Geometry Semester 1 Model Problems (California Essential Standards) Short Answer GE 1.0 1. List the undefined terms in Geometry. 2. Match each of the terms with the corresponding example a. A theorem. b. A conjecture. c. An axiom (postulate). d. An undefined term. e. Inductive Reasoning. f. Deductive Reasoning. U. Every student in Mr. Smith s class drew a quadrilateral, measured the interior angles, and found the sum. They discovered that all of their sums were the same and generalized this was true for every quadrilateral. V. Shelly used a piece of paper as a model of a plane and described it in her notes. W. Rueben found a property of rhombi, but does not know if it can be proven to be true. X. Yasmine noticed that all rectangles have opposite sides congruent and then found a way to prove this was true. Y. Rachel used a property of parallelograms to calculate the lengths of the sides of a polygon on her homework. Z. Andrew noticed that the relationship between corresponding angles has not been proven, but is accepted as true and is used to prove that other angle relationships are true. GE 16.0 3. Given: AB. What is the first step in constructing the perpendicular bisector to AB? a. Draw a line segment connecting points E and F. b. From point C, draw an arc that intersects the line at points A and B. c. Draw a line segment connecting points A and B. d. From points A and B, draw equal arcs that intersect at points E and F. 4. Darla is constructing an equilateral triangle. Which of the following could be her first step? a. c. b. d.
5. 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 6. Amina is bisecting an angle. Which of the construction diagrams shown below best represents the beginning of Amina s construction? a. c. b. d. GE 7.0 7. In the diagram, lines l and m are parallel. What relationship exists between angles A and B? What postulate or theorem supports that relationship? 8. Lines l and m are shown in each diagram. In which diagram MUST lines l and m be parallel? a. b. c. d.
9. For the quadrilateral shown, what is the value of x? 10. In the diagram, quadrilateral TRAP is a trapezoid in which TP RA. What is the value of x? 11. In the diagram shown, what is the value of x? 12. Quadrilateral ABCD is circumscribed by a circle, as shown in the diagram to the right. What is the measure of C? GE 5.0 13. Given: ABCD is a rhombus and AC bisects DB. Prove: AED AEB Which theorem or postulate could be used to prove that AED AEB? 14. In the diagram, A Dand B E. What additional information would be enough to prove that ABC DEF?
GE 4.0 15. If LMN and PQR have sides LM PQ and MN QR, which pair of angles would need to be congruent to be sufficient to prove that LMN PQR? 16. In quadrilateral QUAD, QU is parallel to DA. If QD is not congruent to UA, then which statement below must also be true? a. m Q+ m A= 180 b. U A c. QUAD is a parallelogram. d. QUAD is a trapezoid. GE 13.0 17. In the diagram shown, m CBD = 95. What is the measure of CDB? 18. In the diagram shown, P is a point on ML. What is the measure of the angle marked X? GE 17.0 19. In the diagram, ABCO is a parallelogram. What are the coordinates of the intersection of the diagonals?
20. In the diagram, ABC is a right triangle. What is the slope of BC? GE 6.0 21. 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? Describe the triangle inequality theorem. GE 2.0 22. 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. What theorem will Eduardo use to reach a contradiction? 23. Use the proof to answer the question below. 24. Given: 2 3 Prove: 1 4 Given: AB BC ; D is the midpoint of AC Prove: ABD CBD STATEMENT REASON 1. AB BC ; D is the midpoint 1. Given of AC 2. AD CD 2. Definition of midpoint 3. BD BD 3. Reflexive Property 4. ABD CBD 4.? Which triangle equality theorem or postulate can be used as a correct reason for step 4? a. Use the proof to answer the question below. STATEMENT REASON 1. 2 3 1. Given 2. 1 2; 3 4 2.? 3. 1 4 3. Transitive Property What reason can be used to justify step 2?
Geometry Model Problems (CA Essential Standards) Short Answer Part 2 GE 3.0 25. GRAM is a parallelogram. If GR = RA, which of the following must also be true? a. GRAM is a square b. GRAM is a rhombus c. GRAM is a rectangle d. GA = RM 26. 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 27. All rectangles are squares. Which of the following diagrams is a counterexample to the statement? a. b. c. d. 28. 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. b. c. d. GE 22.0 29. If ABC is rotated 90 clockwise about the origin to form A B C, what would be the coordinates of A? 30. The coordinates of the vertices of JKL, are J(-2, -1), K(1, 3), L(4, -3). If JKL is translated 2 units down and 4 units to the right to create J K L, what are the coordinates of the vertices of J K L?
31. If quadrilateral DEFG is reflected across the y-axis, it would create quadrilateral D E F G. What are the coordinates of point G? GE 4.0 32. Which of the following facts would be sufficient to prove that ABC is similar to DBE? a. CE = BE b. ACE is a right angle c. AC is parallel to DE d. A B 33. Which two must be similar? a. Two isosceles right triangles b. Two isosceles trapezoids c. Two rhombi d. Two rectangles 34. Which triangles must be similar? a. b. c. d. GE 14.0 35. A diagram from a proof of the Pythagorean Theorem is shown. Write an equation that represents the area of the entire square in two ways. On the left side, express the area as the product of the length and the width. On the right, represent the sum of the areas of the triangles and the smaller square. Then use the equation to prove the theorem. GE 15.0 36. A right triangle s hypotenuse has length 11. If one leg has length 6, what is the length of the other leg?
37. 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) GE 20.0 38. The right triangle in the diagram has one side with a length of 5 3. What is the length of the side marked x? GE 18.0 39. In the figure shown, sin A 0.4, cos A 0.5, and tan A 0.9. What is the approximate length of BC? 40. In the figure shown, if and cos A? 4 tan A =, what are sin A 3 41. 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? sin 65 0.9 cos 65 0.4 tan 65 2.1
GE 19.0 42. Triangle JKL is shown in the diagram. Which equation should be used to find the length of LJ? a. sin 28 LJ = 54 b. sin 28 = 54 LJ c. cos 28 LJ = 54 54 d. cos 28 = LJ 43. 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? sin 70 0.94 cos 70 0.34 tan 70 2.75 44. In the diagram, m B= 75 and AC = 11.9 in. Which equation could be used to find BC? a. x = 11.9(tan 75 ) b. x = 11.9(sin 75 ) c. d. 11.9 x = tan 75 11.9 x = sin 75 GE 21.0 45. In the circle shown, the measure of BC = 60, and the measure of ABD = 62. What is the measure of CD?
46. 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? 47. In the circle shown, what is the measure of angle 1? 48. 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? 49. A square is circumscribed about a circle. What is the ratio of the circumference of the circle to the perimeter of the square? a. 1 4 b. 1 2 c. 2 π d. π 4 GE 17.2 50. What are the coordinates of the center of the circle? What is the radius of the circle? What is the equation of the circle shown in the diagram? GE 12.0 51. A hexagon with angles that measure (5x), (5x 35), (5x), (6x), (5x), and (2x + 72). What is the value of x? 52. The sum of the measures of 3 of the exterior angles of a pentagon is 210. If the remaining exterior angles are congruent, what is the measure of each?
53. The measure of each exterior angle of a regular polygon is 18. How many sides does the polygon have? 54. The measure of each interior angle of a regular polygon is 157.5. How many sides does the polygon have? 55. Write an expression in simplest form that represents the sum of the measures of a and b. 56. What is the measure of the interior angle of a regular polygon with 10 sides? GE 10.0 57. A rectangle that is 12 feet wide has a perimeter of 40 feet. What is the area of the rectangle? 58. Each side of a triangle measures 4 m. What is the area of the triangle? (Leave the answer in simplified radical form). 59. Quadrilateral ABCD is a rhombus. If AC = 10 inches and BD = 8 inches, what is the area of ABCD? 60. The diagram shows a trapezoid with a height of 4 cm. What is the area of the trapezoid? GE 9.0 61. 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?
62. 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? GE 8.0 63. 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). 64. The prism shown has a base in the shape of a right triangle. What is the lateral surface area of the prism? 65. What is the volume of the prism shown? 66. 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? GE 11.0 67. 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?