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PART II REVIEW – TRIANGLES & CONGRUENCE Are the triangles congruent? If yes, justify with a congruence theorem and write the congruence statement. Otherwise, write “no” (if they are definitely not congruent) or “not enough information” if you don’t have enough information. You can add information that is not marked if you know it is true. Yes/No? _________________ Yes/No? _________________ Yes/No? _________________ Statement: ______________ Statement: ______________ Statement: ______________ Yes/No? _________________ Yes/No? _________________ Yes/No? _________________ Statement: ______________ Statement: ______________ Statement: ______________ Yes/No? _________________ Yes/No? _________________ Yes/No? _________________ Statement: ______________ Statement: ______________ Statement: ______________ 12. Yes/No? _________________ Yes/No? _________________ Yes/No? _________________ Statement: ______________ Statement: ______________ Statement: ______________ Multiple Choice. Select the one best answer for each question. 13. In the diagram of ∆KLM, mÐL = 70, mÐM = 50 , and MK is extended through N. What is the measure of ÐLKN ? A) 60° B) 120° C) 180° D) 300° 14. Lines s and t intersect as shown. Which angles are vertical? A) Ð9 and Ð10 B) Ð7 and Ð9 C) Ð8 and Ð10 D) Ð7 and Ð10 15. The diagram at the right shows a pair of congruent triangles, with ÐADB @ ÐCDB and ÐABD @ ÐCBD. Which statement must be true? A) ÐADB @ ÐCBD B) ÐABC @ ÐADC C) AB @ CD D) AD @ CD 16. If ∆ ABC @ ∆ JKL @ ∆ RST , then AC must be congruent to A) JL B) JK C) ST D) RS 17. In the diagram below, four pairs of triangles are shown. Congruent corresponding parts are labeled in each pair. Using only the information given in the diagrams, which pair of triangles can not be proven congruent? A) B) C) D) 18. In the diagram of ∆ABC, AB is extended to point D. If mÐCAB = x + 40 , mÐACB = 3x +10 , mÐCBD = 6x , what is mÐCAB? A) 13 B) 25 C) 53 D) 65 19. In ∆ABC, mÐA = x , mÐB = 2x + 2 , and mÐC = 3x + 4 . What is the value of x? A) 29 B) 31 C) 59 D) 61 20. In the diagram of ∆AGE and ∆OLD, ÐGAE @ ÐLOD and AE @ OD. To prove ∆ ABGE @ ∆ OLD by SAS, what additional information is needed? A) GE @ LD B) AG @ OL C) ÐAGE @ ÐOLD D) ÐAEG @ ÐODL 21. If the measure of an angle is represented by x, which expression represents the measure of its complement? A) 180 – x B) 90 + x C) 90 – x D) 180 22. mÐANG = 35° . What is the measure of an angle supplementary to ÐANG ? A) 35° B) 55° C) 325° 23. What does CPCTC mean? A) B) C) D) Corresponding parts of Congruent Triangles are Corresponding Corresponding parts of Congruent Triangles are Congruent Congruent parts of Congruent Triangles are Congruent Congruent parts of Corresponding Triangles are Congruent D) 145° 24. Use appropriate symbols and letters to name these geometric figures. a. _____________ b. _______________ G c. _______________ D E F D B C 25. Sketch the following: a. acute angle ÐWMK d. _______________ m b. straight angle ÐBCD c. d. ⊙ O , with diameter AB e. A pair of complementary angles f. A pair of supplementary angles g. an exterior angle h. Vertical angles i. An obtuse angle 26. AB and CD intersect at E. If mÐAEC = 5x - 30 and mBED x 50, find, mCEB. (Create a sketch to help you) 27. Solve for a. 28. Solve for x. Triangles are drawn with certain measures indicated. Are all triangles with these measures congruent? Indicate why/why not. 29. 30. 31. 1.6" 1" 23° 1.5" 78° 2.3" Yes/No? ___________ Why/Why Not? 63° Yes/No? ___________ Why/Why Not? Yes/No? ___________ Why/Why Not? Solve for x and/or y in each of the figures below. Be sure to show the set-ups for each problem. 32. 33. x+6 7x + 10 x + 20 x = ________, y = _________ 35. After you find x, find mÐA . x = ________ 34. x = ____________ , mÐA = _______________ x = ________, y = _________ 36. After you find x, find x = ____________ , mÐA . 37. After you find x, find mÐA = _______________ mÐA . C A x+15° 2x+45° B x = ____________ , mÐA = ______________ 38. Given two side lengths of a triangle, state what numbers the third side must be between. a, 7, 10 between _________ and _________ b, 2.8, 13.5 between _________ and _________ c, 262, 145 between _________ and _________ 39. Suppose the two triangles at the right are congruent. a. Write a congruence statement for the two triangles. _________________ b. Name the angle included by the sides AC and BC . ________________ c. Name the side included between ÐA and ÐC . ___________________ 40. Based on the congruence statement and the diagram, fill in the missing measures. ∆ DEF @ ∆ DSR mÐF = _____________ RD = _____________ mÐS = _____________ ED = _____________ mÐEDF = _____________ RS = _____________ mÐRDE = _____________ mÐRDS = _____________ 41. Given: AE bisects BD , ÐB @ ÐD Prove: ∆ ABC @ ∆ DBC Statements Reasons 42. Given: C is the midpoint of AE and BD Prove: ∆ ABC @ ∆ EDC Statements 43. Given ÐL @ ÐN , Prove: ML @ ON Reasons ÐLOM @ ÐNMO Statements Reasons 44. Given: LM bisects Prove: JM @ KM ÐJLK , LJ @ LK Statements 45. Reasons Given: AC @ AD , CB @ DB Prove: ÐCAB @ ÐDAB Statements Reasons ANSWERS – PART II 1. Not enough info. We need at least one pair of sides to be congruent. 2. Yes, AAS, DIJG @ DHGJ 3. Yes, AAS; DKOL @ DMNL 4. No, the sides do not correspond 5. Yes, ASA; DWVX @ DZUY 6. Yes; SAA, DACD @ DDBA 7. Yes, SSS or SAS; DFGE @ DIGH 8. Yes, H-L (hypotenuse-leg); DJLK @ DNOM 11. Yes, SAS; DCAD @ DCEB 9. Yes, SAS; DPQS @ DRSQ 10. not enough info 12. Yes, H-L theorem; DLMK @ DONK 13. B 14. D 24a. ED 15. D 16. A 24b. GF 24c. 25a. 17. D 19. A ÐDBC or ÐCBD or ÐB b. f. 18. D c. g. 20. B 21. C 23. B 24d. DE or ED or m d. ÐKIJ 22. D h. e. i. 26. 5x - 30 = x + 50 ® 4x = 80 ® x = 20 ® mÐCEB = 110° 27. 2a + 2(78) + 2(52) = 360 ® a = 50° 28. x + 90 + 4x = 160 ® 5x + 90 = 160 ® 5x = 70 ® x = 14 29. No. We only have two angles. We need at least on side to know all triangles with these measure are @ 30. Yes. SSS 32. 33. 31. Yes. SAA x + 6 + 7x +10 + x + 20 = 180 ® 9x + 36 = 180 ® x = 16 y +105 = 180 ® y = 75° 34. x = 63° 35. y = 54° x + 60 = 105 ® x = 45° 3x - 6 + 8x + 4 = 130 ® 11x - 2 = 130 ® x = 12 36. mÐA = 3(12) - 6 = 30° 37. x + 37 + x + 67 = 90 ® 2x +114 = 90 ® x = -12 mÐA = -12 + 37 = 25° x +15 + 2x + 45 = 90 ® 3x + 60 = 90 ® x = 10 mÐA = 10 +15 = 25° 38a. 3 and 17 38b. 10.7 and 16.3 39a. ∆ ABC @ ∆ A' B'C' 39b. 40. mÐF = 36° RD = 15 mÐEDF = 92° RS = 21 ÐC mÐS = 52° mÐRDE = 88° 38c. 121 and 407 39c. AC ED = 14 mÐRDS = 92° 41. Statements 1. AE bisects BD 2. ÐB @ ÐD 3. ÐACB @ ÐECD 4. BC @ DC 5. ∆ ABC @ ∆ DBC Reasons 1. Given 2. Given 3. Vertical angles are congruent 4. A bisector cuts a segment into 2 congruent segments 5. ASA 42. Statements 1. C is the midpoint of AE and BD 2. BC @ DC , AC @ EC 3. ÐACB @ ÐECD 4. ∆ ABC @ ∆ EDC Reasons 1. Given 2. Midpoints cut segments into 2 congruent segments 3. Vertical angles are congruent 4. SAS 43. Statements 1. ÐL @ ÐN , ÐLOM @ ÐNMO 2. MO @ OM 3. ∆ LMO @ ∆ NOM 4. ML @ ON Reasons 1. Given 2. Reflexive property of congruence 3. AAS 4. CPCTC 44. Statements 1. LM bisects ÐJLK , LJ @ LK 2. ÐJLM @ ÐKLM 2. MO @ OM 4. ∆ JLM @ ∆ KLM 5. JM @ KM Reasons 1. Given 2. An angle bisector cuts and angle into 2 congruent angles. 2. Reflexive property of congruence 4. SAS 5. CPCTC 45. Statements 1. AC @ AD , CB @ DB 2. AB @ AB 3. ∆ ABC @ ∆ ABD 4. ÐCAB @ ÐDAB Reasons 1. Given 2. Reflexive property of congruence 3. SSS 4. CPCTC