# Geometry 21 - More Midterm Practice

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1 Class: Date: Geometry 21 - More Midterm Practice 1. What are the names of three planes that contain point A? 6. If T is the midpoint of SU, what are ST, TU, and SU? A. ST = 7, TU = 63, and SU = 126 B. ST = 80, TU = 80, and SU = 160 C. ST = 18, TU = 18, and SU = 36 D. ST = 63, TU = 63, and SU = 126 A. planes ABDC, ABFE, and ACHF B. planes ABDC, ABFE, and CDHG C. planes CDHG, ABFE, and ACHF D. planes ABDC, EFGH, and ACHF 2. Name the intersection of plane ACG and plane BCG. A. AC B. BG C. CG D. The planes need not intersect. 3. Plane ABC and plane BCE be the same plane. A. must B. may C. cannot 7. Find the measures of PMN and NMR if MN bisects PMR. The measure of PMR is 136. Draw a sketch that shows the given information. Explain your answer. 8. If AB is opposite AC and AC what can you conclude? Explain. is opposite AD, 9. A piece of wood was cut into the wedge shape shown. Name the planes represented by the right-hand and left-hand sides of the wedge. Explain why the two planes are not the same plane. 4. If EF = 2x 12, FG = 3x 15, and EG = 23, find the values of x, EF, and FG. The drawing is not to scale. A. x = 10, EF = 8, FG = 15 B. x = 3, EF = 6, FG = 6 C. x = 10, EF = 32, FG = 45 D. x = 3, EF = 8, FG = If EG = 25, and point F is 2/5 of the way between E and G, find the value FG. The drawing is not to scale. 10. BD bisects ABC. m ABC = 7x. m ABD = 3x Find m DBC. A. 108 B. 72 C. 180 D. 42 A B. 10 C. 15 D. 20 1

2 11. Find the values of x and y. 14. a. Write the following conditional in if-then form. b. Write its converse in if-then form. c. Determine the truth value of the original conditional and its converse. Explain why each of them is true or false, and provide a counterexample(s) for any false statement(s). On a number line, the points with coordinates 2 and 5 are seven units apart. 15. Given: 1 and 2 are complementary, and 2 and 3 are complementary. Prove: 1 3 A. x = 15, y = 17 B. x = 112, y = 68 C. x = 68, y = 112 D. x = 17, y = What are the converse, inverse, and contrapositive of the following true conditional? What are the truth values of each? If a statement is false, give a counterexample. If a figure is a rectangle, then it is a parallelogram. 13. Complete the two-column proof. Given: 1 2, m 1 = 130 Prove: m 3 = Given: 1 and 2 are supplementary, and 1 3. Prove: 3 and 2 are supplementary. Drawing not to scale 1 2, m 1 = 130 m 2 = 130 m 2 = m 3 m 3 = 130 a. b. c. d. 17. Write a two-column proof. Given: 7y = 8x 14; y = 6 Prove: x = 7 2

3 18. Find m Q. The diagram is not to scale. 22. Find the value of x for which p is parallel to q, if m 1 = (9x) and m 3 = 117.The diagram is not to scale. A. 66 B. 56 C. 124 D The expressions in the figure below represent the measures of two angles. Find the value of x. f Ä g. The diagram is not to scale. A. 108 B. 13 C. 117 D Find the value of x. The diagram is not to scale. Given: SRT STR, m SRT = 28, m STU = 2x A. 15 B. 17 C. 16 D Find the values of x and y. The diagram is not to scale. A. 28 B. 30 C. 14 D Find the value of x for which l is parallel to m. The diagram is not to scale. A. x = 51, y = 63 B. x = 77, y = 63 C. x = 63, y = 77 D. x = 77, y = Justify the statement algebraically. In a triangle, if the sum of the measures of two angles is equal to the measure of the third angle, then the triangle is a right triangle. A. 28 B. 56 C. 84 D Given QRS TUV, QS = 3v + 2, and TV = 7v 6, find the length of QS and TV. A. 2 B. 9 C. 8 D. 20 3

4 26. Given ABC PQR, m B = 3v + 4, and m Q = 8v 6, find m B and m Q. A. 22 B. 11 C. 10 D Given m 1 = m 2, what can you conclude about the lines l, m, and n? Explain. 30. Write a two-column proof. Given: 2 and 5 are supplementary. Prove: l Ä m 28. What other information do you need in order to prove the triangles congruent using the SAS Congruence Postulate? 31. Find the value of x. The diagram is not to scale. A. BAC DAC B. AC BD C. CBA CDA D. AC BD 29. Two sides of an equilateral triangle have lengths x + 2 and 2x Which could be the length of the third side: 14 x or 2x + 4? A. 2x + 4 only B. both 14 x and 2x + 4 C. 14 x only D. neither 14 x nor 2x + 4 A. x = 60 B. x = 21 C. x = 15 D. none of these 32. Find the value of x. The diagram is not to scale. Given: RS ST, m RST = 7x 54, m STU = 8x A. 14 B. 152 C. 16 D. 19 4

5 33. Which two triangles are congruent by ASA? MR bisects QO, and MQP ROP. A. MQP and MPN B. MPQ and RPO C. MNP and ONP D. none 34. Supply the reasons missing from the proof shown below. Given: AB AC, BAD CAD Prove: AD bisects BC Statements Reasons 1. AB AC 1. Given 2. BAD CAD 2. Given 3. AD AD 3. Reflexive Property 4. BAD CAD 4.? 5. BD CD 5.? 6. AD bisects BC 6. Definition of segment bisector A. ASA; Corresp. parts of are. B. SAS; Reflexive Property C. SSS; Reflexive Property D. SAS; Corresp. parts of are. 35. Two sides of a triangle have lengths 6 and 8. What lengths are possible for the third side? Explain. 5

6 36. BE is the bisector of ABC and CD is the bisector of ACB. Also, XBA YCA. Which of AAS, SSS, SAS, or ASA would you use to help you prove BL CM? 40. Determine which triangles in the figure are congruent by AAS. A. AAS B. SSS C. SAS D. ASA 37. Write a two-column proof to show that OQR MQR. Given: NO PM, 1 2, and OR MR 41. Separate and redraw BDG and AEF. Identify any common angles or sides. 42. Write a two-column proof. Given: BC EC and AC DC Prove: BA ED 38. P, Q, and R are three different points. PQ = 3x + 2, QR = x, RP = x + 2, and x > 0. List the angles of PQR in order from largest to smallest and justify your response. 39. In the figure, 1 2, 3 4, and TK TL. Prove that RTK STL. 43. m A = 9x 7, m B = 7x 9, and m C = 28 2x. List the sides of ABC in order from shortest to longest. A. AB; AC; BC B. BC ; AB; AC C. AC; AB; BC D. AB; BC ; AC 6

7 44. Write a two-column proof: Given: BAC DAC, DCA BCA Prove: BC CD 48. Where can the perpendicular bisectors of the sides of a right triangle intersect? I. inside the triangle II. on the triangle III. outside the triangle A. I only B. II only C. I or II only D. I, II, or II 49. In ABC, centroid D is on median AM. AD = x + 4 and DM = 2x 4. Find AM. A. 13 B. 4 C. 12 D Write a proof. 50. What is the range of possible values for x? Given: BC DA, 1 2, and CF AF Prove: CFE AFE 46. Find the length of the midsegment. The diagram is not to scale. A. 0 < x < 18 B. 0 < x < 5 C. 7 < x < 18 D. 5 < x < Name the second largest of the four angles named in the figure (not drawn to scale) if the side included by 1 and 2 is 11 cm, the side included by 2 and 3 is 16 cm, and the side included by 3 and 1 is 14 cm. A. 24 B. 0 C. 42 D Use indirect reasoning to explain why a quadrilateral can have no more than three obtuse angles. A. 3 B. 4 C. 2 D. 1 7

8 52. In FGH, draw median FJ from F to the side opposite F. 55. Find m A. The diagram is not to scale. A. 107 B. 117 C. 63 D Mei has a large triangular stone that she wants to divide into four smaller triangular stepping stones in a pathway. Explain why cutting along each midsegment creates four congruent stepping stones. 56. This jewelry box has the shape of a regular pentagon. It is packaged in a rectangular box as shown here. The box uses two pairs of congruent right triangles made of foam to fill its four corners. Find the measure of the foam angle marked. 54. Find the value of x. The diagram is not to scale. A. 54 B. 36 C. 18 D Use less than, equal to, or greater than to complete this statement: The measure of each exterior angle of a regular 10-gon is the measure of each exterior angle of a regular 6-gon. A. cannot tell B. less than C. greater than D. equal to 58. A nonregular hexagon has five exterior angle measures of 55, 58, 69, 57, and 55. What is the measure of the interior angle adjacent to the sixth exterior angle? A. 104 B. 66 C. 114 D. 124 A. 90 B. 45 C. 35 D

9 59. Find the values of the variables in the parallelogram. The diagram is not to scale. 62. For the parallelogram, if m 2 = 4x 20 and m 4 = 3x 11, find m 1. The diagram is not to scale. A. 9 B. 16 C. 164 D. 174 A. x = 53, y = 31, z = 96 B. x = 53, y = 53, z = 127 C. x = 31, y = 53, z = 96 D. x = 31, y = 53, z = For what values of x and y must this quadrilateral be a parallelogram? Find the lengths of the sides. The diagram is not to scale. 60. In the parallelogram, m KLO = 78 and m MLO = 42. Find m KJM. The diagram is not to scale. A. 110 B. 120 C. 78 D In the parallelogram, m QRP = 32 and m PRS = 84. Find m PQR. The diagram is not to scale. 64. For a regular n-gon: a. What is the sum of the measures of its angles? b. What is the measure of each angle? c. What is the sum of the measures of its exterior angles, one at each vertex? d. What is the measure of each exterior angle? e. Find the sum of your answers to parts b and d. Explain why this sum makes sense. 65. Given: SV Ä TU and SVX UTX Prove: VUTS is a parallelogram A. 84 B. 116 C. 32 D. 64 9

10 Geometry 21 - More Midterm Practice Answer Section 1. A 2. C 3. B 4. A 5. C 6. D 7. [4] m PMN = m NMR = 68. Because MN bisects PMR, m PMN = m NMR. By the Angle Addition Postulate, m PMN + m NMR = m PMR. By Substitution, m PMN + m PMN = m PMR, or 2m PMN = m PMR. Thus, m PMN = 1 2 m PMR = 1 (136) = 68, and m NMR = 68 as well. 2 [3] finds correct angle measures and gives explanation, but no diagram [2] finds correct angle measures and draws correct diagram, but incomplete or incorrect explanation [1] finds correct angle measures only 8. AB = AD. Sample explanation: Both rays have endpoint A and extend in the direction away from point C. 9. BFEC and BFDA. Explanations may vary. Sample: If the two planes were the same plane, then points A, B, C, D, E, and F would all be in the same plane and the piece of wood would be flat rather than wedge-shaped. 10. D 11. A 12. Converse: If a figure is a parallelogram, then it is a rectangle. The converse is false. A parallelogram that does not have four 90º angles is not a rectangle. Inverse: If a figure is not a rectangle, then it is not a parallelogram. The inverse is false. A parallelogram with angles that are not all 90º angles is not a rectangle, but it is a parallelogram. Contrapositive: If a figure is not a parallelogram, then it is not a rectangle. The contrapositive is true. 1

11 13. [4] a. Given b. Substitution Property c. Vertical Angles Theorem d. Substitution Property [3] three parts correct [2] two parts correct [1] one part correct 14. a. On a number line, if points have coordinates 2 and 5, then they are seven units apart. b. On a number line, if points are seven units apart, then they have coordinates 2 and 5. c. The original conditional is true by the Ruler Postulate. The converse is false. The points 0 and 7 are seven units apart, but their coordinates are not 2 and [4] By the definition of complementary angles, m 1 + m 2 = 90 and m 2 + m 3 = 90. By the Transitive Property of Equality (or Substitution Property), m 1 + m 2 = m 2 + m 3. By the Subtraction Property of Equality, m 1 = m 3, and 1 3 by the definition of congruent angles. OR equivalent explanation [3] one step missing OR one incorrect justification [2] two steps missing OR two incorrect justifications [1] correct steps with no explanations 16. [4] 1 and 2 are supplementary, because it is given. So, m 1 + m 2 = 180 by the definition of supplementary angles. 1 3 because it is given. So, m 1 = m 2 by the definition of congruent angles. By the Substitution Property, m 3 + m 2 = 180, so by the definition of supplementary angles, 3 and 2 are supplementary. OR equivalent explanation [3] one step missing OR one incorrect justification [2] two steps missing OR two incorrect justifications [1] correct steps with no explanations 17. [4] 7y = 8x 14; y = 6 Given 42 = 8x 14 Substitution Property 56 = 8x Addition Property of Equality 7 = x Division Property of Equality x = 7 Symmetric Property of Equality OR equivalent proof. [3] one step missing OR one incorrect justification [2] two steps missing OR two incorrect justifications [1] correct steps with no explanations 18. B 19. D 20. B 2

12 21. m 1 + m 2 + m 3 = 180. Given m 1 + m 2 = m 3, by substitution, m 3 + m 3 = m 3 = 180, and m 3 = 90. Thus, 3 is a right angle, and the triangle is a right triangle. 22. B 23. D 24. A 25. C 26. C 27. l and m are both perpendicular to n. Explanation: Because l and m are parallel, 1 and 2 are supplementary by the Same-Side Interior Angles Postulate. It is given that m 1 = m 2, so 180 = m 1 + m 2 = m 1 + m 1 = 2m 1, and m 1 = 90 = m 2. Because 1 and 2 are right angles, l is perpendicular to n and m is perpendicular to n. 28. B 29. C 30. [4] Statements Reasons 1. 2 and 5 are supplementary and 5 are supplementary 4. l Ä m 1. Given 2. Vertical angles 3. Substitution 4. Converse of Same-Side Interior Angles Postulate [3] correct idea, some details inaccurate [2] correct idea, some statements missing [1] correct idea, several steps omitted 31. B 32. A 33. B 34. D 35. Let x be the length of the third side. By the Triangle Inequality Theorem, 6 + x > 8, > x, and 8 + x > 6. Solving each inequality, x > 2, x < 14, and x > 2, respectively; or 2 < x < D Statement Reason NO PM 1. Given Given 3. OQN MQP 3. Vertical angles are congruent. 4. OQN MQP 4. AAS 5. OQ MQ 5. Corresp. parts of 6. OR MR 6. Given 7. QR QR 8. OQR MQR 7. Reflexive Property 8. SSS 3

13 38. R, Q, P. Sample: Because x = QR > 0, x < x + 2 < 3x + 2, thus QR < RP < PQ. The largest angle ( R) is opposite PQ; the next largest angle ( Q) is opposite RP. 39. Answers may vary. Sample: RTL STK by ASA, so R S. Supplements of congruent angles are congruent, so RKT SLT. RTK STL by AAS. 40. PQT RQT 41. No angle or side is common to both BDG and AEF. 42. [4] Statement Reason 1. BC EC and AC DC 1. Given 2. BCA ECD 2. Vertical angles are congruent. 3. BCA ECD 3. SAS 4. BA ED 4. Corresp. parts of are. [3] correct idea, some details inaccurate [2] correct idea, not well organized [1] correct idea, one or more significant steps omitted 43. A 44. [4] Statement Reason 1. BAC DAC and DCA BCA 1. Given 2. CA CA 2. Reflexive Property 3. CBA CDA 3. ASA 4. BC CD 4. Corresp. parts of are. [3] correct idea, some details inaccurate [2] correct idea, not well organized [1] correct idea, one or more significant steps omitted 4

14 45. [4] Statement Reason 1. BC DA 1. Given Given 3. BEC DEA 3. Vertical angles are congruent. 4. BEC DEA 4. AAS 5. CE AE 5. Corresp. parts of 6. CF AF 7. EF EF 8. CFE AFE 6. Given 7. Reflexive Property 8. SSS [3] correct idea, some details inaccurate [2] correct idea, not well organized [1] correct idea, one or more significant steps omitted 46. C 47. Assume a quadrilateral has more than three obtuse angles. Then it has four angles, each with a measure greater than 90. Their sum is greater than 360, which contradicts the fact that the sum of the measures of the angles of a quadrilateral is 360. Thus a quadrilateral can have no more than three obtuse angles. 48. B 49. C 50. D 51. D Cutting along each midsegment produces four triangles with sides that are half the length of the original sides. All three sides of each of the four triangle have the same measure, therefore by SSS, the triangles are congruent. 54. B 55. D 56. D 57. B 58. C 59. C 60. B 5

15 61. D 62. C 63. x = 4, y = 3; 12, [4] a. 180(n 2) b. 180(n 2) n c. 360 d. 360 n e. 180(n 2) n n 180(n 2) = n 180n = n = 180n n = 180 This sum makes sense because an interior angle and its adjacent exterior angle are supplementary. [3] parts a d correct; small error in part e [2] parts a d correct [1] three correct answers 65. [4] Because SVX UTX, SV TU because corresponding parts of congruent triangles are congruent. It is given that SV Ä TU. Therefore quadrilateral VUTS is a parallelogram because if one pair of opposite sides of a quadrilateral is both congruent and parallel, then the quadrilateral is a parallelogram. [3] correct idea, some details inaccurate [2] correct idea, not well organized [1] correct idea, one or more significant steps omitted 6

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