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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