Geometry Third Quarter Study Guide 1. Write the if-then form, the converse, the inverse and the contrapositive for the given statement: All right angles are congruent. 2. Find the measures of angles A, B, and C. 3. The measure of an angle is 64 o. What is the measure of its complement? What is the measure of its supplement? 4. Write an equation of the line that passes through point P (2, -3) and is perpendicular to the line x y = 4. 5. Using the diagram, give the coordinates of M if it is a midpoint. 6. A board 24 inches long is cut into two pieces in the ratio Find the length of each piece. 7. Given: ABC is isosceles with base AC, BD bisects Prove: ABD CBD B 8. Name five Theorems or Postulates that you can use to prove that two triangles are congruent. 9. If and are diameters in P, find m.
10. Line l passes through the points ( 3, 1) and (2, 5). If j l and k j, what is the slope of k? Explain your reasoning. 11. Identify the property that makes the statement true. If MP = PQ and PQ = QR, then MP = QR. 12. For each set of numbers, determine whether the numbers represent the lengths of the sides of an acute triangle, a right triangle, an obtuse triangle, or no triangle. a) b) 26, 28, 51 c) 18, 38.5, 42.5 13. Find the values of x and y. 14. Find the measure of exterior angle 15. Find a, b, and h. 16., and and bisect each other. Which triangle congruence theorem or postulate could you use to prove that HML KMJ? Explain. 17. Find RS in C. Explain your reasoning. 18. Determine whether the figures are similar. 19. Name a ray from Q through P. 20. A photo needs to be enlarged from an original with a length of 9 inches and a width of 7 inches to a size where the new width is 14 inches. What is the new length? What is the scale factor?
21. Given: bisects RST. Find QR if and 22. Find the value of x. (not drawn to scale) 23. Define a chord of a circle. 24. In the diagram, are midsegments of triangle ABC. Find the values of the variables if. 25. Line l is the perpendicular bisector of. Find m M. 26. The length of one ramp is 16 feet. The vertical rise is 14 inches. Estimate the ramp s horizontal distance and its ramp angle. 27. Draw a Venn diagram showing the relationship between squares, rectangles, rhombuses, parallelograms, and quadrilaterals. 28. Solve for x, given that. Is equilateral? 29. 1 and 2 are complementary, and 2 and 3 form a linear pair. If m 1 =, what is m 3? Explain your reasoning. 30. Given the following statements, can you conclude that Becky plays basketball on Wednesday night? (1) If it is Wednesday night, Becky goes to the gym. (2) If Becky goes to the gym, she plays basketball.
31. Would HL, ASA, SAS, AAS, or SSS be used to justify that the pair of triangles is congruent? 32. Which lines, if any, can be proved parallel given the following diagram? For each conclusion, provide the justification. 33. Can the measurements 9.7 meters, 1.1 meters, and 6.9 meters be the lengths of the sides of a triangle? 34. Given the following, determine whether quadrilateral XYZW must be a parallelogram. Justify your answer.. 35. 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. 36. Draw and label the angles and the sides of the two special triangles: 45 o -45 o -90 o and 30 o -60 o -90 o. 37. True or False: The median and altitude of a triangle can never be the same line segment. 38. The altitude of an equilateral triangle is 6. What is the length of each side? Find the area of the triangle. 39. According to the Parallel Postulate, if there is a line and a point not on the line, then how many parallels to the given line can be drawn through the point? 40. is tangent to O at A (not drawn to scale). Find the length of the radius r, to the nearest tenth. 41. Each interior angle of a regular n-gon has a measure of 156 o. Find the value of n. 42. If is an altitude of PQR, what type of triangle is PQR?
43. Find the geometric mean of 8 and 12. 44. Find the number of sides of a convex polygon if the measures of its interior angles have a sum of 2340. 45. Solve the right triangle: and find, b, and c. 46. Given that PQR ~ PST, explain why. 47. Given that a b, what is the value of x? 48. Find the m G. (The figure may not be drawn to scale.) 49. Find the values of the variables. 50. Find the value of x. 51. A triangle has the given vertices. Classify the triangle by its sides. Then determine if it is a right triangle. 52. List all of the important characteristics of each quadrilateral. a. square b. rectangle c. parallelogram d. rhombus e. trapezoid f. kite
53. 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. 54. Which lines, if any, can be proved parallel given the following diagram? 55. Point S is between points R and T. P is the midpoint of. RT = 20 and PS = 4. Draw a sketch to show the relationship between the specified segments. Find ST. 56. Given: is the perpendicular bisector of. Name three things that you can conclude. 57. m SQR = ( ) and m PQR = ( ) and m SQP = 70. 58. Solve the right triangle. Find m SQR and m PQR. 59. List each type of quadrilateral for which the statement is always true: The diagonals are congruent. 60. In RSTU, RS is 3 centimeters shorter than ST. The Perimeter of Find RS and ST. RSTU is 42 centimeters. 61. In the diagram,,,, and. Is there enough information given to show that quadrilateral ABCD is an isosceles trapezoid? Explain. 62. If p q, solve for x.
63. If one angle of a triangle is larger than another angle, then the side opposite the larger angle is longer than the side opposite the smaller angle. Use this fact to help you list the sides of triangle STU in order from least to greatest. (The figure may not be drawn to scale.) 64. Find the value of x. a) b) c) 65. Write the standard equation of a circle with a center (3, -2) and a point on the circle (23, 19). 66. form a linear pair.. Find. 67. Find in the diagram, if and. 68. Find the area of the isosceles triangle with side lengths 17 meters, 17 meters, and 30 meters. 69. Calculate the slope of the line. Does it matter which points are used? Why or why not? 70. A building casts a shadow 260 meters long. At the same time, a pole 3 meters high casts a shadow 15 meters long. What is the height of the building? 71. Tell whether each pair of triangles is similar. Explain your reasoning.
72. In and In and State whether the triangles are similar, and if so, write a similarity statement. 73. Identify the hypothesis and conclusion of the statement: If today is Friday, then yesterday was Thursday. 74. The ratios of the side lengths of triangle ABC are 7:9:12 (AB:AC:BC). Solve for x. 75. True or False: If a quadrilateral is a parallelogram, then opposite angles are complementary. 76. In the diagram, is a radius of circle R. Is tangent to circle R? Explain. 77. Find the appropriate symbol to place in the blank. (not drawn to scale) AB AC 78. Graph the equation: 2 2 ( x 5) ( y 3) 9 79. Find the value of x. 80. Find the value of x. a) b) c)
Geometry Third Quarter Study Guide Answer Key 1. If the angles are right angles, then they are congruent. If the angles are congruent, then they are right angles. If the angles are not right angles, then they are not congruent. If the angles are not congruent, then they are not right angles. 2. m A = 103, m B = 77, m C = 46 3. 26 o, 116 o 4. y x 1 5. 6. 7. 1) ABC is isosceles with base AC, BD bisects B (Given) 2) AB BC (Definition of isosceles triangle) 3) ABD CBD (Definition of angle bisector) 4) BD BD (Reflexive Property of Segment Congruence) 5) ABD CBD( SAS ) 8. SSS, SAS, HL, AAS, ASA 9. 140 10.. The slope of line l is =. Since j l, the slope of j is also. Since k j, the slope of k is the negative reciprocal of, which is. 11. Transitive Property of Equality 12. A. acute triangle, B. obtuse triangle, C. right triangle 13. x = 11, y = 14. 89 15. a = 14, b =, h = 16. SAS Congruence Postulate. Since and bisect each other, and. because they are vertical angles. Since you know that 2 pairs of sides and the included angle are congruent, you can use the SAS Congruence Postulate to prove the triangles are congruent. 17. RS = 7. In a circle, two chords that are equidistant from the center are congruent (Theorem 10.6). 18. The figures are not similar. 19. 20. new length = 18 inches; scale factor = 2 21. 34 22. 6 23. A chord of a circle is a segment with endpoints on the circle. 24. X=8, Y= 2, Z=15 25. 64 26. 7.7, 61 o 27. Diagrams vary. 28. x = 8; no 29.. Since 1 and 2 are complementary, the sum of their measures is. So, m 2 =. Since 2 and 3 form a linear pair, they are supplementary and the sum of their measures is. So, m 3 = =. 30. yes 31. AAS 32., Consecutive Interior Angles Converse 33. No 34. Yes. If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. 35. a. True b. If a quadrilateral is not a rectangle, then it is not a parallelogram. False 36. 37. False 38., 12 3 39. Exactly one 40. 10.5 41. n 15 42. A right triangle 43. 4 6 44. 15 45.
46. Since PST ~ PQR, PST Q and PTS R. Since the pairs of angles are corresponding angles, by the Corresponding Angles Converse Postulate. 47. 71 48. 85 o 49. a 20, b 22 50. 124 51. Right scalene triangle 52. 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 53. a. 1080, b. 135, c. 45 54. No lines can be proved parallel from the given information. 55. 12 56. Any three of the following: ; ; =, = 57. m SQR = 20 and m PQR = 50 58. x =, y = 59. square, rectangle 60. 12, 9 61. Yes, enough information is given to show ABCD is an isosceles trapezoid. ABCD is a trapezoid because so. and are not congruent so is not parallel to. By definition ABCD is a trapezoid. The diagonals of trapezoid ABCD are congruent because. So, ABCD is an isosceles trapezoid by Theorem 8.16. 62. 12 63. t, n, s 64. a) 67 o b) 70 o c) 56 o 65. ( x 3) ( y 2) 29 2 2 2 66. 107 67. 68. 69. ; no; the slope ratio is the same for any two points on a line. 70. 52 meters 71. Yes; The two right angles are congruent, and since parallel lines are given the alternate interior angles are congruent, so the triangles are similar by the AA Similarity Postulate 72. not similar 73. hypothesis: today is Friday, conclusion: yesterday was Thursday 74. 6 75. False 76. No; for to be tangent to circle R and would have to be perpendicular. Using the converse of the Pythagorean Theorem you find that, therefore is not a right triangle and and are not perpendicular. 77. 78. Graph 79. X= 150 o 80. a) x=5 b) x=12 c) x= 4