Final Review Problems Geometry AC Name
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1 Final Review Problems Geometry Name SI GEOMETRY N TRINGLES 1. The measure of the angles of a triangle are x, 2x+6 and 3x-6. Find the measure of the angles. State the theorem(s) that support your equation. x = In, m = 30 and the exterior angle at is 120. Which is the longest side of the triangle? State the theorem(s) that support your equation. 3. In the diagram,. If is twice as large as E, find E. x = 30 E 4. In, is twice the measure of and the exterior angle at vertex measures 120. What is the measure of? State the theorem(s) that support your equation. < = 40
2 5. The measures of the three angles of a triangle are in the ratio of 1:4:5. What is the number of degrees in the smallest angle? State the theorem(s) that support your equation. x = The measure of the vertex angle of an isosceles triangle is twice the measure of the base angle. Find the measure of a base angle. State the theorem(s) that support your equation. x = In the diagram of, bisects and E bisects. If the measure of E = 70 and the measure of = 40, find m. Explain. < = 40 E 8. In an isosceles right triangle, the measure of one acute angle is 2x+5. Find the value of x. State the theorem(s) that support your equation. x = 20
3 9. In the diagram, E = 5x+20 and E = 3x+60. Find x. State the theorem(s) that support your equation. x = 20 E 10. If each base angle of an isosceles triangle is 15 more than the vertex angle, find the measure of the vertex angle. x = In the diagram of,. If = 80 and = x, find the value of x. State the theorem(s) that support your equation. x = The angles of a triangle are in the ratio 3:5:7. Find the measure of the smallest angle. k = 12 < = 36
4 13. In the diagram,. If = 2x+18 and = 4x-18, find the value of x. State the theorem(s) that support your equation. x = Which set of numbers can represent the lengths of the sides of a triangle? State the theorem(s) that support your equation. a. 3, 3, 6 b. 3, 4, 7 c. 4, 7, 10 d. 4, 4, In PQR, P = 51 and Q = 57. Which expression is true? State the theorem(s) that support your equation. a. QR>PQ b. PR> PQ c. PQ=QR d. PQ>QR PRLLEL LINES 16. In the diagram EF, x = 70 and y = 105. Find the measure of z. State the theorem(s) that support your equation. x = 70, y = 105, z = 35 x y E z F
5 17. In the diagram,. If EF = 65 and GHF = 45, find EGH. State the theorem(s) that support your equation. E <EGH = 110 G F H 18. In the diagram,, E, = 2x, E = 3x. Find the value of x. State the theorem(s) that support your equation. E x = In the diagram,. If GH = 2x+10 and GH = 3x-20. Find the value of x. State the theorem(s) that support your equation. E G x = 38 H 20. In the diagram, & intersect at E, and & are drawn. If, E = 100 and E = 30, find E. < = 30; < = 50 F E
6 21. In the diagram, 1 and 2 are supplementary. Which is always true? State the theorem(s) that support your equation. p a. l p b. l m 1 c. l m d. m p l QURILTERLS 2 m 22. Which statement is FLSE? a. square is a rectangle b. square is a rhombus. c. rhombus is a square. d. rectangle is a parallelogram. 23. Which figure does NOT always have congruent diagonals? a. Rectangle b. Isosceles trapezoid c. Square d. Rhombus 24. If the diagonals of a quadrilateral are perpendicular and NOT congruent, the quadrilateral may be a(n). a. Rhombus b. Rectangle c. Isosceles trapezoid d. Square 25. Which statement is always true? a. The diagonals of a parallelogram are congruent. b. The diagonals of a parallelogram are perpendicular. c. The diagonals of a parallelogram bisect the angles. d. The diagonals of a parallelogram bisect each other.
7 26. The length of the shorter diagonal,, of rhombus is 8 and = 60. Find the length of a side of the rhombus. State the theorem(s) that support your equation. = In, is on, E is on, and E. If E = 8, find. = The sides of a triangle are 6, 8, 10. What is the perimeter of the triangle formed by joining the midpoints of these sides? What is type of triangle is it? What is the area of the triangle? State the theorem(s) that support your equation. P = 12; right triangle; = In the diagram, equilateral has a perimeter of 18. Points R, S, T are midpoints of the sides. What is the length of RS? = 6; RS = 3 R S T
8 30. In rhombus,, = 4x-2 and = 3x+3. Find x. State the theorem(s) that support your equation. x = In parallelogram PQRS, Q: R = 1:4. Find m Q. State the theorem(s) that support your equation. <Q = In parallelogram, = 60. Find m. State the theorem(s) that support your equation. < = In parallelogram, = (3x-40) and = (7x-100). Find x. x = 15
9 34. In quadrilateral, = 120, = 82 and = 93. Find m. State the theorem(s) that support your equation. < = The diagonals of a rhombus are 24 and 10. Find the length of a side. State the theorem(s) that support your equation In rectangle, = 3x-15 and = 7x-55. Find x. State the theorem(s) that support your equation. x = In the diagram of rhombus, = 50. Find m. State the theorem(s) that support your equation. x = 65
10 38. In rectangle, & intersect at point E. If E = 20 and = 2x+30, find x. State the theorem(s) that support your equation. x = The length of a rectangle is three times its width, and the perimeter is 32. Find the area of the rectangle. = In the diagram of rhombus, = 80. Find m. State the theorem(s) that support your equation. < = The lengths of the bases of a trapezoid are 4 and 8. If the length of the altitude is 3, find the area of the trapezoid. = 18
11 42. In the diagram, is equilateral and EF is a rhombus. If is the midpoint of and the perimeter of is 12, what is the perimeter of EF? State the theorem(s) that support your equation. P = 8 E F 43. What is the number of degrees in the measure of each exterior angle of a regular polygon of 18 sides? State the theorem(s) that support your equation. n = The lengths of the bases of an isosceles trapezoid are 7 and 15. Each leg makes an angle of 45 with the longer base. Find the length of the altitude x = 4, leg x = 4 2, and the median of the trapezoid x = If x+15 and 2x+27 represent the number of degrees in the measures of two consecutive angles of a parallelogram, find the value of x. State the theorem(s) that support your equation. x = 46
12 SIMILR POLYGONS 46. The lengths of the side of a triangle are 2, 5, and 6. If the length of the longest side of a similar triangle is 18, find the perimeter of the larger triangle. P = In the diagram of, is on and E is on, such that E. If = 6, = 4 and = 15, find E. State the theorem(s) that support your equation. x = 9; E = 6 E 48. In the diagram of, E. If = 10, = 6 and E = 7.5, find. State the theorem(s) that support your equation. = 12.5 E 49. What positive number is the mean proportional (geometric mean) between 4 and 9? x = 6
13 50. The lengths of the side of a triangle are 5, 7, and 8. If the longest side of a similar triangle is 24, find the perimeter of the larger triangle. P = The lengths of corresponding sides of two similar polygons are in the ratio 2:5. If the perimeter of the larger polygon is 100, what is the perimeter of the smaller polygon? State the theorem(s) that support your equation. P = In, is a point on, E is a point on and E. If = 4, = 6, and = 9, find E. E = In the diagram, E,, = 4, E = 3 and = 6. Find the length of. State the theorem(s) that support your equation. = 8 E
14 54. In the, E joins points and E on and, respectively. E and E is one-fourth as long as. The ratio of the perimeter of E to the perimeter of is a b. 1 2 c d. 1 4 RIGHT TRINGLES 55. The altitude drawn to the hypotenuse of a right triangle divides the hypotenuse into two segments of lengths 3 and 12. What is the length of the altitude? State the theorem(s) that support your equation. a. 36 b. 18 c. 6 d If the ratio of the corresponding sides of two similar triangles is 4:9, then the ratio of their perimeters is a. 16:81 b. 8:27 c. 4:9 d 2:3 57. Which polygons are LWYS similar? Why? a. Equilateral triangles; all congruent angles b. Parallelograms c. Trapezoids d. Rectangles 58. In square, the length of a side is 3. Find the length of. = In right, is the altitude to hypotenuse. If = 6 and = 9, find. x = 4
15 60. In right, is the altitude to hypotenuse. If = 3 and = 9, find. = The length of a diagonal of a square is5 2. What is the length of a side? s = The length of a side of a square is 2. What is the length of the diagonal? d = The altitude drawn to the hypotenuse of a right triangle divides the hypotenuse into segments of lengths 4 and 9. Find the length of the altitude. State the theorem(s) that support your equation. x = In a right triangle, one leg is 7 and the hypotenuse is 10. Find the length of the other leg. State the theorem(s) that support your equation. b = 51
16 65. The diagonals of a rhombus have lengths of 8 and 6. Find the length of a side of the rhombus In rectangle, = 12 and = 16. Find. State the theorem(s) that support your equation therefore, In square, the length of a side is 3. Find the length of. x = In right, altitude is drawn to the hypotenuse. If = 4 and = 5, find. State the theorem(s) that support your equation. x = In right, is a right angle. ltitude bisects the hypotenuse at. If = 10, find. State the theorem(s) that support your equation. x = 5
17 70. Which set of numbers could be the lengths of the sides of a right triangle? State the theorem(s) that support your equation. a. 2, 6, 40 b. 2, 18, 20 c. 4, 6, 40 d. 4, 36, rectangle has a diagonal of length 10 and one side of length 6. What is the perimeter of the rectangle? P = 28; triangle 72. In the diagram, altitude FG is drawn in EF. If E = 8, G = 4 and E = 60, what is the length of EF? EF = 8 F 73. onstruct the median of a triangle. G E 74. onstruct a triangle. 75. onstruct a triangle. 76. onstruct a rhombus with a 30 angle. 77. onstruct a trapezoid with a 30 angle and a 135 degree angle. 78. onstruct the circumscribed circle of an obtuse isosceles triangle. Perp. is of sides 79. onstruct all three altitudes of an obtuse triangle. 80. Using constructions divide a given segment into a ratio of 3:2.ivide a seg into 5 = parts 81. onstruct a segment that is the geometric mean between segments and.
18 IRLES 82. The radius of a sphere is 25. plane intersects the sphere in a circle whose radius is 24. What is the distance between the center of the sphere and the plane? x = raw 2 circles so that the number of common tangents is 1. Internally tangent circles 84. Tangents PX & PY are drawn to circle O from point P. If PO = 12 and XPY is a right angle what is the length of the diameter? Explain. d = 12 2
19 85. Given: E ; is tangent to the circle at E Find the measure of each and give a reason or explanation. a. 1 x = 30 b. 5 x = 30 c. = x = 60 d. 2 x = 70 e. 3 x = 30 f. 4 x = 50 g. 6 x = 80 h. x = 50 i. E x = 100 j. Name 2 similar triangles. Triangle E and Triangle
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