Geometry Final Exam - Study Guide

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1 Geometry Final Exam - Study Guide 1. Solve for x. True or False? (questions 2-5) 2. All rectangles are rhombuses. 3. If a quadrilateral is a kite, then it is a parallelogram. 4. If two parallel lines are intersected by a transversal, then alternate interior angles are congruent. 5. If two parallel lines are intersected by a transversal, then consecutive interior angles are supplementary. 6. A triangle has side lengths of 7, 9, and 11. Decide whether it is an acute, right, or obtuse triangle. 7. How long is a string reaching from the top of a 13-ft pole to a point on the ground that is 8 ft from the bottom of the pole? Give an exact answer and an approximation to 3 decimal places. For Questions 8 & 9: Decide if the argument is valid or invalid. Explain your reasoning. 8. If a figure is a rhombus, then it is a parallelogram. A square is a rhombus. Therefore, a square is a parallelogram. 9. If a figure is a quadrilateral, then it is a polygon. I have drawn a figure that is a polygon. Therefore, the figure I drew is a quadrilateral. 10. Solve using the diagram and the given measurements. (Note: The triangle is not drawn to scale.) B = 46, a = Does the figure have line symmetry, rotational symmetry, or both? 12. Find AB and BC in the situation shown. AB = x + 16, BC = 5x + 10, AC = Complete the conditional statement to make a true statement. If and are supplementary and, then.

2 14. A tree 19 feet tall casts a shadow which forms an angle of 49 with the ground. How long is the shadow to the nearest hundredth? 15. An equilateral triangle has a side length of. Find its area. 16. If and are diameters in P, find m. 17. Are the lines with the equations and parallel, perpendicular, or skew? Explain your answer. 18. Find the value of x, to the nearest whole number. (not drawn to scale) 19. The expressions given represent the measures of the acute angles of a right triangle. Find the measure of each acute angle. 20. Given:, is supplementary to, and is supplementary to Prove: Statements: Reasons: Find the measure of an acute angle that satisfies the given equation. Round your answers to the nearest tenth Find EF. 24. The vertices of are. Reflect the triangle in the line. Graph the triangle and its image.

3 25. Find the measure of 1. True or False: (questions 26 & 27) 26. If two angles are supplements of the same angle, then they must be equal in measure. 27. Vertical angles are always complementary. 28. Decide whether the solids are similar. 29. Is the biconditional "A quadrilateral is a square if and only if it is a rectangle" True or False? Explain your reasoning. 30. Can you use the SAS Congruence Postulate to prove that the two triangles are congruent? Explain your reasoning. 31. Use the figure to find the measure of 32. "If an obtuse angle is bisected, then two acute angles are obtained." Decide whether the statement and its converse are true. If false, explain. 33. To find the height of a tower, a surveyor positions a transit that is 2 meters tall at a spot 40 meters from the base of the tower. She measures the angle of elevation to the top of the tower to be 46. What is the height of the tower, to the nearest meter? 34. Given that ABC DEF, solve for x and y. 35. The translation vector is =. If the image of A is find the coordinates of point A. 36. Writing: Explain the difference between a rhombus and a rectangle. 37. Find the geometric mean of 6 and Define complementary angles.

4 39. Write and find sin B. 40. A square is inscribed in a circle of radius. Find the area of the square. 41. Find the value of x. 42. Find the surface area to the nearest tenth of a square inch of a sphere that has a radius of 17.3 inches. 43. A polyhedron has 6 faces and 7 vertices. How many edges does it have? Explain your answer. 44. Solve for a and b. 45. Identify the longest side of. 46. Theorem 8.18 says that if a quadrilateral is a kite, then its diagonals are perpendicular. Is the converse true? Justify your reasoning. 47. Find the value of x. 48. In the diagram, Z is a centroid of triangle ABC. 49. Find m and m C. Find the length of BF. 50. Find the volume of a right cone with slant height of 52 cm and radius of 20 cm. Express in terms of.

5 51. Given: ABC is an equilateral triangle; D is the midpoint of Prove: ABD CBD Statements: Reasons: 52. Find the area of the shaded region. 53. Find the value of x if m = 50 and (Assume that the ends of the figure are semicircles.) m = Determine the coordinates of the midpoint of and find the approximate distance for the points and 55. Identify the quadrilateral which has one pair of parallel sides. 56. Find the length of the altitude drawn to the hypotenuse. 57. Find the volume of the right cone 58. Suppose two triangles are similar and one pair of sides of the triangles are congruent. Are the triangles necessarily congruent? Explain your reasoning. 59. An airplane is flying at an elevation of 1500 feet. What is the airplane's angle of elevation from the runway when it is 5000 feet from the runway? Explain. 60. Find the value of x. 61. The points in a coordinate plane are reflected in the y-axis. In general, every point (x, y) is mapped onto what point? 62. Find the distance between the points (3, 1) and ( 1, 2). 63. One side of a kite is 2 cm less than 5 times the length of another. If the perimeter is 80 cm, find the length of each side of the kite.

6 64. Given is tangent to R at S, find RT. 65. Find the measure of the missing angle. 66. True or False: If a quadrilateral is a parallelogram, then consecutive angles are supplementary. For Questions 67 & 68: State one additional fact that is needed for the pair of triangles to be congruent. Identify the postulate or theorem you would then use to prove the congruence What is the length of the diagonal of a square with side lengths 70. The figure shown is a cylindrical solid with a circular cylindrical hole drilled out of the center. Find the surface area of the resulting solid. 71. Find x. 72. Are the two triangles (not drawn to scale) similar? If so, explain why they are. 73. The right prism below has bases which are equilateral triangles of side length 4 cm. Its height is 5 cm. Find its surface area. 74. Find the area of the shaded sector to the nearest whole number.

7 75. Sketch the graph of the equation. Label the coordinates of the center. 76. If the ratio of the radii of two spheres is 1:4, what is the ratio of the surface areas of the two spheres? 77. Find the measure of an interior angle and the measure of an exterior angle for the regular polygon. 27-gon 78. Given: Trapezoid ABCD with midsegment. If and, find the length of. 79. Find the volume of the figure to the nearest tenth. 80. Find RS in C. Explain your reasoning. 81. form a linear pair.. Find. 82. Identify the property that makes the statement true. If MP = PQ and PQ = QR, then MP = QR. 83. In the diagram, are midsegments of triangle ABC. Find the values of the variables if. 84. Would HL, ASA, SAS, AAS, or SSS be used to justify that the pair of triangles is congruent? 85. If the diagonals of a parallelogram are equal in length, then the parallelogram is also what type of figure?

8 86. In what type of trapezoid are the base angles congruent? 87. Writing: Explain the differences between the median, the altitude, and the angle bisector drawn from one of the vertices of a triangle. 88. Given that a b, what is the value of x? (The figure may not be drawn to scale.) 89. A line is perpendicular to and passes through point (6, 2). Write its equation. 90. In and In and State whether the triangles are similar, and if so, write a similarity statement. 91. Find the value of x to the nearest tenth. 92. Name the transformation. 93. Use the given angle measures to decide whether lines a and b are parallel., 94. In,. The length of is two times the length of. Find the lengths of all three sides of the triangle if the perimeter of the triangle is 40 inches. 95. A board 12 inches long is cut into two pieces in the ratio Find the length of each piece. 96. Find the number of sides of a convex polygon if the measures of its interior angles have a sum of Refer to the figure. Given:, BE ED What is the relationship (,, or =) between m BAE and m DAE? 98. Can the measurements 3.1 meters, 2 meters, and 2.5 meters be the lengths of the sides of a triangle? 99. Use a special right triangle to find the tangent of the given angle: "If, then I will go to the game." What is the underlined portion called in this conditional statement?

9 Geometry Study Guide for Final - Answer Section False 3. False 4. True 5. True 6. It is an acute triangle. 7. ft; ft 8. valid; Law of Syllogism 9. invalid; converse error (The figure could have been a triangle.) line symmetry 12. AB = 21, BC = ft 15. sq. units parallel; Slopes are equal and y-intercepts are different True 27. False 28. similar 29. False; the conditional "If a quadrilateral is a rectangle, then it is a square" is false. Not all rectangles are squares. 30. No. In order to use the SAS Congruence Postulate, the sides that form the congruent angles must be congruent. In the diagram, the right angles are congruent, but only 1 pair of sides that forms the right angle are congruent Statement is true, converse is false. An acute angle bisected produces acute angles, also m 34. x = 10.8, y = (13, 8) 36. Sample answer: A rhombus is a quadrilateral with four congruent sides while a rectangle is a quadrilateral with four right angles Two angles are complementary if the sum of their measures is sq. units E = 11; 11, because F + V = E + 2 and = m Z m X Answer: No. The converse says that if a quadrilateral has diagonals that are perpendicular, then it is a kite. However, a rhombus has diagonals that are perpendicular, but a rhombus cannot be a kite since its opposite sides are congruent m = 288, m C =

10 midpoint = distance = 55. trapezoid No; it is possible for two non-corresponding sides of two similar triangles to be congruent without the triangles being congruent. 59. About. so 60. x = ( x, y) units cm, 33 cm True 67. Possible answers: ASA; AAS; SAS 68. Possible answers: ASA; AAS; SAS Yes; corresponding angles are equal in measure and ratios of corresponding sides are all equal : about 166.7, about RS = 7. In a circle, two chords that are equidistant from the center are congruent (Theorem 10.6) Transitive Property of Equality ASA 85. A rectangle 86. an isosceles trapezoid 87. Sample answer: The median is the segment from the vertex to the midpoint of the opposite side, the altitude is the segment from the vertex that is perpendicular to the line containing the opposite side, and the angle bisector is the ray that divides the angle at the vertex in half not similar Rotation 93. Yes 94. AC = 16 in., BC = 16 in., AB = 8 in m BAE m DAE 98. Yes The hypothesis

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