Indicate whether the statement is true or false.

Transcription

1 MATH 121 SPRING PRACTICE FINAL EXAM Indicate whether the statement is true or false. 1. Given that point P is the midpoint of both and, it follows that. 2. If, then. 3. In a circle (or congruent circles) containing two unequal chords, the shorter chord is nearer the center of the circle. 4. 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. 5. If ray BD bisects, then. 6. When the midpoints of the sides of a quadrilateral are joined in order, the quadrilateral formed is always a square. 7. Given that quadrilateral quadrilateral MNPQ,. a. True Copyright Cengage Learning. Powered by Cognero. Page 1

2 b. False 8. For a given circle, the length of a 90 arc is one-third the circumference of the circle. 9. Given that, it follows that. 10. For a rectangle with base length of 1 foot and height 6 inches, the area is In, and are supplementary. 12. If the diagonals of a rhombus have lengths 6 and 8, then the perimeter of the rhombus is When lines and intersect at point, it follows that. 14. According to the Angle-Addition Postulate, m m = m. 15. If a sector of a circle has arc measure 120, then its area equals one-third the area of the circle. 16. Where m = x and and are tangents to the circle, m. Copyright Cengage Learning. Powered by Cognero. Page 2

3 17. In the figure, and are known as vertical angles. 18. Where x is the measure of an angle and 0 < x < 90, the angle is an obtuse angle. 19. Points A, B, and C are said to be collinear if they lie on a line. 20. If the diagonals of a quadrilateral are perpendicular, the quadrilateral must be a square. Indicate the answer choice that best completes the statement or answers the question. 21. Considering the definitions and postulates of geometry, which of the following is a correct statement? a. An angle has more than one angle-bisector. b. A line segment has two midpoints. c. A line segment has two endpoints. d. A plane contains exactly three noncollinear points. 22. For (not shown), it follows that: a. b. c. d. and are supplementary. 23. The owner and a partner in a small business share profits in the ratio 2:1. For a month in which the business realizes a profit of $15,600. what is the owner s share of the profit? a. $5,200 b. $7,800 c. $10,400 d. None of These 24. Suppose that lines r and s are both perpendicular to line t. Then r is parallel to s if: a. r and s are collinear b. r and s intersect c. r and s are coplanar d. None of These 25. If the area of a regular octagon is 25, find the area of the regular octagon whose sides are twice those of the first octagon. a. 50 b. 25 c. 50 d. None of These 26. For a triangle whose perimeter measures 36 units, the radius of the inscribed circle is 3.Find the area of the triangle. a. 27 b. 54 c. 108 d. None of These 27. In,,, and. find. a. 20 b. 40 c. 60 d. 80 Copyright Cengage Learning. Powered by Cognero. Page 3

4 28. If, then it can be proved that: a. b. is a right angle c. bisects d. None of These 29. 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 30. Leaving home, the driver of a car travels 10 miles in the direction N 53 E. In what direction must the driver travel to return home? a. S 37 W b. N 37 W c. S 53 W d. None of These 31. If is circumscribed about the circle in such a way that m = 90 and m = 110, which is true? a. cannot be a right angle b. m m c. m = 200 d. 32. Solve the proportion for x. a. b. c. d. None of These 33. For a construction problem, which instrument could you use? a. calculator b. protractor c. tape measure d. compass 34. 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 4

5 35. For a quadrilateral to be a rectangle, how must the diagonals be related? a. perpendicular b. congruent c. bisect each other d. congruent and bisect each other 36. In trapezoid RSTV, and. If RS = 10 in, ST = 12 in, VT = 18 in, and RW = 8 in, find the area of RSTV. a. 112 b. 100 c. 120 d. None of These 37. Which statement is true? a. Any two equilateral triangles are similar. b. Any two equilateral triangles are congruent. c. Any two rectangles are similar. d. Any two rectangles are congruent. 38. Given lines and m with, it follows that: a. b. c. d. None of These 39. Given that and, it follows that: a. b. c. d. 40. In, m is 26 larger than m. Find m. a. 77 b. 103 c. 116 d. 126 Enter the appropriate word(s) to complete the statement. 41. If and, then: 42. If two planes intersect, they intersect in a(n): 43. If and are supplementary and, then must be a(n): 44. A property of geometry that is accepted as true without proof is a(n): 45. If ray BD bisects, then m 46. Assuming that (i) and (ii) are true statements, what conclusion can you deduce? (i) If x = 5, then x 3. (ii) x < In trapezoid RSTV,. If m, m, m, and m, find the value of the expression. Copyright Cengage Learning. Powered by Cognero. Page 5

6 48. In the figure, the perpendicular-bisector of. If and, find WV. 49. Find the sum of the interior angles for an octagon. 50. Which method for proving triangles congruent is used only for proving right triangles congruent? 51. For a square whose length of apothem is a, find an expression containing a that represents the area A of the square. 52. In, m and m. Determine the value of x. 53. In the figure, secants and intersect the circle at points R and S respectively. If m = 36 and m : m = 4:1, find m. 54. For, M and N are the midpoints of the indicated sides. If and, find the length of. 55. Write the formula for the area A of any quadrilateral that has perpendicular diagonals of lengths and. 56. It is given that. If,,, and, find y. 57. In, m = 37 and m = 69. Find m. 58. When the midpoints of the sides of a kite RSTV are joined in order, quadrilateral MNPQ is formed. Being as specific as possible, what type of quadrilateral is MNPQ? 59. In kite WXYZ, WX = WZ and XY = YZ. If and m = 76, find m. Copyright Cengage Learning. Powered by Cognero. Page 6

7 60. Consider the noncollinear points A, B, C, and D. By using two points at a time, find the total number of lines that they determine. 61. If m = 37, find m. 62. To determine the measure of, which expression should be calculated. (m m ) or (m m )? 63. If m = 50, then what fraction of the area of the circle represents the area of the sector shown? Copyright Cengage Learning. Powered by Cognero. Page 7

8 64. One property of proportions takes the form, If, then. Use this property to complete this statement. Because, it follows that: 65. In,,, and. Which angle, if any, measures 90? 66. Where,, and, find. 67. In the figure, bisects of. Complete the proportion that follows.? 68. Suppose that you have proved that and, in turn, that. What reason would you use to further conclude that? 69. Having proved that by the reason SSS, what reason allows you to conclude that? 70. A triangle has a perimeter of 40 and area of 60. Using A = rp, find the length of radius r for the inscribed circle for this triangle. 71. Given: Provide the statements for this proof. S1. R1. Given S2. R2. Distributive Law (FOIL) S3. R3. Substitution Proerty of Equality S4. R4. Addition Property of Equality 72. Given: and are supplementary Supply missing statements and missing reasons for this proof. Copyright Cengage Learning. Powered by Cognero. Page 8

9 S1. and are supplementary R1. S2. R2. If the exterior sides of 2 adjacent angles form a straight line, the angles are supplementary. S3. R3. Two angles that are supplementray to the same angle are congruent. 73. Provide missing statements and missing reasons for the following proof. Given: ; diagonals and intersect at point P and S1. R1. Given S2. R2. S3. and R3. S4. R4. S5. R5. ASA S6. R Provide the missing statements and missing reasons for the following proof. Given: and ; and S1. R1. Given S2. R2. S3. R3. S4. R4. CASTC Copyright Cengage Learning. Powered by Cognero. Page 9

10 76. Given: In, ; is a right angle. and are complementary. Provide all statements and reasons for this proof. 77. In the figure, A-B-C. Explain why and must be supplementary. 78. Provide missing statements and reasons for the following proof. Given: bisects S1. R1. Given S2. R2. S3. R3. Angle-Addition Postulate S4. R4. or Copyright Cengage Learning. Powered by Cognero. Page 10

11 79. Supply missing statements and missing reasons for the following proof. Given: so that bisects ; also, S1. so that bisects R1. S2. R2. S3. R3. Given S4. R4. If 2 angles of one triangle are congruent to 2 angles of a second triangle, then the third angles of these triangles are also congruent. 80. Supply missing statements and missing reasons in the following proof. Given: in the figure shown S1. R1. S2. R2. S3. R3. Vertical angles are congruent. S4. R Provide all statements and all reasons for the following proof. Given:,,, and Quad. ABCD is a parallelogram Copyright Cengage Learning. Powered by Cognero. Page 11

12 82. Supply missing statements and missing reasons for the following proof. Given: and transversal p; is a right angle is a right angle S1. and transversal p R1. S2. R2. S3. R3. Congruent measures have equal measures. S4. R4. S5. R5. Substitution Property of Equality S6. R6. Definition of a right angle 83. In the figure provided,. Explain why it is necessary that is also congruent to 84. Explain why the angle-bisector method is justified. Consider that the given angle,, is to be bisected by the constructed ray,. Copyright Cengage Learning. Powered by Cognero. Page 12

13 85. Provide the missing statements abd nissing reasons for the proof of this theorem. If both pairs of opposite sides of a quadrilateral are congruent, the quadrilateral is a parallelogram. Given: Quad. MNPQ; and MNPQ is a parallelogram S1. R1. Given S2. Draw diagonal R2. S3. R3. Identity S4. R4. SSS S5. R5. S6. R6. If 2 lines are cut by a trans. so that alternate interior angles are congruent, these lines are parallel. S7. R7. S8. R8. S9. R Use the given drawing and information to prove Theorem (AAS). Provide all statements and reasons. Given:,, and 87. Provide missing statements and missing reasons for the proof of the following theorem. The diagonals of a rhombus are perpendicular. Copyright Cengage Learning. Powered by Cognero. Page 13

14 Given: Rhombus ABCD; diagonals and intersect at point X S1. R1. S2. R2. The diagonals of a rhombus ( ) bisect each other. S3. R3. S4. R4. All sides of a rhombus are congruent. S5. R5. SSS S6. R6. S7. R Supply missing statements and missing reasons for the following proof. Given: Rectangle MNPQ with diagonals and S1. R1. S2. and are rt. R2. S3. R3. S4. R4. The diagonals of a rectangle are congruent. S5. R Provide the missing reasons for the following proof. Given: M is the midpoint of ; also, S1. M is the midpoint of R1. S2. R2. S3. R3. S4. R4. S5. R5. Copyright Cengage Learning. Powered by Cognero. Page 14

15 90. Supply missing statements in the following proof. Given: and S1. R1. Given S2. R2. If 2 parallel lines are cut by a transversal, corr. angles are congruent. S3. R3. Given S4. R4. Same as #2. S5. R5. Transitive Property of Congruence S6. R6. If 2 lines are cut by a transversal so that corresponding angles are congruent, then these lines are parallel. Copyright Cengage Learning. Powered by Cognero. Page 15

16 Answer Key 1. True 2. True 3. False 4. True 5. True 6. False 7. True 8. False 9. False 10. False 11. True 12. False 13. True 14. True 15. True 16. True 17. True 18. False 19. True 20. False 21. c 22. d 23. c 24. c 25. d 26. b 27. d Copyright Cengage Learning. Powered by Cognero. Page 16

17 28. a 29. c 30. c 31. b 32. c 33. d 34. b 35. d 36. a 37. a 38. b 39. b 40. b line 43. right angle 44. postulate 45. m 46. x HL 51. A = x = BC = 20 Copyright Cengage Learning. Powered by Cognero. Page 17

18 55. A = rectangle (m m ) {2,3,4,5} Means-Extremes Property 69. CPCTC S1. S2. S3. S R1. Given S2. and are supplementary. S S1. ; diagonals and intersect at point P R2. The opposite sides of a parallelogram are parallel. R3. If 2 parallel lines are cut by a transversal, the alternate interior angles are congruent. R4. The opposite sides of a parallelogram are congruent. S S1. and ; and R2. Substitution Property of Equality R3. SSS Copyright Cengage Learning. Powered by Cognero. Page 18

19 S In, the longest side lies opposite the largest angle, right angle R. That is,. Similarly, in, the longest side lies opposite the largest angle, right angle TUS. It follows that. By the Transitive Property of Inequality,. 76. Proof: S1. R1. Given S2. is a right angle. R2. Given S3. R3. Definition of right angle S4. R4. Substitution Prop. of Equality S5. R5. Subtraction Property of Equality S6. and are comp. R6. Definition of complementary angles 77. In the figure, is a straight angle. By definition,. But by the Angle-Addition Postulate. Then by substitution. By definition, and are supplementary. 78. S1. Given R2. Definition of angle-bisector S3. S4. R4. Substitution Property of Equality 79. R1. Given S2. R2. Definition of angle-bisector S3. S S1. in the figure shown R1. Given R2. If 2 parallel lines are cut by a trans, the alternate interior angles are congruent. S3. S4. R4. AA 81. S1., R1. Given S2. R2. If 2 coplanar lines are to the same line, they are. S3., and R3. Given S4. R4. Same as reason 2. S5. Quad. ABCD is a parallelogram R5. Definition of parallelogram 82. R1. Given R2. If 2 parallel lines are cut by a trans, corresponding angles are congruent. S3. R4. Given S5. S6. is a right angle 83. Given that, we know that and that. Because vertical angles are congruent, we also have. Then is congruent to by the reason AAS. Copyright Cengage Learning. Powered by Cognero. Page 19

20 84. The first arc is drawn to intersect the sides of at 2 points, say E and F. Thus,. From E and F, arc of the same length are drawn to intersect at a point, say G. Then. Because. Then by SSS. It follows that, by CPCTC. 85. S1. Quad. MNPQ; and R2. Through 2 points, there is exactly one line. S3. S4. R5. CPCTC S6. R7. CPCTC R8. Same as reason 6 S9. MNPQ is a parallelogram R9. Definition of parallelogram 86. 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 87. S1. Rhombus ABCD; diagonals and intersect at point X R1. Given R3. Identity S4. S5. R6. CPCTC S7. R7. If 2 lines meet to form congruent adjacent angles, these lines are perpendicular. 88. S1. Rectangle MNPQ with diagonals and R1. Given R2. All angles of a rectangle are right angles. R3. Identity S4. S5. R5. HL 89. R1. Given R2. Definition of midpoint R3. Given R4. Identity R5. SSS 90. S1. S2. S3. S4. S5. S6. Copyright Cengage Learning. Powered by Cognero. Page 20