Int. Geometry Units 1-6 Review 1

Transcription

1 Int. Geometry Units 1-6 Review 1 Things to note about this review and the Unit 1-6 Test: 1. This review packet covers major ideas of the first six units, but it does not show examples of all types of problems.. Tests and quizzes are another excellent place to study. 3. The test will be a mixture of multiple choice, free response, and proofs. 1. Given that = 1, then what is the value of x? (U1L, U1L3) x 5x. is the midpoint of. What is? (U1L, U1L3) y+ 5y-7 3. OG bisects OH. If the m OG = 4x + 18 and m OH = 10x +, then classify OH as acute, right, obtuse, or straight. (U1L4) 4. Given the diagram and m G = 8x 5. The value of x can be represented in the form 5 where p and q have no common factors. What is p + q? (U1L4) 6 G ( x+7) 46 Questions 5-8: Let s = today is Saturday, r = I ran 5 miles, and p = I write poetry. Translate the following statements into symbols. (U1L5, U1L6) 5. It is not Saturday or I write poetry. 6. Today I ran 5 miles and today is Saturday 7. If I write poetry, then I did not run 5 miles. 8. Today is Saturday if and only if I do not write poetry.

2 Int. Geometry Units 1-6 Review Questions 9-10: Given that p is true, q is true, and r is false, determine the truth value. (U1L5) 9. ~ ( p q) r 10. r ( q p) 11. Suppose p ~ q is true. What can you say about the truth values of p and q? (U1L5) Questions 1-19: The Venn diagram shows the results of a survey about types of ice cream that they enjoy eating. How many people enjoy eating.? (U1L6) 1. None 13. hocolate and Vanilla 14. Strawberry or hocolate 15. Rocky Road or Vanilla 16 Rocky Road and Strawberry 17. Vanilla and hocolate and Strawberry. 18. Only Vanilla 19. hocolate but not Vanilla 0. Given the statement If p, then q, which of the following are equivalent statements? (U1L7) a) p only if q b) p if q c) ll p are q d) p if and only if q e) p imples q 1. Write the two conditionals of the biconditional n angle is a right angle if and only if it measures 90 o. (U1L7). a) Given the statement If you live in Las Vegas, then you live in Nevada. Write the converse, inverse, and contrapositive, and state the truth value. b) Which statements are logically equivalent? (U1L7 & U1L8)

3 Int. Geometry Units 1-6 Review 3 Questions 3-4: etermine if the argument pattern is valid and state the pattern name. (U1L9) 3. If the butler committed the crime, then the maid didn t do it. The maid didn t do it. Therefore, the butler committed the crime. 4. b a; c b c a Questions 5-6: Given the statements, write a conclusion to create a valid argument. If it is not possible then write none. (U1L9) 5. If you pass this test, then you will get an for the course. You do not pass the test. 6. ll Senators are over 30 years old. (hint: rewrite this in if-then form) Jose is 5 years old. 7. The supplement of an angle is 15 more than 6 times the complement of the angle. What is the measure of the angle? (UL5) 8. 1 and are supplementary angles, and 1 and 3 are vertical angles. If m = 7, then find m 3. (UL5) 9. etermine if (UL6) ( 6x+4) ( 1x-6) 30 We have a theorem that states Two lines are perpendicular if and only if they form congruent adjacent angles. xplain why this cannot be applied to the picture below knowing that points,, and are not collinear. (UL6) 1

4 Int. Geometry Units 1-6 Review Given the conditional If two coplanar lines are perpendicular to the same line, then the two lines are parallel. (a) reate an appropriate diagram (b) Write the given information. (c) Write the prove statement. o NOT prove. (hint: remember to label your picture, it will help with parts b and c) (UL7) 3. What are skew lines? (U3L1) Questions 33-35: Use the diagram to list a pair of: (U3L1) 33. lternate interior angles 34. orresponding angles 35. Same-side interior angles Solve for the variables. 37. Solve for the variables. (U3L & U3L3) (U3L & U3L3) ( 5y+4) ( 6x) y ( x-5) 84 ( 3y+10) 38. Give the value of the variable to make and. (U3L4) ( 3x-5) ( 5y+15) 85

5 Int. Geometry Units 1-6 Review ind the value of x. 40. In the figure, for which (U3L5 & U3L6) value of x is l m? (U3L4 & U3L5) 3y 78 l 7y 5y 35 m 41. ind the value of x. 4. and. (U3L & U3L5 & U3L6) ind the value of y. (U3L & U3L5) a a b b y ind the value of x. (U3L5 & U3L6) The angles of a quadrilateral are in the ratio 5:7:9:3. What is the measure of the largest angle? (U3L5 & U3L6)

6 Int. Geometry Units 1-6 Review Two exterior angles of a triangle measure 134 o and 145 o. What is the measure of the 3 rd exterior angle of the triangle? (U3L5 & U3L6) Questions 46-49: In each of the diagrams below, name all pairs of congruent triangles you can identify (without drawing more segments or naming more points). (a) Write down the triangle congruence statements and (b) explain why the triangles are congruent (SSS, SS, etc). If there are not any pairs of triangles in a given diagram that are congruent, state none. (U4L-5 & U4L7) (a) (b) (a) (b) I G (a) (b) (a) (b) H 50. ind the value of x, y, and z that makes. (U4L1) 10y z 5x

7 Int. Geometry Units 1-6 Review ind the value of y. 5. In equilateral triangle, (U4L-5 & U3L-3 & U3L6) find the value of y. (U4L6 & U3L6) 5 5 5x-18 x+30 6 y y 53. In triangle. m = 90, =, and the ratio of m to m is to 3. What is the value of x? (U4L6 & U3L5) 54. ind the value of x. 55. What is the perimeter of (U4L6 & U3L & U3L5-6) the triangle? (U4L6) 134 5x+6 3x x-8 80

8 Int. Geometry Units 1-6 Review 8 Questions 56-58: etermine if the statement is true or false from the diagram. (U5L1) 56. m 3 > m = > In triangle, it is known that > and <. Order the angles from least to greatest measure. (U5L) 60. If the figure was drawn to scale then which length would be the shortest? (U5L) c 6 d e 50 a 68 b Two sides of a triangle are 56 and 74. What is the possible range of values of the 3 rd side to form a triangle? (U5L3) 6. Three sides of a triangle are 1, (k+3), and (k+8). What are the possible values of k? (U5L3) 63. Two sides of an isosceles triangle are 5 and 7. What is the perimeter of the triangle? (U5L3) 64. What is the difference between an altitude and a median of a triangle? (U5L4) Questions 65-68: omplete each statement. (U5L4) 65. If is on the bisector of, then is equidistant from and. 66. If is equidistant from and, then lies on the.

9 Int. Geometry Units 1-6 Review If is on the perpendicular bisector of H, then is equidistant from and. G 68. If is equidistant from and G, then lies on. 69. a) ind the value of x and y given that ~. (U5L5) b) What is the scale factor? (U5L5) c) Hence, or otherwise, what is? (U5L5) H 5x ( 3y-30) x irections 70-73: Tell whether the triangles are similar or not similar. If they are similar state, the postulate or theorem used and write a similarity statement. (U5L6-7) G 15 J 1 H 55 I G H I J

10 Int. Geometry Units 1-6 Review Solve for x. 75. ind the values of x and y. (U5L8) (U5L8 & U5L6 & U5L1) 9 x 4 x x y 76. Solve for x in simplest 77. ind the length of the radical form. (U6L1 & U6L) altitude from (U6L) 5x Questions 78-80: lassify the triangle as acute, obtuse, right. If a triangle cannot be formed, then write not a triangle. (U6L3 & U5L3) , 48, ,, , 7, 6 Questions 81-83: Solve for the variable. nswer in simplest radical form x x 7 45 x

11 Int. Geometry Units 1-6 Review 11 Questions 84-9: omplete the following proofs. 84. Given: RP = TQ PS = QS 85. Given: XY WY; m 1 m Prove: SRT STR Prove: WY is not perpendicular to WZ S X Y 1 W P Q Z R T Z 86. Given: Given: S T; m 4 m 5 Prove: Prove: S does not bisect SR R 1 S 3 4 T 88. Given: PR QS, SR QR, 89. Given: ; and PQ QR Prove: O O Prove: QT RT P S T O Q R

12 Int. Geometry Units 1-6 Review Given: Given: PS QT and PQ ST Prove: SV VR = TV VQ Prove: SU SP QP = QR R P Q P T S 1 V T S Q R U 9. Given: m > m, and 93. Prove m 1 > m 3 Prove: > nswers: 1. x = V W (x = 4 produces negative lengths). = Obtuse (m OH = 17 ) 4. p + q = 3 (x = WY V 5. ~s p 6. r s 7. p ~r 8. s ~p 9. alse 10. alse 11. p = true; q = false a and c and e

13 Int. Geometry Units 1-6 Review If an angle is a right angle, then it measures 90 o. If an angle measures 90 o, then it is a right angle.. a) original True onverse If you live in Nevada, then you live in Las Vegas. alse Inverse If you don t live in Las Vegas, then you don t live in Nevada. alse ontrapositive If you don t live in Nevada, then you don t live in Las Vegas. True b) original and contrapositive; inverse and converse 3. Invalid; onverse rror 4. Valid; Law of Syllogism 5. None (converse error) 6. Therefore, Jose is not a Senator. (Law of ontrapositive Inference) o 8. m 3 = yes (x=8) 30. The theorem states two lines are. but there are 3 lines shown in the diagram. j k 31. a) b) Given: Lines l and k are coplanar; l j and l k l 1 c) Prove: k j 3. Non-coplanar lines (two lines not on a same plane can never intersect, but they are not considered parallel) and and and x = 16; y = x = 38; y = x = 30; y = x = x = x = y = x = largest angle = 135 o rd exterior angle = 81 o 46. ; HL 47. ; SSS 48. None (SS) 49. None () 50. x = 5, y = 8, z = y = y = x = x = True 57. alse (notation) 58. True 59. m < m < m 60. c < 3 rd b side < V oth are segments originating from a vertex of a triangle, but the altitude is perpendicular to the line containing the opposite side while the median goes to the midpoint of the opposite side.

14 Int. Geometry Units 1-6 Review and 66. angle bisector of 67. and H 68. Perpendicular bisector of G 69. a) x = 7, y = 40 b) 3: c) = ~, SSS~ 71. GJ~ IH, ~ 7. no similar triangles 73. G~ HIJ, SS~ 74. x = x = 6, y = Wj 76. V l k 77. altitude = Right triangle 79. Not a triangle 80. cute triangle 81. x = x = x = 6 Note with the proofs, there are multiple solutions to these problems. l 84. Statement Reason 1. RP = TQ and PS = QS 1. Given. RP + PS = TQ + QS. ddition Po 3. RS = RP + PS and TS = TQ + QS 3. Segment ddition Post. 4. RS = TS 4. Substitution Po 5. SRT STR 5. Isosceles Triangle Theorem 85. Temp. assume that WZ WY and we know XY WY so by the definition of perpendicular lines 1 and are right angles. ll right angles are congruent so 1, but this contradicts the fact that m 1 m so our assumption is false and WY is not perpendicular to WZ. 86. Statement Reason Given.. corresponding angles are congruent Given If a line is parallel to one of two parallel lines, it must be parallel to the other one as well 87. Temp. assume S does bisects SR which means by definition m 1 = m. We know S T and when lines are parallel corresponding angles and alternate interior angles are congruent so m 1 = m 4 and m = m 5. This means by transitivity m 4 = m 5, but this contradicts the given that m 4 m 5 so our assumption is false and S does not bisect SR.

15 Int. Geometry Units 1-6 Review Temp. assume QT = RT which means by the Isosceles Triangle Theorem m SQR = m PRQ. We know SR QR and PQ QR so by definition PQR and SRQ are right angles and all right angles are congruent meaning PQR SRQ. y the reflexive poe QR=QR which means PQR SRQ by S. y the definition of congruent triangles PR=QS which contradicts the fact that PR QS so our assumption is false and QT RT. 89. Statement Reason 1. and 1. Given. =. reflexive property 3. Δ Δ 3. SSS congruence 4. O O 4. definition of a congruent triangles 5. O = O 5. reflexive property 6. ΔO Δ O 6. SS congruence 7. O O 7. definition of a congruent triangles Statement Reason Given. SVQ TVR. vertical angles are congruent 3. ΔSVQ ~ Δ TVR 3. ~ SV VQ 4. efinition of similar triangles 4. = TV VR 5. SV VR = TV VQ 5. means-extreme property (cross multiplication) Statement Reason 1. PQ ST 1. Given. SU TU. Triangle Proportionality Theorem = SP TR 3. PS QT 3. Given 4. QP TU 4. Triangle Proportionality Theorem = QR TR 5. QP SU 5. transitivity (or substitution) = QR SP

16 Int. Geometry Units 1-6 Review Statement Reason Given. m 1 = m. lternate Interior ngle Theorem 3. m > m 3. Given 4. m > m 1 4. Substitution Po 5. > 5. In a triangle the longer side is opposite the larger angle. 93. Statement Reason 1. m 1 > m 1. xterior ngle Inequality Theorem m > m 3. xterior ngle Inequality Theorem 3. m 1 > m 3 3. Transitive (or prop of inequalities)