5. Given: RSOT is a parallelogram 6. Given: TVAX is a rectangle MS TP Prove: TXV VAT

Transcription

1 1 Honors Geometry Name: First Semester Final Exam Review Part I: Proofs 1. Given: AB BC 2. Given: RHG KHJ AE EC OH is median to GJ Prove: AD DC OH is altitude to GJ Prove: P M 3. Given: ZY bisects SX 4. Given: ABDE is a parallelogram ST WX BC is the base of isosceles ΔBCD SX bisects ZY Prove: ACDE is an isosceles trapezoid Prove: SZW XYT 5. Given: RSOT is a parallelogram 6. Given: TVAX is a rectangle MS TP Prove: TXV VAT Prove: MOPR is a parallelogram 7. Given: NPRS is a parallelogram 8. Given: AB AC, BD DC with diagonals SP and NR O. Prove: ADB ADC TO plane of parallelogram NPRS Prove: ΔSTP is isosceles 9. Given: GH GK HM KM Prove: HMKJ is a kite 10. Given: AD BC 11. Given: FJ is the base of an isosceles DAB CBA FG JH, O is the midpoint of MF Prove: ΔABE is isosceles K is the midpoint of MJ Prove: OH KG 12. Given: AC AD, FC DG 13. Given: ΔKOR is equilateral 1 2 KOPR is a parallelogram BE CD KMOR is a parallelogram Prove: FCDG is an Prove: ΔJMP is equilateral isosceles trapezoid

2 2 14. Given: BA AC 15. Given: A D DC AC C bisects BE DC BA Prove: AB ED Prove: B D 16. Given: AB CB 17. Given: 1 2 BD AC 3 4 Prove: D is not the Prove: ΔABE ΔDCE midpoint of AC Part II: Final Exam Review. You may need to draw the diagram to solve the problem. 1. Find x. 2. Answer always, sometimes, or never. a) If a triangle is obtuse it is isosceles. b) The bisector of the vertex angle of a scalene is perpendicular to the base. c) If one of the diagonals of a quadrilateral is the perpendicular bisector of the other the quadrilateral is a kite. d) If A, B, C and D are non-coplanar, AB BC, and AB BD, then AB is perpendicular to the plane determined by B, C, and D. e) Two parallel lines determine a plane. f) Planes that contain two skew lines are parallel. g) Supplements of complementary angles are congruent. 3. FGHJ is a parallelogram, FG = x + 5, GH = 2x + 3, m G = 40, m J = 4x Find: m F, perimeter of FGHJ 4. ABCD is a parallelogram, m A = 3x + y, m D = 5x + 10, m C = 5y Find m B. 5. The measure of the supplement of an angle exceeds three times the measure of the complement of the angle by 12. Find the measure of half of the supplement. 6. Write the most descriptive name for each figure: a) A four-sided figure in which the diagonals are perpendicular bisectors of each other. b) A four-sided figure in which the diagonals bisect each other. c) A triangle in which there is a hypotenuse. d) A four-sided figure in which the diagonals are and all sides are. 7. If one of two supplementary angles is 16 less than three times the other find the measure of the larger. 8. Two consecutive angles of a parallelogram are in the ratio of 7 to 5. Find the measure of the larger. 9. Find m 1 if a b. 10. Given: ΔFJH is isosceles with base JH, K and G are midpoints, FK = 2x + 3, GH = 5x 9, JH = 4x Find: The perimeter of ΔFHJ

3 3 11. a) How many points determine a line? b) How many non-collinear points determine a plane? c) Collinear means? d) Coplanar means? 12. Fill in each blank with line, segment, or ray. a) A has one endpoint. b) A has a definite length. c) A can be bisected. d) A has two endpoints. e) A has no endpoints. f) A has no midpoints. g) The union of two opposite rays is called a. 13. SR = RQ = QT and T is the midpoint of SM. a) If SR = 20, then RT = b) If TM = 45, then RQ = c) If QT = 12, then QM = d) If RT = 8, then SM = 14. Use the figure at the right. a) m DQM = b) m DQY = c) m TQM = d) m TQS = e) m SQY = f) m DQT = g) Name two right angles. Q h) Name two obtuse angles that have QD as a side. i) Name three acute angles that have QM as a side. j) bisects MQS. 15. Name the following parts of isosceles ABC. 16. Name the following parts for right TMR. a) legs a) right angle b) vertex angle b) hypotenuse c) base angles c) legs 17. Classify each triangle according to its angle measures: a) 17 o, 80 o, 83 o b) 45 o, 45 o, 90 o c) 60 o, 60 o, 60 o d) 25 o, 65 o, 90 o e) 10 o, 10 o, 160 o 18. Use the figure at the right for the following true/false statements. a) AC and DE intersect in B. b) E, B, F determine a plane. c) The plane determined by C, B, D contains E. d) C and B determine a plane. e) Only one plane contains D, B, E. f) Only one plane contains DB and C. g) C, B, H, and E are coplanar. h) C, B, E, and D are coplanar. i) The intersection of planes m and n is BA.

4 20. If m R = 2x + 7 and the measure of the supplement of R = x + 8, find m R. 19. Use the diagram to find: 21. If m NOM = 37 and m MOP = 73 find m NOP. a) EIB b) BIC c) HIB I 22. In this figure how many angles are adjacent to RST? 23. Find the measure of the supplement of A if it is five times the measure of A. 24. If the measure of the complement of the angle is 10 less than ½ the measure of the supplement of the same angle find the measure of the angle, its complement, and its supplement. 25. Find the complement and the supplement of a 26 o 17 angle. 26. If EF, EG, and EH are coplanar and EF is between EG and EH, then m FEG + m = m. 27. Complete each statement: a) A triangle which has three congruent sides is an triangle. b) The of a statement says the opposite of the original statement. c) A triangle which has two congruent sides is an triangle. d) Two triangles are congruent if their are congruent. e) A geometric figure is congruent to itself by the property. f) If two sides of a triangle are congruent then the opposite these sides are. g) Every equiangular triangle is an triangle. h) A always contains the phrase if and only if. i) Any point on the of a segment is equidistant from the endpoints of the segment. 28. Write the reason each pair of triangles is congruent. SAS, ASA, SSS, AAS, HL, or none. a) b) c) d) e) f) 29. Find the missing angles and, where possible, the missing sides. a) b) 4

5 30. Classify each statement as true or false. a) If 1 2, then BA BC. b) If BCA BAC, then BA BC c) If BA = BC, then BD is the perpendicular bisector of AC. d) If EA = EC, then BD is the perpendicular bisector of AC. e) If BA = BC and EA = EC, then B, E, and D, are collinear. 31. Given: KM NO 32. Given: MN PO Find: m N Find: all pairs of 's. 33. If 4 9, then 36. A pair of interior angles on the same side of 34. If a b, then m 1 = the transversal is: 35. If a b, then m 3 = 37. A pair of corresponding angles is: 38. A pair of alternate interior angles is: 39. If DE DF, m F = 40. Find x and m D , m B = 100 o, and m C = 35 o. 42. Find x and m A. Find: m 1, m 2, m 3, m 4, and m A. 43. If AE BD, BF CE, 44. l XY, m 2 = m 1 = 100 o, then m DBF = 5

6 6 45. Given: a b, x = 46. If KM NO, which angles are supplementary? 47. m 1 = m + m is an exterior angle of which triangle? 49. If PQ ST, m P = 48 o, m PRQ = 110 o, 50. l m, find m 1, m 2, m 3, m 4, m 5. find m Q, m S, m T. 51. If EFG is a right angle, FH HG, 52. If l m, m 2 = 5x-9, m 6 = 2x+18, m 2 = 40 o, then m E = then find x and m S, T, and W are the midpoints of the sides of this triangle. a) If DE = 18, then WT = b) If WS = 11, then 22 = c) If WT = 5, ST = 8, SW = 7, then the perimeter of ΔDEF = 54. ABCD is a square. 55. EFGH is a rhombus. Fill in all angle measures. Fill in all angle measures. 56. JKLM is a rectangle. 57. NOPQ is a parallelogram. Fill in all angle measures. Fill in all angle measures. 58. RSTU is an isosceles trapezoid. Fill in all angle measures.

7 7 59. A, B, C, and D are midpoints of VY,VW,WX, and XY. If VX = 20 and CD = 3, find BA, WY, AD, and BC. 60. If two sides of a triangle have measures 5 and 12, then the third side is between and. 61. If m A > m B > m C in triangle ABC, which side is largest? 62. Parallelogram ABCD. If m A = 5x-20 and m C = 3x, find x and m B. 63. If the perimeter of the parallelogram in #62 is 120 and AB = 4x+20 and BC = 6x-10, find x, AB, and AD. 64. The coordinates of A and B are (2,7) and (-3,5), respectively. Find the coordinates of the midpoint of AB. Find the slope of AB. 65. Show using slope that ABC is a right triangle if A(4,6), B(1,2), and C(5,-1). 66. If M is the midpoint of AB and A(-3,-6) and M(7,9), find the coordinates of B. 67. An exterior angle of a regular convex polygon is 15 o. How many sides does the polygon have? 68. What is the sum of the measures of the angles of a convex hexagon? If three of the angles are 120 o, a fourth is 84 o, and the fifth is twice the sixth, find the measure of the sixth angle. 69. Find the number of diagonals of a convex dodecagon. 70. Find the measure of each interior angle of a regular 32-gon. 71. Find the number of sides of a regular polygon if the measure of each interior angle is eleven times the measure of each exterior angle. 72. Write the converse, inverse, and contrapositive of: If it is warm today, then H-F will win. 73. A(2,6), B(8,-2), and C(-10,4). 74. Use the same points and find the slope Find the slope of the median to side: of the altitude to side: a) AB a) AB b) BC b) BC c) AC c) AC 75. If M(-7,2) is the midpoint between A(8,1) and B, find the coordinates of B. 76. Prove ABCD is a rectangle if A(-2,3), B(8,3), C(8,1), and D(-2,1). 77. Find the slope of AC so that m BCA is a right angle if B(-2,4) and C(6,1). 78. Given: KITE is a kite and KT is the bisector of EI. KI = 6x + 2y 2, IT = 2x + 3y TE = 4x + 2y 3, KE = 3x + 4y. Find: x, y, and the perimeter of KITE 79. Given: PQR STV. PQ = x 2, SV = 6, ST = 2x + 15, TV = 3 x. Find : a) All possible x values. b) The perimeter of PQR. c) Is PQR scalene, isosceles, or equilateral 80. What conclusion can be drawn from the following: c f, g b, p f, c b

8 Answers 1) 1. given 2. Draw AC; 2 pts det line 3. BD bis AC; If 2 pts on a line are =dist from seg endpts, then bis. 4. AD AC; If a pt lies on bis, then it is = dist from seg endpts. 2) 1. given 2. OH GJ; If altitude, then 3. OHJ & OHG Rt s; If, then Rt. 4. OHJ OHG; If Rt, then 5. OH OH; Reflexive 6. GH JH; if median, then div into 2 seg 7. ΔGHO ΔJHO; SAS 8. G J; CPCTC 9. PHO MHO; If 2 ;s sub from 2 s then diff 10. GHM JHP; ; If 2 ;s added to 2 s then sums 11. ΔGHM ΔJHP; ASA 12. P M; CPCTC 3) 1. Given 2. RS RX & RZ RY; If bis, then 2 seg 3. RT RW; If 2 seg sub from 2 seg, then diff 4. ZRW YRT & ZRS YRX; Vert s 5. ΔTRY Δ ZRW; SAS 6. RTY RWZ; CPCTC 7. ΔXRY ΔSRZ; SAS 8. S X; CPCTC 9. SZW XYT; No choice thm 4) 1. Given 2. ED AB; If ogram, then opp sides 3. ACDE is a trap; If 1 pr opp sides then trap 4. AE BD; If ogram, then opp sides 5. BD DC; If isos Δ, then legs 6. AE DC; transitive 7. ACDE is isos trap, If trap has 2 legs, then isos. 5) 1. Given 2. RS TO; If ogram, then opp sides 3. RS TO; If ogram, then opp sides 4. RSM PTO; If lines, then alt ext s 5. ΔRSM ΔOTP; SAS 6. MR PO; CPCTC 7. RMS OPT ; CPCTC 8. MR PO; If 1 pr opp sides & then ogram 6) 1. given 2. XT VA; If rect, then opp sdies 3. TV TV; reflexive 4. XV TA; If rect, then diag 5. ΔXTV ΔAVT; SSS 6. TXV VAT; CPCTC 7) 1. given 2. TO TO; Reflexive 3. TO OP, OS; If plane, then all lines in plane thru foot 4. TOP & TOS Rt s; If, the Rt s 5. TOP TOS; If rt s, then 6. SO PO; If parallelogram, then diag bis each other 7. ΔTOS ΔTOP; SAS 8. TS TP; CPCTC 9. ΔSTP is isos; If 2 sides, then isos. 8) 1. list 2 possibilities 2. ADB ADC; assume one poss. to be true 3. Draw AD; 2 pts det a line 4. AD AD; reflexive 5. ΔADB ΔADC; SAS, 6. AB AC; CPCTC 7. AB not AC, given 8. Assumption was false, contradiction 9. ADB not ADC, only remaining poss. 9) 1. Given 2. Draw HK; 2 pts det a line 3. GJ bis HK; If 2 pts on a line are =dist from seg endpts, then line is bis to the seg 4. HMKJ is a kite; If 1 diag bis to other diag, then kite. 10) 1. Given 2. AB AB; Reflexive 3. ΔDAB ΔCBA; SAS 4. DBA CAB; CPCTC 5. ΔABE is isos; If 2 s, then isos. 11) 1. Given 2. F J; If isos, then base s 3. FH JG; If a seg is added to 2 seg, then sums 4. MF MJ; If isos, then legs 5. OF KJ; If big seg, then like div 6. ΔOFH ΔKJG; SAS 7. OH KG; CPCTC 12) 1. Given 2. FCDG is a trap; If 1 pr opp sides, then trap 3. 1 supp FCD & 2 supp GDC; If 2 s form str s (AFD), then supp. 4. FCD GDC; If 2 s supp to s, then 5. FCDG is an isos trap; If trap has lower s, then isos. 13) 1. Given 2. ΔKOR is =angular; If Δ=lateral, then =angular 3. RKO KRO KOR; If =angular, then all s 4. M KRO & P RKO; If ogram, then opp sides 5. KR MO & KR OP; If ogram, then opp sides 6. JKR M & JKR P; If lines, then corr s 7. JKR RKO & JRK ORK; Transitive 8. KR KR; Reflexive 9. ΔKJR ΔROK; ASA 10. KJR ROK; CPCTC 11. J M P; Transitive 12. ΔJMP is =angular; If all s, then =angular 13. ΔJMP is =lateral, If Δ =angular, then also =lateral. 14) 1. Given 2. BAC & DCA are Rt s; If, then Rt. 3. BAC DCA; If Rt s then 4. AC AC; Reflexive 5. ΔBAC ΔDCA; SAS 6. B D; CPCTC 15) 1. Given 2. BC CE; If bisector, then 2 seg 3. BCA DCE; Vert s 4. ΔBCA ΔDCE; AAS 5. AB ED; CPCTC 16) 1. list 2 possibilities 2. D is the midpoint of AC; assume one poss. to be true3. AD DC; midpoint splits seg into 2 segs 4. BD BD; Reflexive 5. DBA, BDC are Rt. Angles, perp. Lines form rt. angles 5. DBA BDC; right angles 6. ΔABD ΔCBD; SAS 7. AB CB; CPCTC 8. AB not CB, given 9. Assumption was false, contradiction 10. D is not mdpt of AC, only remaining poss. 17) 1. Given 2. AE ED; If sides, then s 3. BEA CED; Vert s 4. ΔABE ΔDCE; ASA 8

9 Part II 1) 145º 2) a) S e) A b) N f) S c) A g) S d) A 3) m F = 140º perimeter = 58 4) 110º 5) 64.5º 6) a) RHOM b) PARA c) RT Δ d) SQUARE 7) 131º 8) 105º 9) 110º 10) 60 11) a) 2 b) 3 non-collinear c) lies on same line d) lies on same plane 12) a) ray e) line b) seg f) line, ray c) seg g) line, angle d) seg 13) a) 40 c) 48 b) 15 d) 24 14) a) 60º f) 90º b) 150º g) SQY, DQT c) 30º h) YQT, DQS d) 30º i) DQM, MQT, MQS e) 90º j) QT 15) a) BC & AB b) B c) A, C 16) a) R b) TM c) TR, RM 17) a) acute, scalene b) right, isosceles c) eq, eq d) right, scalene e) obtuse, isosceles 18) a) T d) F g) F b) T e) F h) T c) T f) T i) T 19) a) 90º b) 15º c) 150º 20) 117º 21) 36º 22) 4 23) 150º 24) 20º, 70º, 160º 25) 63º 43 & 153º 43 26) FEH, GEH 27) a) equilateral b) converse c) isosceles d) corr s / sides e) reflexive f) s g) equilateral h) bi-conditional i) perpendicular bisector 28) a) SSS d) SAS b) ASA e) ASA c) SAS f) HL 29) a) m ABC = 55º m ACB = 55º m BAC = 70º AB = 8 b) m BAC = 75º m BCA = 60º m D = 75º ; sides not possible 30) a) F d) F b) T e) T c) F 31) 65º 32) ) p q 34) 16 (alt ext) or 13 (corr) 35) 15 (corr) or 14 (alt int) 36) 3 & 6 or 4 & 5 37) 1 & 5 or 2 & 6 3 & 7 or 4 & 8 38) 4 & 6 or 3 & 5 39) 65º 40) 59º 41) 50º, 50º, 85º, 95º, 45º 42) 52.5º, 22.5º 43) 80º 44) 80º 45) 45º O & KLO 46) N & MLN 47) 3, 4 48) BDC 49) 22º, 22º, 48º 9

10 10 50) 55º, 50º, 55º, 50º, 130º 51) 40º 52) 9, 36º 53) a) 9 b) FE c) 40 54) All corner angles are 45º All angles where diagonals intersect are 90º 55) F G E ) All corner angles are 15º and 75º All angles where diagonals intersect are 30º and 150º 57) Corner angles are 15º and 42º Corner angles are 93º and 30º All angles where diagonals intersect are 45º and 135º 58) Lower base angles are 15º and 42º Upper base angles are 108º and 15º All angles where diagonals intersect are 30º and 150º 59) 3, 6, 10, 10 60) 7, 17 61) BC 62) 10, 150º 63) 5, 40, 20 H 64) (-½, 6), 2/5 65) mab = 4/3 mbc = -3/4 mac = -7 AB BC B is a rt So ΔABC is rt Δ 66) (17, 24) 67) 24 68) 92 (non-convex) 69) 54 70) ) 24 72) conv If H-F will win, then it is warm today inv If it is not warm today, then H-F will not win contra If H-F will not win, then it is not warm today. 73) a) -2/15 b) 5/3 c) -7/12 74) a) 3/4 b) 3 c) -6 75) (-22, 3) 76) AD undef BC undef AB 0 DC 0 AC -1/5 BD 1/5 AC and BD not opp reciprocals 77) BC -3/8 AC 8/3 BC and AC are opp reciprocals 78) x = 4, y = 5, perimeter of KITE = ) a) -3 b) 21 c) isosceles 80) p g Exam Notes 110 minutes 35 multiple 2pts = 70pts 12 short 4pts = 48pts choose 2 out of 3 direct 8pts each = 16 pts one indirect 8pts = 8pts Total = 142pts