT90011 NAT: NCTM 9-12.GEO.1.a
STA: MI.MIGLC.MTH.06.9-12.G1.2.1 | MI.MIGLC.MTH.06.9-12.G1.2.2 |
MI.MIGLC.MTH.06.9-12.G1.4.1 | MI.MIGLC.MTH.06.9-12.G1.4.2 | MI.MIGLC.MTH.06.9-12.G1.4.3
TOP: Lesson 8.4 Properties of Rhombuses, Rectangles, and Squares
KEY: solve | angle | rhombus
BLM: Application NOT: 978-0-618-65613-4
81. ANS:
a valid definition
PTS: 1
DIF: Level B
REF: MGEO0004
NAT: NCTM 9-12.GEO.1.c | NCTM 9-12.GEO.1.a | NCTM 9-12.REA.2
STA: MI.MIGLC.MTH.06.9-12.L3.1.3 | MI.MIGLC.MTH.06.9-12.L3.3.2 |
MI.MIGLC.MTH.06.9-12.L3.3.3
TOP: Lesson 8.4 Properties of Rhombuses, Rectangles, and Squares
KEY: counterexample | definition
BLM: Comprehension
NOT: 978-0-618-65613-4
82. ANS:
valid; Law of Syllogism
PTS: 1
DIF: Level C
REF: MIM20402
NAT: NCTM 9-12.GEO.1.c | NCTM 9-12.COM.3 | NCTM 9-12.REA.3
STA: MI.MIGLC.MTH.06.9-12.L3.1.2 | MI.MIGLC.MTH.06.9-12.L3.2.3 |
MI.MIGLC.MTH.06.9-12.L3.3.1 | MI.MIGLC.MTH.06.9-12.L3.3.3
TOP: Lesson 8.4 Properties of Rhombuses, Rectangles, and Squares
KEY: conditional | logic
BLM: Evaluation
NOT: 978-0-618-65613-4
83. ANS:
No; for example, a rhombus with sides of length 6 cm and angles measuring 30°, 150°, 30°, and 150° is not
congruent to a square with sides of length 6 cm, even though all four pairs of corresponding sides are
congruent. The necessary additional information needed to prove congruence would be either three pairs of
corresponding angles congruent or two pairs of adjacent corresponding angles congruent.
PTS:
TOP:
KEY:
NOT:
1
DIF: Level B
REF: BS022543
STA: MI.MIGLC.MTH.06.9-12.G2.3.2
Lesson 8.4 Properties of Rhombuses, Rectangles, and Squares
write | quadrilateral | congruent | proof
BLM: Evaluation
978-0-618-65613-4
17
ID: A
84. ANS:
Part A
AB =
−6 − −8
2
+ 3 − −3
2
BC =
−3 − −6
2
+ 2− 3
CD =
−5 − −3
2
+ −4 − 2
AD =
−5 − −8
2
+ −4 − −3
2
= 2 10
=
2
10
= 2 10
2
=
10
Part B
slope of AB =
3 − −3
= 3
−6 − −8
slope of BC =
2− 3
1
= −
−3 − −6
3
slope of CD =
−4 − 2
= 3
−5 − −3
slope of AD =
−4 − −3
1
= −
−5 − −8
3
Part C The quadrilateral is a rectangle. Answers will vary.
PTS: 1
DIF: Level B
REF: MC100258
NAT: NCTM 9-12.GEO.2.a | NCTM 9-12.GEO.1.a
STA: MI.MIGLC.MTH.06.9-12.A1.2.9 | MI.MIGLC.MTH.06.9-12.G1.1.5 |
MI.MIGLC.MTH.06.9-12.G1.4.2 | MI.MIGLC.MTH.06.9-12.G2.3.4
TOP: Lesson 8.4 Properties of Rhombuses, Rectangles, and Squares
KEY: square | slope | identify | distance | coordinate | parallelogram | rhombus | quadrilateral
BLM: Application NOT: 978-0-618-65613-4
85. ANS:
1. ABCD is a rhombus.
1. Given
2. ABCD is a parallelogram.
2. Definition of a rhombus
3. AB ≅ CD; BC ≅ AD
3. Opposite sides of a parallelogram are congruent.
4. AC ≅ AC
4. Reflexive Property
5. ∆ACB ≅ ∆CAD
5. SSS Congruence Postulate
PTS:
NAT:
STA:
TOP:
KEY:
1
DIF: Level B
REF: MLGE0303
NCTM 9-12.GEO.1.c | NCTM 9-12.REA.3 | NCTM 9-12.REA.4
MI.MIGLC.MTH.06.9-12.G1.4.2 | MI.MIGLC.MTH.06.9-12.G2.3.2
Lesson 8.4 Properties of Rhombuses, Rectangles, and Squares
rhombus | diagonal | proof
BLM: Analysis
NOT: 978-0-618-65613-4
18
ID: A
86. ANS:
Sample answer: A rhombus is a quadrilateral with four congruent sides while a rectangle is a quadrilateral
with four right angles.
PTS: 1
DIF: Level B
REF: BS022063
STA: MI.MIGLC.MTH.06.9-12.G1.4.1 | MI.MIGLC.MTH.06.9-12.G1.4.2 |
MI.MIGLC.MTH.06.9-12.G1.4.3
TOP: Lesson 8.4 Properties of Rhombuses, Rectangles, and Squares
KEY: rectangle | rhombus | quadrilateral | TEKSc1
BLM: Comprehension
NOT: 978-0-618-65613-4
87. ANS:
False; the conditional "If a quadrilateral is a rectangle, then it is a square" is false. Not all rectangles are
squares.
PTS: 1
DIF: Level B
REF: MIM20411
NAT: NCTM 9-12.REA.1 | NCTM 9-12.COM.3 | NCTM 9-12.PRS.4
STA: MI.MIGLC.MTH.06.9-12.L1.1.3 | MI.MIGLC.MTH.06.9-12.L3.2.1 |
MI.MIGLC.MTH.06.9-12.A1.2.4 | MI.MIGLC.MTH.06.9-12.A1.2.5 | MI.MIGLC.MTH.06.9-12.A1.2.6 |
MI.MIGLC.MTH.06.9-12.A1.2.7 | MI.MIGLC.MTH.06.9-12.A1.2.8
TOP: Lesson 8.4 Properties of Rhombuses, Rectangles, and Squares
KEY: conditional | logic
BLM: Comprehension
NOT: 978-0-618-65613-4
88. ANS:
a. True
b. Converse: If a quadrilateral is a rectangle, then it is a square.
Inverse: If a quadrilateral is not a square, then it is not a rectangle.
Contrapositive: If a quadrilateral is not a rectangle, then it is not a square. The contrapositive is true.
PTS: 1
DIF: Level C
REF: MGEO0049
TOP: Lesson 8.4 Properties of Rhombuses, Rectangles, and Squares
KEY: inverse | converse | contrapositive BLM: Application NOT: 978-0-618-65613-4
89. ANS:
a. True
b. If a quadrilateral is not a rectangle, then it is not a parallelogram. False
PTS: 1
DIF: Level B
REF: MGEO0050
TOP: Lesson 8.4 Properties of Rhombuses, Rectangles, and Squares
KEY: inverse | conditional
BLM: Application NOT: 978-0-618-65613-4
90. ANS:
a. True
b. If a quadrilateral is not a rhombus, then it is not a square. True
PTS: 1
DIF: Level B
REF: MGEO0051
TOP: Lesson 8.4 Properties of Rhombuses, Rectangles, and Squares
KEY: conditional | contrapositive
BLM: Application NOT: 978-0-618-65613-4
19
ID: A
91. ANS:
invalid; converse error (ABCD could be a rhombus.)
PTS: 1
DIF: Level C
REF: MIM20405
NAT: NCTM 9-12.COM.3 | NCTM 9-12.GEO.1.c | NCTM 9-12.REA.3
STA: MI.MIGLC.MTH.06.9-12.L3.1.2 | MI.MIGLC.MTH.06.9-12.L3.2.1 |
MI.MIGLC.MTH.06.9-12.L3.2.3 | MI.MIGLC.MTH.06.9-12.L3.2.4 | MI.MIGLC.MTH.06.9-12.L3.3.1 |
MI.MIGLC.MTH.06.9-12.L3.3.3
TOP: Lesson 8.4 Properties of Rhombuses, Rectangles, and Squares
KEY: logic
BLM: Evaluation
NOT: 978-0-618-65613-4
92. ANS:
1
1
Equations may vary. Sample equations are given; y = − x – 3, y = 2x + 2, y = − x + 2.
2
2
PTS: 1
DIF: Level B
REF: MIM20647
NAT: NCTM 9-12.GEO.4.a | NCTM 9-12.GEO.2.a
STA: MI.MIGLC.MTH.06.9-12.A3.1.2 | MI.MIGLC.MTH.06.9-12.A3.1.4 |
MI.MIGLC.MTH.06.9-12.G1.4.2 | MI.MIGLC.MTH.06.9-12.G2.3.4
TOP: Lesson 8.4 Properties of Rhombuses, Rectangles, and Squares
KEY: line | graph | slope | parallel | perpendicular
BLM: Application
NOT: 978-0-618-65613-4
93. ANS: B
PTS: 1
DIF: Level C
REF: MLGE0294
NAT: NCTM 9-12.GEO.1.a
STA: MI.MIGLC.MTH.06.9-12.G1.4.1 | MI.MIGLC.MTH.06.9-12.G1.4.2 |
MI.MIGLC.MTH.06.9-12.G1.5.2
TOP: Lesson 8.5 Use Properties of Trapezoids and Kites
KEY: isosceles trapezoid | leg
BLM: Application NOT: 978-0-618-65613-4
94. ANS: C
PTS: 1
DIF: Level B
REF: MHST0015
STA: MI.MIGLC.MTH.06.9-12.G1.4.1 | MI.MIGLC.MTH.06.9-12.G1.4.2
TOP: Lesson 8.5 Use Properties of Trapezoids and Kites
KEY: property | isosceles trapezoid
BLM: Comprehension
NOT: 978-0-618-65613-4
20
ID: A
95. ANS: A
PTS: 1
DIF: Level B
REF: MHST0016
NAT: NCTM 9-12.GEO.1.a
STA: MI.MIGLC.MTH.06.9-12.G1.4.1 | MI.MIGLC.MTH.06.9-12.G1.4.2 |
MI.MIGLC.MTH.06.9-12.G1.4.3
TOP: Lesson 8.5 Use Properties of Trapezoids and Kites
KEY: trapezoid | midsegment
BLM: Application NOT: 978-0-618-65613-4
96. ANS: B
PTS: 1
DIF: Level A
REF: MLGE0045
STA: MI.MIGLC.MTH.06.9-12.G1.4.1 | MI.MIGLC.MTH.06.9-12.G1.4.2 |
MI.MIGLC.MTH.06.9-12.G1.4.3
TOP: Lesson 8.5 Use Properties of Trapezoids and Kites
KEY: identify | kite
BLM: Knowledge NOT: 978-0-618-65613-4
97. ANS: D
PTS: 1
DIF: Level B
REF: HLGM0492
STA: MI.MIGLC.MTH.06.9-12.G1.4.1 | MI.MIGLC.MTH.06.9-12.G1.4.2 |
MI.MIGLC.MTH.06.9-12.G1.4.3
TOP: Lesson 8.5 Use Properties of Trapezoids and Kites
KEY: side | parallel | quadrilateral | kite
BLM: Comprehension
NOT: 978-0-618-65613-4
98. ANS: D
PTS: 1
DIF: Level B
REF: MC100231
STA: MI.MIGLC.MTH.06.9-12.G1.4.2 | MI.MIGLC.MTH.06.9-12.G2.3.4
TOP: Lesson 8.5 Use Properties of Trapezoids and Kites
KEY: vertex | coordinate geometry | isosceles trapezoid
BLM: Application
NOT: 978-0-618-65613-4
99. ANS:
an isosceles trapezoid
PTS:
STA:
TOP:
BLM:
100. ANS:
20
1
DIF: Level A
REF: MHST0018
MI.MIGLC.MTH.06.9-12.G1.4.1 | MI.MIGLC.MTH.06.9-12.G1.4.2
Lesson 8.5 Use Properties of Trapezoids and Kites
KEY: property | isosceles trapezoid
Knowledge NOT: 978-0-618-65613-4
PTS: 1
DIF: Level B
REF: MHGM0064
NAT: NCTM 9-12.GEO.1.a
STA: MI.MIGLC.MTH.06.9-12.G1.4.1 | MI.MIGLC.MTH.06.9-12.G1.4.2 |
MI.MIGLC.MTH.06.9-12.G1.4.3
TOP: Lesson 8.5 Use Properties of Trapezoids and Kites
KEY: midsegment | trapezoid
BLM: Application NOT: 978-0-618-65613-4
101. ANS:
3 cm, 1 cm
PTS: 1
DIF: Level B
REF: MLGE0328 NAT: NCTM 9-12.GEO.1.a
STA: MI.MIGLC.MTH.06.9-12.G1.4.1 | MI.MIGLC.MTH.06.9-12.G1.4.2 |
MI.MIGLC.MTH.06.9-12.G1.4.3 | MI.MIGLC.MTH.06.9-12.G1.5.2
TOP: Lesson 8.5 Use Properties of Trapezoids and Kites
KEY: perimeter | side | kite
BLM: Application NOT: 978-0-618-65613-4
21
ID: A
102. ANS:
110°
PTS: 1
DIF: Level B
REF: MLGE0097 NAT: NCTM 9-12.GEO.1.a
STA: MI.MIGLC.MTH.06.9-12.G1.4.1 | MI.MIGLC.MTH.06.9-12.G1.4.2 |
MI.MIGLC.MTH.06.9-12.G1.4.3 | MI.MIGLC.MTH.06.9-12.G1.4.4
TOP: Lesson 8.5 Use Properties of Trapezoids and Kites
KEY: angle measure | kite
NOT: 978-0-618-65613-4
103. ANS:
Sample answer: Since it is given that EFGH is a kite, then EF ≅ EH and GF ≅ GH because a kite has two
pairs of congruent adjacent sides. Also, EG ≅ EG by the Reflexive Property. So, by the SSS Postulate,
∆EFG ≅ ∆EHG. Therefore, by the definition of congruent triangles, ∠F ≅ ∠H.
PTS: 1
DIF: Level B
REF: BS022277
NAT: NCTM 9-12.REA.4 | NCTM 9-12.REA.3 | NCTM 9-12.GEO.1.c
STA: MI.MIGLC.MTH.06.9-12.G1.4.2 TOP: Lesson 8.5 Use Properties of Trapezoids and Kites
KEY: kite | proof | congruent angles | congruent triangles | SSS BLM: Analysis
NOT: 978-0-618-65613-4
104. ANS:
Sample answer: Since it is given that EFGH is a kite, then EF ≅ EH and GF ≅ GH because a kite has two
pairs of congruent adjacent sides. Also, EG ≅ EG by the Reflexive Property. So, by the SSS Postulate,
∆EFG ≅ ∆EHG. Therefore, by the definition of congruent triangles, ∠FEG ≅ ∠HEG.
PTS: 1
DIF: Level B
REF: BS022278
NAT: NCTM 9-12.GEO.1.c | NCTM 9-12.REA.4 | NCTM 9-12.REA.3
STA: MI.MIGLC.MTH.06.9-12.G1.4.2 TOP: Lesson 8.5 Use Properties of Trapezoids and Kites
KEY: kite | proof | congruent angles | SSS | congruent triangles BLM: Analysis
NOT: 978-0-618-65613-4
105. ANS:
Sample answer: (–3, 0), (0, 3), (7, 0), (0, –7)
PTS: 1
DIF: Level B
REF: BS022284
NAT: NCTM 9-12.GEO.4.a | NCTM 9-12.GEO.2.a
STA: MI.MIGLC.MTH.06.9-12.G1.4.2 | MI.MIGLC.MTH.06.9-12.G2.3.4
TOP: Lesson 8.5 Use Properties of Trapezoids and Kites
KEY: vertices | coordinate geometry | trapezoid | diagonal
BLM: Application
NOT: 978-0-618-65613-4
106. ANS:
Answers may vary. A sample answer is given. Contrasting properties: In a kite, there are two pairs of
adjacent sides of equal length; in a parallelogram, there are two pairs of opposite sides of equal length.
Matching properties: Kites and parallelograms are both quadrilaterals.
PTS: 1
DIF: Level B
REF: MIM10695
TOP: Lesson 8.5 Use Properties of Trapezoids and Kites
KEY: compare | Venn diagram | property
BLM: Comprehension
NOT: 978-0-618-65613-4
22
ID: A
107. ANS:
Notice in the figure that the portion of the triangle folded down has sides of length
perimeter of the trapezoid is 1 +
1
. Therefore, the
2
1 1 1
1
+ + = 2 units.
2 2 2
2
PTS: 1
DIF: Level C
REF: MIM10696
STA: MI.MIGLC.MTH.06.9-12.G1.4.1 | MI.MIGLC.MTH.06.9-12.G1.5.2
TOP: Lesson 8.5 Use Properties of Trapezoids and Kites
KEY: equilateral triangle | perimeter | trapezoid
BLM: Synthesis
NOT: 978-0-618-65613-4
108. ANS:
Figures may vary. Sample answer: For the figure shown, AB = AD = 5 and BC = CD = 13. Therefore,
since two distinct pairs of consecutive sides are equal in measure, quadrilateral ABCD is a kite.
PTS: 1
DIF: Level B
REF: MGEO0038
NAT: NCTM 9-12.GEO.4.a | NCTM 9-12.GEO.2.a
STA: MI.MIGLC.MTH.06.9-12.A1.2.9 | MI.MIGLC.MTH.06.9-12.G1.1.5 |
MI.MIGLC.MTH.06.9-12.G1.4.2 | MI.MIGLC.MTH.06.9-12.G2.3.4
TOP: Lesson 8.5 Use Properties of Trapezoids and Kites
KEY: prove | coordinate geometry | kite
BLM: Application NOT: 978-0-618-65613-4
23
ID: A
109. ANS:
AB= (6 − 0) 2 + 0 − 0
2
= 6; BC= (5 − 6) 2 + 2 − 0
2
=
5 ; CD= (1 − 5) 2 + 2 − 2
2
= 4;
2
AD= (1 − 0) 2 + 2 − 0 = 5;
0−0
2−2
= 0; slope of CD:
= 0; Since the slopes of AB and CD are equal, ABCD is a
slope of AB:
6−0
1−5
trapezoid. Since BC = AD, ABCD is an isosceles trapezoid.
PTS: 1
DIF: Level B
REF: MIM30148
NAT: NCTM 9-12.PRS.4 | NCTM 9-12.GEO.2.a
STA: MI.MIGLC.MTH.06.9-12.L1.1.3 | MI.MIGLC.MTH.06.9-12.L1.1.4 |
MI.MIGLC.MTH.06.9-12.L1.3.1 | MI.MIGLC.MTH.06.9-12.A1.2.4 | MI.MIGLC.MTH.06.9-12.A1.2.5 |
MI.MIGLC.MTH.06.9-12.A1.2.6 | MI.MIGLC.MTH.06.9-12.A1.2.7 | MI.MIGLC.MTH.06.9-12.A1.2.8 |
MI.MIGLC.MTH.06.9-12.A1.2.9 | MI.MIGLC.MTH.06.9-12.A3.1.4 | MI.MIGLC.MTH.06.9-12.G1.1.5 |
MI.MIGLC.MTH.06.9-12.G1.4.2 | MI.MIGLC.MTH.06.9-12.G2.3.4
TOP: Lesson 8.5 Use Properties of Trapezoids and Kites
KEY: isosceles trapezoid | coordinate geometry | proof
BLM: Analysis
NOT: 978-0-618-65613-4
110. ANS:
JK= (3 − 0) 2 + 0 − 0
2
2
2
= 3; KL= (6 − 3) 2 + 2 − 0 = 13; LM= ( − 3 − 6) 2 + 2 − 2 = 9;
0−0
2−2
2
2
JM =
−3 − 0 + 2 − 0 = 13; slope of JK:
= 0; slope of LM:
= 0; Since the slopes of JK
3−0
−3 − 6
and LM are equal, JKLM is a trapezoid. Since KL = JM, JKLM is an isosceles trapezoid.
PTS: 1
DIF: Level B
REF: MIM30149
NAT: NCTM 9-12.GEO.2.a | NCTM 9-12.PRS.4
STA: MI.MIGLC.MTH.06.9-12.L1.1.3 | MI.MIGLC.MTH.06.9-12.L1.1.4 |
MI.MIGLC.MTH.06.9-12.L1.3.1 | MI.MIGLC.MTH.06.9-12.A1.2.4 | MI.MIGLC.MTH.06.9-12.A1.2.5 |
MI.MIGLC.MTH.06.9-12.A1.2.6 | MI.MIGLC.MTH.06.9-12.A1.2.7 | MI.MIGLC.MTH.06.9-12.A1.2.8 |
MI.MIGLC.MTH.06.9-12.A1.2.9 | MI.MIGLC.MTH.06.9-12.A3.1.4 | MI.MIGLC.MTH.06.9-12.G1.1.5 |
MI.MIGLC.MTH.06.9-12.G1.4.2 | MI.MIGLC.MTH.06.9-12.G2.3.4
TOP: Lesson 8.5 Use Properties of Trapezoids and Kites
KEY: isosceles trapezoid | coordinate geometry | proof
BLM: Analysis
NOT: 978-0-618-65613-4
111. ANS:
The value of a is 119. The value of c can be found from the fact that sides AB and DC are parallel. That
means that c° + 124 ° = 180°, so c = 180 − 124 = 56. Another way to find the value of c is to use the fact
that the vertex angles of the trapezoid must total 360°: 119 + 124 + c + 61 = 360, so
c = 360 − 119 + 124 + 61 = 56.
PTS: 1
DIF: Level B
REF: MC100117 NAT: NCTM 9-12.GEO.1.a
STA: MI.MIGLC.MTH.06.9-12.G1.4.1 | MI.MIGLC.MTH.06.9-12.G1.4.2 |
MI.MIGLC.MTH.06.9-12.G1.4.4
TOP: Lesson 8.5 Use Properties of Trapezoids and Kites
KEY: angle measure | trapezoid
BLM: Analysis
NOT: 978-0-618-65613-4
24
ID: A
112. ANS: B
PTS: 1
DIF: Level B
REF: HLGM0476
STA: MI.MIGLC.MTH.06.9-12.G1.4.2 | MI.MIGLC.MTH.06.9-12.G2.3.4
TOP: Lesson 8.6 Identify Special Quadrilaterals
KEY: coordinate geometry | quadrilateral
BLM: Comprehension
NOT: 978-0-618-65613-4
113. ANS: C
PTS: 1
DIF: Level B
REF: MGEO0011
NAT: NCTM 9-12.GEO.2.a | NCTM 9-12.GEO.1.a
STA: MI.MIGLC.MTH.06.9-12.A1.2.9 | MI.MIGLC.MTH.06.9-12.G1.1.5 |
MI.MIGLC.MTH.06.9-12.G1.4.2 | MI.MIGLC.MTH.06.9-12.G2.3.4
TOP: Lesson 8.6 Identify Special Quadrilaterals
KEY: slope | identify | distance formula | quadrilateral
BLM: Comprehension
NOT: 978-0-618-65613-4
114. ANS: C
PTS: 1
DIF: Level A
REF: MHST0017
STA: MI.MIGLC.MTH.06.9-12.G1.4.1 | MI.MIGLC.MTH.06.9-12.G1.4.2 |
MI.MIGLC.MTH.06.9-12.G1.4.3
TOP: Lesson 8.6 Identify Special Quadrilaterals
KEY: property | quadrilateral
BLM: Knowledge NOT: 978-0-618-65613-4
115. ANS: A
PTS: 1
DIF: Level A
REF: MIM10065
NAT: NCTM 9-12.GEO.1.a
TOP: Lesson 8.6 Identify Special Quadrilaterals
KEY: quadrilateral | identify
BLM: Knowledge NOT: 978-0-618-65613-4
116. ANS: A
PTS: 1
DIF: Level A
REF: MIM10066
NAT: NCTM 9-12.GEO.1.a
TOP: Lesson 8.6 Identify Special Quadrilaterals
KEY: quadrilateral | identify
BLM: Knowledge NOT: 978-0-618-65613-4
117. ANS: C
PTS: 1
DIF: Level B
REF: MIM20279
NAT: NCTM 9-12.GEO.1.a
STA: MI.MIGLC.MTH.06.9-12.G1.4.3
TOP: Lesson 8.6 Identify Special Quadrilaterals
KEY: classify | quadrilateral
BLM: Comprehension
NOT: 978-0-618-65613-4
118. ANS: D
PTS: 1
DIF: Level B
REF: MIM20280
NAT: NCTM 9-12.GEO.1.a
STA: MI.MIGLC.MTH.06.9-12.G1.4.3
TOP: Lesson 8.6 Identify Special Quadrilaterals
KEY: classify | quadrilateral
BLM: Comprehension
NOT: 978-0-618-65613-4
119. ANS: D
PTS: 1
DIF: Level B
REF: MIM20281
NAT: NCTM 9-12.GEO.1.a
STA: MI.MIGLC.MTH.06.9-12.G1.4.3
TOP: Lesson 8.6 Identify Special Quadrilaterals
KEY: classify | quadrilateral
BLM: Comprehension
NOT: 978-0-618-65613-4
120. ANS:
trapezoid, quadrilateral
PTS: 1
DIF: Level A
REF: MLGE0401 NAT: NCTM 9-12.GEO.1.a
TOP: Lesson 8.6 Identify Special Quadrilaterals
KEY: square | rectangle | parallelogram | rhombus | trapezoid | quadrilateral
BLM: Knowledge NOT: 978-0-618-65613-4
121. ANS:
square
PTS: 1
DIF: Level A
REF: MPMC0708
STA: MI.MIGLC.MTH.06.9-12.G1.4.1 | MI.MIGLC.MTH.06.9-12.G1.4.2 |
MI.MIGLC.MTH.06.9-12.G1.4.3
TOP: Lesson 8.6 Identify Special Quadrilaterals
KEY: identify | quadrilateral
BLM: Knowledge NOT: 978-0-618-65613-4
25
ID: A
122. ANS:
The following characteristics for each quadrilateral might be indicated.
a. four congruent sides, four right (congruent) angles, opposite sides parallel, congruent
diagonals, diagonals are the perpendicular bisectors of each other.
b. opposite sides congruent and parallel, four right (congruent) angles, congruent
diagonals, diagonals bisect each other
c. opposite sides congruent and parallel, opposite angles congruent, diagonals bisect each
other
d. four congruent sides, opposite sides parallel, opposite angles congruent, diagonals are
the perpendicular bisectors of each other
e. one pair of opposite sides parallel
f. two pairs of adjacent sides congruent
PTS: 1
DIF: Level B
REF: BS022547
STA: MI.MIGLC.MTH.06.9-12.G1.4.1 | MI.MIGLC.MTH.06.9-12.G1.4.2 |
MI.MIGLC.MTH.06.9-12.G1.4.3
TOP: Lesson 8.6 Identify Special Quadrilaterals
KEY: square | rectangle | parallelogram | rhombus | trapezoid | quadrilateral | kite
BLM: Comprehension
NOT: 978-0-618-65613-4
123. ANS:
rhombus
PTS: 1
DIF: Level A
REF: MIM20273 NAT: NCTM 9-12.GEO.2.a
STA: MI.MIGLC.MTH.06.9-12.A1.2.9 | MI.MIGLC.MTH.06.9-12.A3.1.4 |
MI.MIGLC.MTH.06.9-12.G1.1.5 | MI.MIGLC.MTH.06.9-12.G1.4.2 | MI.MIGLC.MTH.06.9-12.G2.3.4
TOP: Lesson 8.6 Identify Special Quadrilaterals
KEY: quadrilateral | coordinate geometry | identify
BLM: Comprehension
NOT: 978-0-618-65613-4
124. ANS:
An isosceles trapezoid; the figure has two parallel sides of unequal length with the other two sides being of
the same length.
PTS: 1
DIF: Level B
REF: MIM20274
NAT: NCTM 9-12.PRS.4 | NCTM 9-12.GEO.2.a
STA: MI.MIGLC.MTH.06.9-12.L1.1.3 | MI.MIGLC.MTH.06.9-12.L1.1.4 |
MI.MIGLC.MTH.06.9-12.L1.3.1 | MI.MIGLC.MTH.06.9-12.A1.2.4 | MI.MIGLC.MTH.06.9-12.A1.2.5 |
MI.MIGLC.MTH.06.9-12.A1.2.6 | MI.MIGLC.MTH.06.9-12.A1.2.7 | MI.MIGLC.MTH.06.9-12.A1.2.8 |
MI.MIGLC.MTH.06.9-12.A1.2.9 | MI.MIGLC.MTH.06.9-12.A3.1.4 | MI.MIGLC.MTH.06.9-12.G1.1.5 |
MI.MIGLC.MTH.06.9-12.G1.4.2 | MI.MIGLC.MTH.06.9-12.G2.3.4
TOP: Lesson 8.6 Identify Special Quadrilaterals
KEY: quadrilateral | coordinate geometry | identify
BLM: Comprehension
NOT: 978-0-618-65613-4
26
ID: A
125. ANS:
A trapezoid; the figure has two parallel sides but none of the sides are of equal length.
PTS: 1
DIF: Level B
REF: MIM20275
NAT: NCTM 9-12.PRS.4 | NCTM 9-12.GEO.2.a
STA: MI.MIGLC.MTH.06.9-12.L1.1.3 | MI.MIGLC.MTH.06.9-12.L1.1.4 |
MI.MIGLC.MTH.06.9-12.L1.3.1 | MI.MIGLC.MTH.06.9-12.A1.2.4 | MI.MIGLC.MTH.06.9-12.A1.2.5 |
MI.MIGLC.MTH.06.9-12.A1.2.6 | MI.MIGLC.MTH.06.9-12.A1.2.7 | MI.MIGLC.MTH.06.9-12.A1.2.8 |
MI.MIGLC.MTH.06.9-12.A1.2.9 | MI.MIGLC.MTH.06.9-12.A3.1.4 | MI.MIGLC.MTH.06.9-12.G1.1.5 |
MI.MIGLC.MTH.06.9-12.G1.4.2 | MI.MIGLC.MTH.06.9-12.G2.3.4
TOP: Lesson 8.6 Identify Special Quadrilaterals
KEY: quadrilateral | coordinate geometry | identify
BLM: Comprehension
NOT: 978-0-618-65613-4
126. ANS:
PTS: 1
DIF: Level B
REF: MIM20654
TOP: Lesson 8.6 Identify Special Quadrilaterals
KEY: quadrilateral | Venn diagram
BLM: Comprehension
NOT: 978-0-618-65613-4
127. ANS:
No. The converse says that if a quadrilateral has diagonals that are perpendicular, then it is a kite. However, a
rhombus has diagonals that are perpendicular, but a rhombus cannot be a kite since its opposite sides are
congruent.
PTS: 1
DIF: Level C
REF: GE0.08.06.ER.01
NAT: NCTM 9-12.REA.3 | NCTM 9-12.GEO.1.c | NCTM 9-12.GEO.1.a | NCTM 9-12.REA.4
STA: MI.MIGLC.MTH.06.9-12.G1.4.1 | MI.MIGLC.MTH.06.9-12.G1.4.2 |
MI.MIGLC.MTH.06.9-12.G1.4.3 | MI.MIGLC.MTH.06.9-12.G1.4.4 | MI.MIGLC.MTH.06.9-12.G2.3.2
TOP: Lesson 8.6 Identify Special Quadrilaterals
KEY: kite | quadrilateral | diagonal | converse
BLM: Analysis
NOT: 978-0-618-65613-4
27
ID: A
128. ANS:
The vertices are P −3, 2 , Q −10, 8 , R −1, 10 , S 6, 4 . First, prove that PQRS is a parallelogram by
showing that both sets of opposite sides have equal slopes and therefore are parallel.
4 − (10)
8− 2
10 − 8
6
2
6
Slope of PQ =
= − . Slope of QR =
= . Slope of RS =
= − .
−10 − (−3)
−1 − (−10)
6 − (−1)
7
9
7
2− 4
2
Slope of SP =
= . Second, prove that PQRS is a rhombus by showing that the parallelogram has
−3 − (6)
9
10 − (2)
= 4.
diagonals with slopes that have a product of –1. Slope of PR =
−1 − (−3)
Slope of QS =
4− 8
1
1
= − . 4 × (− ) = –1
6 − (−10)
4
4
PTS: 1
DIF: Level B
REF: MLGE0027
TOP: Lesson 8.6 Identify Special Quadrilaterals
KEY: coordinate geometry | rhombus | proof
NOT: 978-0-618-65613-4
129. ANS:
The vertices are H −4, − 2 , I 3, 5 , J 7, 6 , K −5, − 6 .
Slope HI =
IJ =
BLM: Analysis
6 − −6
−6 − −2
−2 − 5
5−6 1
= 1, slope IJ =
= , slope JK =
= 1, and slope KH =
= 4.
−4 − 3
3−7 4
7 − −5
−5 − −4
6−5
2
+ 7−3
2
=
17, KH =
−2 − −6
2
+ −4 − −5
2
=
17
HI and JK are opposite sides of HIJK with the same slopes. IJ and KH are opposite sides of HIJK with
different slopes. Therefore, exactly two sides, HI and JK , are parallel. Therefore, HIJK is a trapezoid.
The length of IJ = 17 and the length of KH = 17.
Therefore, since IJ ≅ KH, by the definition of an isosceles trapezoid,
HIJK is an isosceles trapezoid.
PTS:
TOP:
KEY:
NOT:
1
DIF: Level B
REF: MLGE0028A
Lesson 8.6 Identify Special Quadrilaterals
coordinate geometry | isosceles trapezoid | proof
BLM: Analysis
978-0-618-65613-4
28
ID: A
130. ANS:
The vertices are H −5, − 1 , I 1, 5 , J 6, 7 , K −7, − 6 .
Slope HI =
HJ =
7 − −6
−6 − −1
−1 − 5
5−7 2
5
= 1, slope IJ =
= , slope JK =
= 1, and slope KH =
= .
−5 − 1
1−6 5
6 − −7
−7 − −5
2
(7 − (−1)) 2 + (6 − (−5)) 2 =
185, IK =
(5 − (−6)) 2 + (1 − (−7)) 2 =
185
HI and JK are opposite sides of HIJK with the same slopes. IJ and KH are opposite sides of HIJK with
different slopes. Therefore, exactly two sides, HI and JK , are parallel. Therefore, HIJK is a trapezoid.
The length of HJ = 185 and the length of IK = 185.
Therefore, since the diagonals HJ ≅ IK have the same length,
HIJK is an isosceles trapezoid.
PTS: 1
DIF: Level B
REF: MLGE0028B
TOP: Lesson 8.6 Identify Special Quadrilaterals
KEY: coordinate geometry | isosceles trapezoid | proof
BLM: Analysis
NOT: 978-0-618-65613-4
131. ANS:
The vertices are C −7, − 9 , D −1, − 1 , E 4, − 3 , F 1, − 7 .
−9 − (−1) 4
−1 − (−3)
−3 − −7
2
4
= , slope DE =
= − , slope EF =
= ,
−7 − (−1) 3
−1 − 4
5
4−1
3
−7 − −9
2
2
1
and slope FC =
= . DE =
−3 − (−1) + 4 − (−1) = 29,
1 − −7
4
Slope CD =
FC =
−9 − −7
2
+ −7 − 1
2
= 2 17
CD and EF are opposite sides of CDEF with the same slopes. DE and FC are opposite sides of CDEF with
different slopes. Therefore, exactly two sides, CD and EF , are parallel. Therefore, CDEF is a trapezoid.
The length of DE = 29 and the length of FC = 2 17.
Therefore, since DE ≠ FC, by the definition of a non-isosceles trapezoid,
CDEF is a non-isosceles trapezoid.
PTS:
TOP:
KEY:
NOT:
1
DIF: Level B
REF: MLGE0028C
Lesson 8.6 Identify Special Quadrilaterals
coordinate geometry | isosceles trapezoid | proof
BLM: Analysis
978-0-618-65613-4
29
ID: A
132. ANS:
The vertices are B −3, − 5 , C 1, − 1 , D 6, 1 , E −5, − 10 .
−5 − (−1)
1 − −10
−1 − 1 2
= 1, slope CD =
= , slope DE =
= 1,
−3 − 1
1−6
5
6 − −5
2
−10 − −5
5
2
and slope EB =
= . CD =
1 − (−1) + 6 − 1 = 29,
−5 − −3
2
Slope BC =
EB =
−5 − −10
2
+ −3 − −5
2
=
29
BC and DE are opposite sides of BCDE with the same slopes. CD and EB are opposite sides of BCDE with
different slopes. Therefore, exactly two sides, BC and DE, are parallel. Therefore, BCDE is a trapezoid.
The length of CD = 29 and the length of EB = 29.
Therefore, since CD ≅ EB, by the definition of an isosceles trapezoid,
BCDE is an isosceles trapezoid.
PTS: 1
DIF: Level B
REF: MLGE0028D
TOP: Lesson 8.6 Identify Special Quadrilaterals
KEY: coordinate geometry | isosceles trapezoid | proof
BLM: Analysis
NOT: 978-0-618-65613-4
133. ANS:
Yes, enough information is given to show ABCD is an isosceles trapezoid. ABCD is a trapezoid because
∠BCA ≅ ∠DAC so BC AD. ∠BAC and ∠DCA are not congruent so BA is not parallel to CD. By definition
ABCD is a trapezoid. The diagonals of trapezoid ABCD are congruent because AC ≅ BD. So, ABCD is an
isosceles trapezoid by Theorem 8.16.
PTS: 1
DIF: Level C
REF: GEO.08.06.SR.03
NAT: NCTM 9-12.GEO.1.a
STA: MI.MIGLC.MTH.06.9-12.G1.4.1 | MI.MIGLC.MTH.06.9-12.G1.4.2
TOP: Lesson 8.6 Identify Special Quadrilaterals
KEY: classify quadrilaterals | isosceles trapezoid | explain
BLM: Analysis
NOT: 978-0-618-65613-4
134. ANS:
ABCD is a parallelogram. Since ∆ABC ≅ ∆CDA and corresponding parts of congruent triangles are
congruent, ∠BAC ≅ ∠DCA and ∠BCA ≅ ∠DAC . Therefore, AB CD and AD CB and ABCD is a
parallelogram.
OR
Since ∆ABC ≅ ∆CDA and corresponding parts of congruent triangles are congruent, AB ≅ CD and AD ≅ CB.
Since both pairs of opposite sides are congruent, ABCD is a parallelogram.
PTS: 1
DIF: Level B
REF: GE0.08.06.WR.02
NAT: NCTM 9-12.GEO.1.a | NCTM 9-12.GEO.1.c | NCTM 9-12.REA.4 | NCTM 9-12.REA.3
STA: MI.MIGLC.MTH.06.9-12.G1.4.1 | MI.MIGLC.MTH.06.9-12.G1.4.2 |
MI.MIGLC.MTH.06.9-12.G1.4.3 | MI.MIGLC.MTH.06.9-12.G2.3.2
TOP: Lesson 8.6 Identify Special Quadrilaterals
KEY: paragraph proof | parallelogram | quadrilateral
BLM: Analysis
NOT: 978-0-618-65613-4
30
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