Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Duality (projective geometry) wikipedia , lookup
History of geometry wikipedia , lookup
Multilateration wikipedia , lookup
Line (geometry) wikipedia , lookup
History of trigonometry wikipedia , lookup
Rational trigonometry wikipedia , lookup
Compass-and-straightedge construction wikipedia , lookup
Trigonometric functions wikipedia , lookup
Integer triangle wikipedia , lookup
Euler angles wikipedia , lookup
Geometry Study Guide for 1st Semester Final Name:_________________________________________________Per_____________Date_____________ For the following three questions: Use the following statement: “If you are a geometry student, then you are cool.” 7. Use the diagram shown below. If XY 4x 8 , YZ 2x 4 , and XZ 40 , then x? X 1. “you are cool” represents what part of the statement? Z Y 2. “If you are cool, then you are a geometry student” represents the: 3. “you are a geometry student” represents what part of the statement. 8. Point Y is between points X and Z on a line segment. Find the length of XY given the following information: XY 3x 4; YZ 4 x 9; XZ 30 4. Write the following statement in If-Then form, and then write its converse. “Three points on the same line are collinear.” 5. Write the following statement in If-Then form, and then identify the hypothesis and the conclusion. “Dogs like to run in the park” 6. Write the following biconditional statement as a conditional statement and its converse. “The radius of a circle is 4 if and only if the diameter is 8.” For the next four questions, Fill in the blank in the given statement: 9. __________________ parts of congruent triangles are congruent. 10. The measure of an ______________ angle of a triangle is equal to the sum of the measures of the two Remote (nonadjacent) interior angles. 11. A plane contains at least _________ noncollinear points. 12. A line contains at least ___________ points. Page 2 Use the given information and the diagram to answer the following four questions. 19. ABD CBD. Name the theorem or postulate that justifies the stated congruence. A Are the given pairs of angles congruent, complementary, supplementary, adjacent, and/or vertical? B 5 6 7 D 8 C 13. 8 and 6 20. ABD CBD. Name the theorem or postulate that justifies the stated congruence. 14. 5 and 7 A 15. 5 and 8 B 16. 7 and 6 In the diagram q r . Answer the following two questions using this information. D C q 8 9 7 r 21. What must be true in order for ABC EDC by the AAS congruence postulate? E B C 17. Are 9 and 8 , congruent, complementary or supplementary? D 18. Is the sum of 9 , 8 and 7 equal to 45º, 90º, or 180º? A Page 3 22. Which other congruence must exist in order for XYZ RSZ by the SSS congruence postulate. 26. Can you use the following length, 5, 8, and 4 to construct a triangle? Why? R Y 27. Can you use the following length, 9, 15, and 4 to construct a triangle? Why? Z S X 23. Which congruence postulate/theorem can be used to prove that the two triangles shown are congruent? 28. Can you use the following length, 25, 11, and 33 to construct a triangle? Why? 29. Can you use the following length, 7, 12, and 19 to construct a triangle? Why? 30. List all three sides in order from smallest to largest. (triangle not drawn to scale) A 45º 24. Which congruence postulate/theorem can be used to prove that the two triangles shown are congruent? C 65º P B 31. List all three angles in order from smallest to largest. (triangle not drawn to scale) M Q A N 25. List the six ways you have learned to prove that two lines are parallel. 13 11 C B 23 Page 4 32. The lengths of the sides of a triangle are x, 11, and 21. What is the possible value of x? 39. In the figure (not drawn to scale), DB bisects mADC , mADB = x+30, and mCDB = 3x-10. Find mADC . A B 33. The lengths of the sides of a triangle are x, 33, and 41. What is the possible value of x? C (x+30)˚ (3x-10)˚ D 34. The lengths of the sides of a triangle are x, 3, and 5. What is the possible value of x? 40. Classify XYZ, by its sides. Y 35. The perimeter of a square ABCD is 80 inches, and CD = 4x + 12. What is the value of x? 9 9 36. The perimeter of a square EFGH is 48 inches, and EF = 6x - 42. What is the value of x? X 37. In the diagram, CD bisects ACB . The measures of the two congruent angles are 2 x 7 and 4 x 9 . Find x. 41. Classify XYZ, by its sides. X D A Z 9 7 6 B C Z Y 5 42. Classify XYZ, by its sides. Z 38. In the diagram, YZ bisects PYX . Given that mXYP 150 , find mZYP . 2 3 Z X P 150 Y Y 3 X Page 5 43. Classify XYZ, by its angles. 47. Use the diagram to find the value of x. Z xº 30˚ 120º xº 60˚ X Y 48. Use the diagram to find the value of x. 44. Classify XYZ, by its angles. xº X 65º 60º 55˚ 55˚ 70˚ Z Y 45. Classify XYZ, by its angles. For the following four questions, use the diagram below. (The diagram is not drawn to scale) b˚ j k+5 a˚ 91˚ 42˚ c˚ k+2 X 49. Which angle is greater, b or c ? 50. Which angle is greater, a or c ? 47˚ Z Y 51. Which angle is greater, a or b ? 52. Is the following statement true? Yes or No 46. Use the diagram to find the value of x. xº 120º a b c 53. Find the value of x and y in the isosceles triangle below. xº 50º 116º yº Page 6 54. In the diagram below, find mA For the following 6 questions, use the diagram below. 62 120 6 A 7 2 3 5 8 1 4 * 55. In the figure below, x is a whole number. What is the smallest possible value of x ? x x 23 58. What type of Angles are 6 and 4. 59. What type of Angles are 7 and 5. 60. What type of Angles are 4 and 1. 56. If two angles are supplementary and the measure of one angle is eight times the measure of the other, what is the measure of the greater angle? What is the measure of the smallest angle? 57. If two angles are complementary and the measure of one angle is four times the measure of the other, what is the measure of the greater angle? What is the measure of the smallest angle? 61. What type of Angles are 8 and 2. 62. What type of Angles are 1 and 8. 63. What type of Angles are 3 and 7. 64. For which value of x are lines r and s parallel? r (4x+35) (6x-15) s Page 7 For the following 5 questions, Use the diagram shown below in which, R // P 75. Find the coordinates of the midpoint of a segment with the following end points, A(5,4), B(-3,2) R 47 9 P 5 8 36º 76. Find the coordinates of the midpoint of a segment with the following end points, J(-1,-9), K(11,-5) 65. Find m8 66. Find m5 77. Fill in the reasons in the following two-column proof: 67. Find m7 Given: NQ MP , NQ // MP Prove: MPN QNP N Q 68. Find m4 69. Find m9 M P Statements 70. 1 and 2 are a linear pair. 1 and 3 are vertical angles. If m3 62 , then m2 71. 1 and 2 are a linear pair. 1 and 3 are vertical angles. If m2 77 , then m1 72. 1 and 2 are a linear pair. 1 and 3 are vertical angles. If m3 56 , then m2 73. Find the distance between the following two points G (3, 0) and H (8,10). 74. Find the distance between the following two points M (1,-3) and N (3, 5). 1. NQ MP Reasons 1. NQ // MP 2. QNP MPN 3. NP NP 4. MPN QNP 2. 3. 4. Page 8 78. Provide the reasons for the following proof. Given: BC CD, AB DE Prove: AC CE 81. Use construction tools to construct a line through P parallel to m. Show construction lines. P A B C D E Statements Reasons BC CD, AB DE BC AB CD AB m BC AB CD DE BC AB AC , CD DE CE AC CE 82. Use construction tools to construct a line through P perpendicular to m. Show construction lines. 79. Copy angle B onto line x with the vertex of the new angle at point Q. P m B 83. Classify MNO. N x Q 8 80. Use construction tools to construct a line through P perpendicular to the given line. Show construction lines. P M 9 10 O [A] Equilateral [B] Isosceles [C] Scalene [D] none of these Page 9 84. Solve for x, given that AB BC . Is ABC equilateral? 88. The measure of B is _____. [A] 89 [B] 79 [C] 99 [D] 101 85. Find the value of x. 89. Given: LMN UVW . Complete the statements. A. UW _____ B. LMN ____ 53° x° 64° 86. Find the measure of exterior angle A. 87. Which of the following could NOT be a measure of an exterior angle? 90. If ABC DEF, AB = 24 feet, m B = 43 , and m F = 31 , which of the following statements is false? [A] D A [B] mA = 106 [C] FD = 24 ft [D] ED = AB 91. Write a two-column proof: Given: BD and AC bisect each other BC AD BC AD C Prove: BCE DAE B A 40 E 65 B [A] 105 [B] 125 [C] 115 [D] 140 C A D Page 10 92. Refer to the figure below. Which of the following statements is true? H 95. Refer to the figure shown. Give a congruence statement for the two triangles and name the theorem or postulate that proves the congruence. I G J I = JI GJ ~ K H J HI JK IJ LK [A] There are no congruent triangles. [B] GIJ JHG by SSS 96. Which postulate or theorem can be used to determine the measure of RT ? [C] GHJ IHJ by SAS [D] GJH IJH by SSS 93. State the postulate or theorem that can be used to conclude that OCD OAB . [A] SSS Congruence Postulate [B] AAS Congruence Theorem [C] ASA Congruence Postulate 94. Refer to the figure shown. Which of the following statements is true? U V W T TV WV UV XV X [A] TUV WXV by SAS [B] TUV VWX by SAS [C] TUV XWV by ASA [D] TUV WXV by ASA [D] SAS Congruence Postulate 97. Given: B E and C F . What other piece of information is needed to show ABC DEF by ASA Congruence Postulate? [A] A D [B] EF FE [C] B = F [D] BC EF L Page 11 98. Given: DCA BCA, B D Prove: AB AD B C 104. Write an indirect proof. Given: m 1 = 122 and m 2 = 121 Prove: a || b a 1 A 2 b D 99. Given the points P(–1, –6), Q(15, 24), and M(7, 9), prove that M is the midpoint of PQ. 100. Refer to the figure below. 105. What is the sum of the measures of the exterior angles of any polygon? 106. Find the measure of the missing angle. ? A. If BC = 15, then LN = ______. B. If AB = 3x + 5 and NM = 2x + 1, then NM = ______. 101. The coordinates of the midpoints of the sides of a triangle are L(0, 1), M(4, 0), and N(2, –2). Find the coordinates of the vertices of the triangle. 72° 116° 108° 107. Find x and y. 131° x° 102. Two sides of a triangle have lengths 8 and 11. What are the possible lengths of the third side x? 103. Two sides of a triangle have lengths 7 and 13. The third side, x, has a length that is ______. [A] 6 < x < 13 [B] < 6 [C] > 20 [D] 6 < x < 20 119° y° 56° Page 12 108. Find AM in the parallelogram if PN 6 and MO 18. M 111. For parallelogram PQLM below, if mPML 83 , then mMLQ ______ . N A P O [A] 83 [B] mQLM [C] 97 [D] mPQM 109. Refer to the figure below. U V 15 29° 24 Y 17° X 41 W Given: UVWX is a parallelogram, mWXV 17 , mWVX 29 , XW 41, UX 24 , UY = 15 A. Find m WVU. B. Find WV. C. Find m XUV. D. Find UW. 112. Consecutive angles in a parallelogram are ALWAYS ________. [A] vertical angles [B] complementary angles [C] supplementary angles [D] congruent angles 113. Choose the statement that is NOT ALWAYS true. For any parallelogram _______. 110. For parallelogram PQLM below, if mPML 83 , then mPQL ______ . [A] opposite sides are congruent [B] the diagonals are perpendicular [C] the diagonals bisect each other [D] opposite angles are congruent [A] 97 [B] 83 [C] mQLM [D] mPQM Page 13 114. If ON 8 x 6, LM 5x 3, NM x 5, and OL 5 y 7, find the values of x and y given that LMNO is a parallelogram. O 118. Use the distance formula to determine whether ABCD below is a parallelogram. N L M 1 3 [B] x = 3; y 1 3 [C] x = 3; y 3 [D] x = 1; y 11 5 [A] x = 1; y 115. Given: ABF DEC and FB || EC Prove: BCEF is a parallelogram F E D 119. Choose the statement that is NOT always true. For an isosceles trapezoid _______. [A] the diagonals are perpendicular [B] the legs are congruent [C] the diagonals are congruent [D] the base angles are congruent 120. For the isosceles trapezoid shown below, the B & C are _______. A B C A B 116. (2, 3) and (3, 1) are opposite vertices in a parallelogram. If (0, 0) is the third vertex, then the fourth vertex is _____. 5 [A] , 2 2 [C] 5, 4 D [B] 1, 2 [D] 1, 1 117. If the diagonals of a parallelogram are perpendicular, then the parallelogram is also what type of figure? C [A] Congruent [B] perpendicular [C] Supplementary [D] Complementary 121. Use slope and/or the Distance Formula to determine the most precise name for the figure: A(– 8, –7), B(–3, –3), C(1, 2), D(–4, –2). [A] rectangle [B] rhombus [C] kite [D] square Page 14 122. Describe the figure using as many of these words as possible: rectangle, trapezoid, square, quadrilateral, parallelogram, rhombus. 123. Describe the figure using as many of these words as possible: rectangle, trapezoid, square, quadrilateral, isosceles trapezoid, parallelogram, rhombus. 127. Given: AC BD If AC = 18 and BD = 15, find the area of kite ABCD. 128. The area of the quadrilateral is _____. [A] 58 sq. units [B] 23.3 sq. units [C] 16.25 sq. units [D] 380 sq. units 129. An expression for the area of kite ABCD is _____. 124. Find the area of a square with side length 15 m. 125. The area of the parallelogram is _____. [A] 40 111 sq. units [B] 340 sq. units [C] 800 sq. units [D] 680 sq. units 126. In rhombus ABCD, AB = 20 and AC = 32. Find the area of the rhombus to the nearest tenth. [A] 379.3 [B] 384.0 [C] 506.0 [D] 499.6 [A] ( BD)( AC ) 2 [B] (AC)(BD) [C] ( AB) (CD) ( BD) 2 [D] (AB)(AD)