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Geometry Model Problems Test (California Essential Standards) GE 1.0 1. Which of the following is a list of the undefined terms in Geometry: a. Point, line, plane. b. Ray, line, line segment. c. Point, line, line segment. d. Ray, line segment, plane. 2. While working with drawings of quadrilaterals, Deanna made the following observation: Opposite angles of parallelograms are congruent. Although she had not proven the statement was true, she planned to ask her class to prove that it was. Deanna’s observation would be best described by which term below? a. A theorem. b. An axiom. c. An undefined term. d. A conjecture. GE 2.0 3. Theorem: A triangle has at most one obtuse angle. Eduardo is proving the theorem above by contradiction. He began by assuming that in ABC, ∠A and ∠B are both obtuse. Which theorem will Eduardo use to reach a contradiction? a. If two angles of a triangle are equal, the sides opposite the angles are equal. b. If two supplementary angles are equal, the angles measure 90 ˚. c. The largest angle in a triangle is opposite the longest side. d. The sum of the measures of the angles of a triangle is 180 ˚. 4. Use the proof to answer the question below. Given: AB ≅ BC ; D is the midpoint of AC Prove: STATEMENT 1. AB ≅ BC ; D is the midpoint 1. Given REASON of AC 2. AD ≅ CD 2. Definition of midpoint 3. BD ≅ BD 4. 3. Reflexive Property 4. ? What reason can be used to prove that the triangles are congruent? a. b. c. d. AAS ASA SAS SSS 5. Given: ∠2 ≅ ∠3 Prove: ∠1 ≅ ∠4 Use the proof to answer the question below. STATEMENT 1. ∠2 ≅ ∠3 2. ∠1 ≅ ∠2; ∠3 ≅ ∠4 3. ∠1 ≅ ∠4 REASON 1. Given 2. ? 3. Transitive Property What reason can be used to justify step 2? a. Vertical angles are congruent. b. Complements of congruent angles are congruent. c. Supplements of congruent angles are congruent. d. Corresponding angles are congruent. GE 3.0 6. GRAM is a parallelogram. If GR = RA, which of the following must also be true? a. GRAM is a square b. GA = RM c. GRAM is a rectangle d. GRAM is a rhombus 7. LATR is a parallelogram. If ∠L ≅ ∠A, and m∠L + m∠A = 180˚, which of the following must also be true? a. LATR is a square b. LATR is a rhombus c. LATR is a rectangle d. LT ⊥ AR 8. All rectangles are squares. Which of the following diagrams is a counterexample to the statement? a. c. b. d. 9. If two triangles have two pairs of angles congruent and one pair of sides congruent , then the two triangles are congruent. Which of the following diagrams is a counterexample to the statement? a. d. c. b. GE 7.0 10. In the diagram, lines l and m are cut by a transversal. If which of the following could be used to prove that are parallel? a. Converse of the Alternate Interior Angles Theorem. b. Converse of the Alternate Exterior Angles Theorem. c. Converse of the Corresponding Angles Postulate. d. Converse of the Consecutive Interior Angles Theorem. , 11. Lines l and m are shown in each diagram. In which diagram MUST lines l and m be parallel? a. b. c. 12. For the quadrilateral shown, what is the value of x? a. 8 b. 16 c. 18 d. 20 13. In the diagram, quadrilateral TRAP is a trapezoid in which . What is the value of x? a. 12.5 b. 8 c. 5 d. 4 14. In the diagram shown, what is the value of x? a. 10 b. 16 c. 18 d. 24 d. 15. Quadrilateral ABCD is circumscribed by a circle, as shown in the diagram to the right. What is the measure of C? a. 70° b. 82° c. 86° d. 94° GE 4.0 16. If and and have sides sufficient to prove that , which of the following would be ? a. b. c. d. 17. Which two must be similar? a. Two isosceles right triangles b. Two isosceles trapezoids c. Two rhombi d. Two rectangles 18. Which triangles must be similar? a. b. c. d. 19. In quadrilateral QUAD, QU is parallel to DA . If QD is not congruent to UA , then which statement below must also be true? a. b. c. QUAD is a parallelogram. d. QUAD is a trapezoid. GS 17.0 20. In the diagram, ABCO is a parallelogram. What are the coordinates of the intersection of the diagonals? c a, b + 2 a b. , b + c 2 b+c c. a , 2 a b+c d. , 2 2 a. GE 5.0 21. Given: ABCD is a rhombus Which theorem or postulate listed below could be used to prove that ? a. b. c. d. and AC bisects DB . Prove: 22. In the diagram below, ∠A ≅ ∠D and ∠B ≅ ∠E . AAA SSA AAS SSS What additional information would be enough to prove that ? a. b. c. d. GE 6.0 23. If two sides of a triangle are 6 inches and 10 inches, what is the smallest whole number length that could be the third side? a. 16 inches. b. 15 inches. c. 5 inches. d. 4 inches. GE 17.0 24. In the diagram, ABCO is a parallelogram. What are the coordinates of the intersection of the diagonals? a. b. c. d. 25. In the diagram, ABC is a right triangle. What is the slope of BC ? a. c. b. d. 31. What is the sum of the measures of ∠a and ∠b? a. b. c. d. GE 13.0 33. 102° 152° 176° 184° In the diagram shown m∠ACB =111˚. What is the measure of ∠CBA ? a. b. c. d. 48° 42° 27° 21° 34. In the diagram shown, what is the measure of the angle marked X? a. 114˚ b. 145˚ c. 156˚ d. 160˚ GE 22.0 35. If ABC is rotated 90˚clockwise about the origin to form A’B’C’, what would be the coordinates of A’? a. A’ (-3, -1) b. A’ (-1, -3) c. A’ (1, 3) d. A’ (3, 1) 36. The coordinates of the vertices of units down and 4 units to the right to create ? a. b. c. d. 37. , , , , , are , , . If is translated 2 , what are the coordinates of the vertices of , , , , If quadrilateral DEFG is reflected across the y-axis, it would create quadrilateral D’E’F’G’. What are the coordinates of point G’ ? a. G’ (6, 2) b. G’ (-6, -2) c. G’ (-2, -6) d. G’ (2, 6) GE 14.0 38. A diagram from a proof of the Pythagorean Theorem is shown. Which expression could be used to complete the expression by representing the areas of the smaller objects on the right side of this equation : ? a. b. c. d. GE 15.0 a. 5 39. A right triangle’s hypotenuse has length 11. If one leg has length 6, what is the length of the other leg? 40. b. 85 c. 157 d. 17 In a basketball game, a player from the home team threw the ball from corner C to a player standing at point E. (E is the midpoint of AD ). Then the player at point E threw the ball to a player at corner B. If the court was 80 feet long and 50 feet wide, how far was the ball thrown? (Leave in simplified radical form) a. 10 41 feet b. 20 41 feet c. 10 89 feet d. 20 89 feet GE 18.0 41. In the figure shown, , , and . What is the approximate length of BC ? a. b. c. d. 42. 4 in. 16 in. 20 in. 36 in. In the figure below, if tan A = 4 , what are 3 and ? a.= sin A 4 3 = ,cos A 7 7 b. = sin A 3 4 = ,cos A 7 7 c. = sin A 3 4 = ,cos A 5 5 d.= sin A 4 3 = ,cos A 5 5 43. A ladder is leaned against a wall at an angle of 65° to the ground. How far off the ground does the ladder touch the wall? a. b. c. d. 12 feet 27 feet 33 feet 63 feet GE 19.0 Triangle JKL is shown in the diagram. Which equation should be used to find the length 44. of LJ ? a. b. c. d. 45. On a swing set, on engineer used a support bar that was 20 feet long. If the support bar forms a 70° angle to the ground, how far apart will the support bars be at the base? a. b. c. d. 46. 18.8 feet 13.6 feet 37.6 feet 55 feet In the diagram, and equation could be used to find BC? a. b. c. d. . Which GE 20.0 47. The right triangle in the diagram has one side with a length of What is the length of the side marked ? . a. 5 b. 15 c. d. GE 21.0 48. = 60 , and the In the circle shown, the measure of BC ? measure of ∠BAD = 62 . What is the measure of CD a. b. c. d. 49. 60˚ 62˚ 64˚ 68˚ In the circle shown, DF and CE are chords intersecting at G. If DG = 9, FG = 4, and EG = 12, what is the length of CG ? 50. a. 3 b. 3.4 c. 5.3 d. 7 In the circle shown, what is the measure of angle 1? a. b. c. d. 51. 14˚ 28˚ 33˚ 71˚ LM is tangent to a circle, whose center is C, at point M. MQ is a diameter. If m∠QNP = 65˚ and m∠NPM = 50˚, what is m∠PMR? a. b. c. d. 52. A square is circumscribed about a circle. What is the ratio of the circumference of the circle to the perimeter of the square? a. GE 8.0 53. 40˚ 50˚ 65˚ 75˚ b. c. d. A cylinder rolls across a table top for 10 complete revolutions. If the diameter of the base is 6 inches, how far did the cylinder travel? (Leave the answer in terms of π). X a. 60π in. 54. b. 90π in. c. 120π in. d. 360π in. The prism shown has a base in the shape of a right triangle. What is the lateral surface area of the prism? a. b. c. d. 55. 36 square centimeters 66 square centimeters 72 square centimeters 84 square centimeters What is the volume of the prism shown? a. b. c. d. 56. 350 cubic millimeters 286 cubic millimeters 280 cubic millimeters 230 cubic millimeters A target for a yard game is made with areas that are alternately painted white and gray, as shown in the diagram. The inner circle is white and has a radius of 1 inch. Each of the other three rings has a radius 1 inch more than the ring before it. What is the area of the white portion of the target? a. b. c. d. 5π square inches 6π square inches 8π square inches 10π square inches GE 10.0 57. A rectangle that is 12 feet wide has a perimeter of 40 feet. What is the area of the rectangle? a. b. c. d. 96 square feet 192 square feet 240 square feet 480 square feet 58. Each side of a triangle measures 4 m. What is the area of the triangle? (Leave the answer in simplified radical form). a. 12 square meters square meters b. c. 4 square meters square meters d. 59. Quadrilateral ABCD is a rhombus. If BC = 5 inches and BD = 6 inches, what is the area of ABCD? a. b. c. d. 60. The diagram shows a trapezoid with a height of 4 cm. What is the area of the trapezoid? a. b. c. d. GE 11.0 61. 15 square inches 24 square inches 30 square inches 48 square inches 22 square centimeters 24 square centimeters 32 square centimeters 42 square centimeters The volume of a right rectangular prism is calculated to be 18 cubic centimeters. If the length, the width, and the height of the prism are all doubled, what would be the volume of the new prism? a. b. c. d. GE 9.0 62. The cylinder shown has a height of 4 cm and the diameter of the base is 10 cm. What is the volume of the cylinder? a. b. c. d. 63. 26 cubic centimeters 36 cubic centimeters 72 cubic centimeters 144 cubic centimeters 400π cubic centimeters 100π cubic centimeters 80π cubic centimeters 40π cubic centimeters The pyramid shown has a square base that measures 8 cm on each side. The slant height of the pyramid is 6 cm. What is the surface area of the prism? a. b. c. d. 128 square centimeters 160 square centimeters 256 square centimeters 288 square centimeters GE 16.0 64. Given: . What is the first step in constructing the perpendicular bisector to ? a. b. c. d. 65. 66. Draw a line segment connecting points E and F. From point C, draw an arc that intersects the line at points A and B. Draw a line segment connecting points A and B. From points A and B, draw equal arcs that intersect at points E and F. Darla is constructing an equilateral triangle. Which of the following could be her first step? a. c. b. d. Marsha is using a straightedge and compass to do the construction shown. Which statement best describes the construction Martha is doing? a. a line through P parallel to line l by constructing two lines perpendicular to the same line b. a line through P parallel to line l by copying an angle c. a line through P perpendicular to line l d. a line through P congruent to line l 67. Amina is bisecting an angle. Which of the construction diagrams shown below best represents the beginning of Amina’s construction? c. a. b. d.