Indicate whether the statement is true or false. 1. If the diagonals of a quadrilateral are perpendicular, the quadrilateral must be a square. 2. If M and N are midpoints of sides and of, then. 3. The reason Identity, which is used to state that a line segment or angle is congruent to itself, is also known as the Reflexive Property of Congruence. 4. The diagonals of any rhombus are perpendicular. 5. If the diagonals of a rhombus have lengths 6 and 8, then the perimeter of the rhombus is 28. 6. The perimeter P of an isosceles triangle with a leg of length a and base of length b is given by. 7. If the midpoints of the sides of are joined to form, then the perimeter of will be twice that of. 8. The reason for the congruence of two triangles due to the congruence of two pairs of angles and their included pair of sides is represented by AAS. 9. If in, then. 10. The method CPCTC can be used to prove that two triangles are congruent. Copyright Cengage Learning. Powered by Cognero. Page 1
11. In, and are supplementary. 12. To construct the midpoint of the horizontal line segment, begin by marking off arcs of equal length from points A and B so that the arcs intersect both above and below. 13. If and, then. 14. Having proved that by SSS, we can then state that. Copyright Cengage Learning. Powered by Cognero. Page 2
15. If and in quadrilateral ABCD, then ABCD is an isosceles trapezoid. 16. The bisector of one angle of a triangle always separates the triangle into two smaller and congruent triangles. 17. When the midpoints of the sides of a quadrilateral are joined in order, the quadrilateral formed is always a square. 18. Given that point P is the midpoint of both and, it follows that. 19. In the Pythagorean Theorem (where ), the number c represents the length of the hypotenuse of a right triangle. 20. If sides and of quadrilateral MNPQ are congruent and also parallel, then MNPQ is a parallelogram. Copyright Cengage Learning. Powered by Cognero. Page 3
Indicate the answer choice that best completes the statement or answers the question. 21. In, M and N are the midpoints of and as shown. Then: a. b. c. d. None of These 22. For, is an exterior angle. Which of the following must be true? a. b. c. d. 23. Two of the angles of a triangle have measures of 50 and 60. Which number cannot be the measure of an exterior angle of this triangle? a. 100 b. 110 c. 120 d. 130 Copyright Cengage Learning. Powered by Cognero. Page 4
24. In isosceles triangle RST,. If,, and the perimeter of is 36, find the value of x. a. x = 5.5 b. x = 6 c. x = 6.5 d. None of These 25. If D is the midpoint of, what name is given to in relation to? a. angle-bisector b. median c. altitude d. hypotenuse 26. If the diagonals of a rhombus measure 10 cm and 24 cm, what is the perimeter of the rhombus? a. 13 cm b. 34 cm c. 52 cm d. 68 cm 27. The midpoints of the sides of rhombus ABCD are joined in order to form quadrilateral MNPQ. Being as specific as possible, what type of quadrilateral is MNPQ? a. parallelogram b. rectangle c. square d. rhombus Copyright Cengage Learning. Powered by Cognero. Page 5
28. is an altitude for. If RV = 5, RS = 7, and WT = 4, find the length of altitude. a. 3 b. 3.5 c. 4 d. None of These 29. Given that and, which method establishes that? a. ASA b. SAS c. AAA d. SSS 30. Both pairs of opposite sides of quadrilateral WXYZ are congruent. Being as specific as possible, what type of quadrilateral is WXYZ? a. parallelogram b. rectangle c. square d. rhombus Copyright Cengage Learning. Powered by Cognero. Page 6
31. In,. Which statement is true? a. is an altitude of. b. is an altitude of. c. is the longest side of. d. The 3 angle-bisectors meet in the exterior of. 32. In, it follows that a. and include b. and include c. and include d. and include 33. In order to justify the construction of the angle-bisector of, we verify that two triangles are congruent by which method? a. SAS b. ASA c. SSS d. HL 34. In, m is 26 larger than m. Find m. a. 77 b. 103 c. 116 d. 126 35. For the angle-measure indicated, which angle cannot be constructed? a. 22.5 b. 30 c. 40 d. 45 36. In, m and m. Find the value of x. Copyright Cengage Learning. Powered by Cognero. Page 7
37. In the figure, the perpendicular-bisector of. If and, find WV. 38. Given kite ABCD with and, the diagonals (if drawn) would be perpendicular. For diagonals and, what other relationship exists? 39. In, diagonals and intersect at point T. If MN = 12.5, NP = 8.7, and QN = 14.6, find QT. 40. What part of is included by and? 41. In addition to being congruent, how are the diagonals of a square related? 42. Where, and. Find. 43. When the midpoints of the sides of a square RSTV are joined in order, quadrilateral MNPQ is formed. Being as specific as possible, what type of quadrilateral is MNPQ? 44. Just as SSS, SAS, ASA, AAS, and HL are methods for proving that triangles are congruent, use three letters to state two methods that are not valid. Copyright Cengage Learning. Powered by Cognero. Page 8
45. In isosceles trapezoid HJKL,. Suppose that points M, N, P, and Q are the midpoints of the sides,,, and respectively. If the length of diagonal is 10 cm, find the perimeter of MNPQ. 46. For, E lies on side so that. For, what name is given to the line segment? 47. In the figure, and bisect each other. If and is larger than, find. Copyright Cengage Learning. Powered by Cognero. Page 9
48. In, M is the midpoint of and N is the midpoint of. If the length of is represented by, what expression (containing x) represents the length of? 49. In rectangle ABCD, AB = 5 and BC = 4. As a square root, find the length of diagonal. Copyright Cengage Learning. Powered by Cognero. Page 10
50. In kite ABCD, AB = AD = y 5, BC = y, and. Find the perimeter of ABCD. 51. For a trapezoid, the lengths of the two bases are a and b. What expression represents the length of the median of the trapezoid? 52. In the figure, and bisects. Name the reason that justifies why. Copyright Cengage Learning. Powered by Cognero. Page 11
53. In, RS = 19, ST = 15, and WT = 10. Find the length of altitude. 54. In trapezoid RSTV,. If m, m, m, and m, find the value of the expression. 55. In the figure, S-T-U-V and. Draw a conclusion regarding and. Copyright Cengage Learning. Powered by Cognero. Page 12
56. The angle shown measures 60. How would you construct an angle that measures 30? 57. has a perimeter of 84 cm. If cm and ST = 25 cm, find the angle of greatest measure in. 58. For a quadrilateral to be cyclic, what requirement must be satisfied? 59. For kite MNPQ, and. If diagonal is drawn, what type of triangles are formed? 60. In the figure, bisects the acute angle of to form isosceles triangle ABD with base. Which side of is longest? Copyright Cengage Learning. Powered by Cognero. Page 13
61. In the figure, the measure of exterior angle BCD is twice the measure of.what type of triangle is? 62. For a parallelogram to be a rhombus, what condition must it satisfy? 63. In a right triangle, the length of the hypotenuse is 13 inches and the length of one of the legs is 12 inches. Find the length of the other leg. 64. Supply all statements and all reasons for the following proof. Given: ; M is the midpoint of and N is the midpoint of Prove: MNAB is a trapezoid Copyright Cengage Learning. Powered by Cognero. Page 14
65. Use the given drawing and information to prove Theorem 3.1.1 (AAS). Provide all statements and reasons. Given:,, and Prove: 66. Use the drawing shown to explain the following theorem. The length of the median of a trapezoid equals one-half the sum of the lengths of the two bases. Given: Trapezoid ABCD with median Prove: [Hint: X is the midpoint of auxiliary diagonal.] 67. Be sure you can construct any the constructions provided on the course website, and that you can justify the constructions. 68. Assuming the Theorem 4.1.1, be able to establish the corollaries 4.1.2, 4.1.3, 4.1.4, and 4.1.5 Copyright Cengage Learning. Powered by Cognero. Page 15
69. Given the definition of a rectangle (see p. 187 or class notes) be able to establish the corollary 4.3.1, 4.3.2. Similarly, given the definition of the square provided in the book, establish the corollary 4.3.3. Copyright Cengage Learning. Powered by Cognero. Page 16
Answer Key 1. False 2. True 3. True 4. True 5. False 6. True 7. False 8. False 9. False 10. False 11. True 12. True 13. True 14. True 15. True 16. False 17. False 18. True 19. True 20. True 21. b 22. b 23. a 24. b 25. b 26. c Copyright Cengage Learning. Powered by Cognero. Page 17
27. b 28. c 29. a 30. a 31. a 32. d 33. c 34. b 35. c 36. x = 11 37. 5 38. bisects 39. 7.3 40. 41. perpendicular-bisectors of each other 42. 58 43. a square 44. SSA and AAA 45. 20 cm 46. altitude 47. 38 48. 49. 50. 34 51. 52. SAS 53. RW = 12 54. Copyright Cengage Learning. Powered by Cognero. Page 18
55. 56. Bisect the given angle. 57. 58. All vertices of the quadrilateral must lie on a circle. 59. isosceles 60. 61. isosceles 62. It has two congruent adjacent sides. 63. 5 inches 64. S1. ; M is the midpoint of and N is the midpoint of R1. Given S2. R2. The line segment that joins the midpoints of 2 sides of a triangle is parallel to the third side of the triangle. S3. MNAB is a trapezoid R3. Definition of trapezoid 65. S1.,, and R1. Given S2. R2. If 2 angles of a triangle are to 2 angles of a 2nd triangle, the 3rd angles are also. S3. R3. ASA 66. Draw diagonal. Where M and X are the midpoints of the sides of, it follows that. With 67. 68. 69. midpoints N and X as shown on the sides of,. Then. But, so. Copyright Cengage Learning. Powered by Cognero. Page 19