Geometry  Review for Final Chapters 5 and 6


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1 Class: Date: Geometry  Review for Final Chapters 5 and 6 1. Classify PQR by its sides. Then determine whether it is a right triangle. a. scalene ; right c. scalene ; not right b. isoceles ; not right d. isoceles ; right 2. Find m KMN. a. 20 c. 101 b. 79 d. 5 1
2 3. A ramp is designed with the profile of a right triangle. The measure of one acute angle is 2 times the measure of the other acute angle. Find the measure of each acute angle. a. 45, 45 c. 22.5, 67.5 b , d. 30, Write a congruence statement for the triangles. a. PRQ TUV c. QRP TUV b. PRQ UTV d. QRP UTV 2
3 5. In the diagram, CDE GHI. Find the value of y. a. y = 51 c. y = 42 b. y = 54 d. y = Find m BDC. a. 68 c. 56 b. 84 d
4 7. Find the value of x. a. 10 c. 5 b. 4 d In the diagram, M is the midpoint of BC. Find the coordinates of M. a. b. Ê 3 2,2 ˆ Á Ê 3 4, 4 ˆ Á 3 Ê 4 c. 3, 3 ˆ Á 4 Ê d. 2, 3 ˆ Á 2 4
5 9. Which statements about the diagram are true? a. BDE is obtuse. d. ABC is obtuse. b. ABC is scalene. e. BDE is right. c. The value of x is 18. f. The value of y is What additional information is needed to prove that ABN DCO using the SAS Congruence Theorem? a. BNA COD and CO BN c. ABN DCO and CO BN b. AN DO and ABN DCO d. AN DO and BAN CDN 11. In a coordinate plane, PQR has vertices P(0, 0) and Q(3a, 0). Which coordinates of vertex R make PQR an isosceles right triangle? Assume all variables are positive. Ê ˆ Ê ˆ 3a 3 3a 2 3a 2 a. 3a, d., 2 Á 2 2 Á Ê 3a b. 2, 3a ˆ e. Ê3a, 3a Á 2 Á ˆ c. Ê Á0, 3aˆ 5
6 12. Find the measure of each acute angle. 13. In the diagram of the basketball hoop below, can you use the HypotenuseLeg Congruence Theorem to prove that WXY WZY? a. yes b. no 14. Are the triangles shown in the Brazilian flag congruent by the ASA Congruence Theorem? a. yes b. no 6
7 15. Are the triangles shown in the foursided die congruent by the AAS Congruence Theorem? 16. a. yes b. no Can the triangles be proven congruent with the information given in the diagram? If so, state the theorem you would use. 17. a. yes; AAS Congruence Theorem b. yes; ASA Congruence Theorem c. no a. yes; AAS Congruence Theorem b. yes; ASA Congruence Theorem c. no 7
8 18. a. yes; AAS Congruence Theorem b. yes; ASA Congruence Theorem c. no 19. Which reason is not necessary to explain how you can find the distance across the lake? a. ASA Congruence Theorem b. Right Angles Congruence Theorem c. SAS Congruence Theorem d. Corresponding parts of congruent triangles are congruent. e. Vertical Angles Congruence Theorem 20. Which reason is not used in a plan to prove that XW ZY? a. HL Congruence Theorem c. Base Angles Theorem b. Reflexive Property of Congruence d. Corresponding parts of congruent triangles are congruent. 8
9 21. Which reason is not used in a plan to prove that 1 2? a. SAS Congruence Theorem c. Vertical Angles Congruence Theorem b. Alternate Interior Angles Theorem d. Corresponding parts of congruent triangles are congruent. Find the coordinates of vertex C for the figure placed in a coordinate plane. 22. a rectangle with width m and length twice its width a. C(2m, m) c. C(m, 2m) b. C(m, m) d. C(2m, 2m) 23. an isosceles triangle a. C(3, 0) c. C(8, 0) b. C(6, 0) d. C(4, 0) 9
10 24. In the diagram, AB passes through the center C of the circle and DC AB. Name two triangles that are congruent. a. DCA ACD c. DCA CBD b. DAC CBD d. not enough information 25. Use the information in the diagram to determine which statements are true. a. You can use the Vertical Angles Congruence Theorem to prove that ABC DEC. b. CAB CDE because corresponding parts of congruent triangles are congruent. c. Point C is the midpoint of AD. d. You cannot make a conclusion using congruent triangles. 26. Which statements are true for ABC with vertices A(0, 0), B(m, m), and C(2m, 0)? (Assume m is positive.) a. The slope of AC is undefined. c. ABC is a right triangle. Ê 3m b. The midpoint of BC is 2, m ˆ d. ABC is isosceles. Á 2. 10
11 Match the numbered statement below with its reason to prove that the sides of the flag are parallel. Given Prove QP RS, QP Ä RS QR Ä SP a. SAS Congruence Theorem b. Converse of Alternate Interior Angles Theorem c. Corresponding parts of congruent triangles are congruent. d. Given e. Reflexive Property of Congruence f. Alternate Interior Angles Theorem QPR SRP QP RS PR RP QPR SRP QR Ä SP Write a proof. 32. Given J is the midpoint of KM,JL KM Prove JKL JML 11
12 33. Given DF DH,FG HG Prove DFG DHG 34. Given B is the midpoint of AE, AC Ä DE Prove ABC EBD 35. Given ST UV, RS Ä TU, RTS and TVU are right angles. Prove RST TUV 12
13 Decide whether there is enough information given in the picture to prove that the triangles are congruent using the SAS Congruence Theorem. Explain your reasoning. 36. ABD, CBD 37. KLN, MNL 38. In the diagram of the house, the length of AB is 15 feet. Explain why the length of BC is the same. 39. Find PF. Explain your reasoning. 13
14 40. A sandwich cut diagonally forms two right triangles, with JK LM. Prove that the two triangles are congruent. 41. Determine whether each statement is true or false. If true, explain your reasoning. If false, give a counterexample. a. If two triangles are congruent, then their perimeters are the same. b. If two triangles have the same perimeter, then the triangles are congruent. Write a coordinate proof. 42. Given Coordinates of vertices of ABC, M is the midpoint of AB, P is the midpoint of AC. Prove MP = 1 2 BC 14
15 43. Find BC. a. BC = 15 c. BC = 24 b. BC = 30 d. BC = Find m BAD. a. m BAD = 46 c. m BAD = 23 b. m BAD = 11.5 d. m BAD = 67 15
16 45. In the diagram, AD bisects BAC. Find BD. a. BD = 12 c. BD = 9 b. BD = 1 d. BD = Write an equation of the perpendicular bisector of the segment with endpoints GÊ Á 2,0 ˆ and H Ê Á 8, 6 ˆ. a. y = 5 3 x + 34 c. y = x 16 3 b. y = 5 3 x 8 d. y = 5 3 x Find the coordinates of the circumcenter of ABC with vertices AÊ Á 4,0 ˆ, B Ê Á8,3ˆ, and C Ê Á 4,3 ˆ. a. Ê Á 6, 14.5 ˆ c. Ê Á 2,1.5 ˆ b. Ê Á 0,1.5 ˆ d. Ê Á 6,1.1 ˆ 16
17 In MNQ, BP = 5x + 4, AP = 7x 11, MP = 6x 6, and QP = 8x 15. Match the point of concurrency P below with its value of x in the diagram. 48. incenter P a. x = 10 d. x = 15 2 b. x = 9 e. x = c. x = 4 f. x = circumcenter P Is there enough information given in the diagram to conclude that point P lies on the perpendicular bisector of LM? Explain your reasoning
18 The diagram shows fire hydrants located at points A and B. Line PQ coincides with Pont Road and is the perpendicular bisector of AB. Will a firefighter at the scene of the car fire have to walk farther to connect the hose to fire hydrant A or fire hydrant B? Explain your reasoning. 53. A pet owner plans to tie up the dog so it can reach the places shown in the diagram. Explain how the owner can determine where to place a stake so that it is the same distance from the dog s food, the doghouse, and the door to the owner s house. 18
19 54. You are drilling a hole in the triangular birdhouse shown. You want the hole to be the same distance from all three sides. Where should you drill the hole? Explain your reasoning. 55. The diagram shows a piece of wood for one side of a shelf. Point P is the center of a hole for a dowel rod that is to be drilled 5 centimeters from sides AB and BC. You draw segment BP. Can you conclude that the segment bisects ABC? Explain. 19
20 Geometry  Review for Final Chapters 5 and 6 Answer Section 1. C 2. C 3. D 4. D 5. C 6. D 7. B 8. A 9. B, C, D, E 10. C, D 11. B, C, E , B 14. B 15. A 16. A 17. C 18. B 19. C 20. C 21. B 22. A 23. B 24. A 25. D 26. B, C, D 27. F 28. D 29. E 30. A 31. B 32. STATEMENTS REASONS 1. J is the midpoint of KM. 1. Given 2. KJ MJ 2. Definition of midpoint 3. JL KM 3. Given 4. LJM and LJK are right angles. 4. Definition of perpendicular lines 5. LJM LJK 5. Right Angles Congruence Theorem 6. LJ LJ 6. Reflexive Property of Congruence 7. JKL JML 7. SAS Congruence Theorem 1
21 STATEMENTS REASONS 1. DF DH 1. Given 2. FG HG 2. Given 3. DG DG 3. Reflexive Property of Congruence 4. DFG DHG 4. SSS Congruence Theorem STATEMENTS REASONS 1. B is the midpoint of AE. 1. Given 2. AC Ä DE 2. Given 3. AB EB 3. Definition of midpoint 4. ABC DBE 4. Vertical Angles Congruence Theorem 5. CAB DEB 5. Alternate Interior Angles Theorem 6. ABC EBD 6. ASA Congruence Theorem STATEMENTS REASONS 1. ST UV, RS Ä TU, RTS and TVU 1. Given are right angles 2. RTS TVU 2. Right Angles Congruence Theorem 3. SRT UTV 3. Corresponding Angles Theorem 5. RST TUV 5. AAS Congruence Theorem 36. no; The congruent angles are not the included angles. 37. yes; Two pairs of sides and the included angles are congruent. 38. By the Linear Pair Postulate and the definition of supplementary angles, m BCA = = 40. Because m A = 40 = m BCA, by definition of congruent angles, A BCA. So, by the Converse of the Base Angles Theorem, AB BC. So, AB = BC = 15 feet ft; Using the Exterior Angle Theorem, m PFY = m ZPF m PYF = = 30. Because m PYF = 30 = m PFY, by the definition of congruent angles, PYF PFY. So, by the Converse of the Base Angles Theorem, PF YP. So, PF = YP = 12 feet. 40. You are given that JK LM and that the triangles are right triangles. So, one pair of corresponding legs is congruent. By the Reflexive Property of Congruence, JL JL. So, by the HL Congruence Theorem, JLM LJK. 41. a. true; Because the triangles are congruent, the corresponding side lengths are congruent. The side lengths have the same measures by the definition of congruence. So, the triangles will have the same perimeter. b. false; Sample answer: ABC with AB = 3 meters, BC = 4 meters, and AC = 6 meters has a perimeter of 13 meters, and DEF with DE = 4 meters, EF = 5 meters, and DF = 4 meters has a perimeter of 13 meters. ABC is not congruent to DEF. Ê 42. Using the Midpoint Formula, M 0, 3n ˆ Á 2 and P Ê Á 2n,0 ˆ. Using the Distance Formula, MP = 5n and BC = 5n B So, MP = 1 2 BC. 2
22 44. C 45. C 46. B 47. A 48. D 49. B 50. no; You would need to know that either LN = MN or LP = MP. 51. yes; Because point P is equidistant from L and M, point P is on the perpendicular bisector of LM by the Converse of the Perpendicular Bisector Theorem. Also, LN MN, so PN is a bisector of LM. Because P can only be on one of the bisectors, PN is the perpendicular bisector of LM. 52. The firefighter will walk the same distance to connect the hose to either fire hydrant; Point P is on the perpendicular bisector of AB, so AP = BP by the Perpendicular Bisector Theorem. 53. The pet owner can copy the positions of the three places, connect the points to draw a triangle, and draw three perpendicular bisectors of the triangle. The point where the perpendicular bisectors meet, the circumcenter, should be the location of the stake. 54. You should drill the hole at the incenter of the birdhouse because the incenter is equidistant from the sides of the triangle by the Incenter Theorem. 55. yes; Point P is in the interior of ABC and it is equidistant from sides AB and BC. So, P lies on the bisector of ABC. 3
23 Geometry  Review for Final Chapters 5 and 6 [Answer Strip] D 3. C 5. C 1. D 4. D 6. C 2.
24 Geometry  Review for Final Chapters 5 and 6 [Answer Strip] B 7. B, C, D, E B A C, D 14. B 11. B, C, E
25 Geometry  Review for Final Chapters 5 and 6 [Answer Strip] 15. A 18. B 21. B 16. A 19. C 22. A 23. B 17. C 20. C
26 Geometry  Review for Final Chapters 5 and 6 [Answer Strip] 24. A 43. B 25. D 27. F 28. D 29. E 30. A 31. B 44. C 26. B, C, D
27 Geometry  Review for Final Chapters 5 and 6 [Answer Strip] 45. C 46. B 48. D 47. A 49. B
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