Name: ________________________ Date:

ID: A Period:

1

Geometry Ch 4 Worksheet

Multiple ChoiceIdentify the choice that best completes the statement or answers the question.

____ 1. If BCDE is congruent to OPQR, then DE is congruent to ? .

a. PQ b. QR c. OP d. OR

____ 2. In the paper airplane, ABCD ≅ EFGH, m∠B = m∠BCD = 90, and m∠BAD = 155. Find m∠GHE.

a. 155 b. 90 c. 35 d. 25

____ 3. Use the information given in the diagram. Tell why PS ≅ QR and ∠PQR ≅ ∠RSP.

a. Reflexive Property, Transitive Propertyb. Transitive Property, Reflexive Propertyc. Given, Reflexive Propertyd. Given, Given

____ 4. If ∆KLM ≅ ∆STU , which of the following can you NOT conclude as being true?

a. ∠L ≅ ∠T b. ∠K ≅ ∠S c. KL ≅ SU d. LM ≅ TU

Name: ________________________ ID: A

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____ 5. ∠BAC ≅ ?

a. ∠NPM b. ∠NMP c. ∠MNP d. ∠PNM

____ 6. Which congruence statement does NOT necessarily describe the triangles shown if ∆DEF ≅ ∆FGH?

a. ∆FDE ≅ ∆HFG c. ∆FED ≅ ∆HGFb. ∆EDF ≅ ∆GFH d. ∆EFD ≅ ∆HGF

____ 7. The two triangles are congruent as suggested by their appearance. Find the value of g. The diagrams are not to scale.

a. 17 b. 15 c. 90 d. 8

____ 8. Given ∆QRS ≅ ∆TUV, QS = 3v + 4, and TV = 6v − 8, find the length of QS and TV.a. 32 b. 4 c. 16 d. 17

Name: ________________________ ID: A

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____ 9. Given ∆ABC ≅ ∆PQR, m∠B = 5v + 2, and m∠Q = 6v − 8, find m∠B and m∠Q.a. 52 b. 68 c. 53 d. 71

____ 10. Justify the last two steps of the proof.

Given: RS ≅ UT and RT ≅ USProve: ∆RST ≅ ∆UTS

Proof:

1. RS ≅ UT 1. Given

2. RT ≅ US 2. Given

3. ST ≅ TS 3. ?

4. ∆RST ≅ ∆UTS 4. ?

a. Reflexive Property of ≅; SSS c. Symmetric Property of ≅; SSSb. Symmetric Property of ≅; SAS d. Reflexive Property of ≅; SAS

____ 11. What other information do you need in order to prove the triangles congruent using the SAS Congruence Postulate?

a. AC ≅ BD c. ∠BAC ≅ ∠DAC

b. ∠CBA ≅ ∠CDA d. AC ⊥ BD

Name: ________________________ ID: A

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____ 12. State whether ∆ABC and ∆AED are congruent. Justify your answer.

a. yes, by SAS onlyb. yes, by SSS onlyc. yes, by either SSS or SASd. No; there is not enough information to conclude that the triangles are congruent.

____ 13. Name the angle included by the sides NM and MP.

a. ∠M b. ∠N c. ∠P d. none of these

____ 14. In each pair of triangles, parts are congruent as marked. Which pair of triangles is congruent by ASA?a. c.

b. d.

Name: ________________________ ID: A

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____ 15. What is the missing reason in the two-column proof?

Given: AC→

bisects ∠DAB and CA→

bisects ∠DCBProve: ∆DAC ≅ ∆ABC

Statements Reasons

1. AC→

bisects ∠DAB 1. Given2. ∠DAC ≅ ∠BAC 2. Definition of angle bisector3. AC ≅ AC 3. Reflexive property

4. CA→

bisects ∠DCB 4. Given5. ∠DAC ≅ ∠BCA 5. Definition of angle bisector6. ∆DAC ≅ ∆BAC 6. ?

a. SAS Postulate c. AAS Theoremb. SSS Postulate d. ASA Postulate

____ 16. From the information in the diagram, can you prove ∆FDG ≅ ∆FDB? Explain.

a. yes, by AAA c. yes, by SASb. yes, by ASA d. no

Name: ________________________ ID: A

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____ 17. Can you use the ASA Postulate, the AAS Theorem, or both to prove the triangles congruent?

a. AAS only c. ASA onlyb. either ASA or AAS d. neither

____ 18. What else must you know to prove the triangles congruent by ASA? By SAS?

a. ∠ADC ≅ ∠CAB; AD ≅ BC c. ∠ACD ≅ ∠CAB; AD ≅ AC

b. ∠ACD ≅ ∠CAB; AD ≅ BC d. ∠ACD ≅ ∠CAB; AB ≅ CD

____ 19. Based on the given information, what can you conclude, and why?

Given: ∠H ≅ ∠L, HJ ≅ JL

a. ∆HIJ ≅ ∆JLK by SAS c. ∆HIJ ≅ ∆LKJ by SASb. ∆HIJ ≅ ∆JLK by ASA d. ∆HIJ ≅ ∆LKJ by ASA

Name: ________________________ ID: A

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____ 20. Name the theorem or postulate that lets you immediately conclude ∆ABD ≅ ∆CBD.

a. AAS b. SAS c. ASA d. none of these

____ 21. If ∠A ≅ ∠D and ∠C ≅ ∠F, which additional statement does NOT allow you to conclude that ∆ABC ≅ ∆DEF?

a. BC ≅ EF c. AB ≅ EF

b. AC ≅ DF d. ∠B ≅ ∠E

Name: ________________________ ID: A

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____ 22. Supply the missing reasons to complete the proof.

Given: ∠N ≅ ∠Q and NO ≅ QO

Prove: MO ≅ PO

Statement Reasons

1. ∠N ≅ ∠Q and

NO ≅ QO1. Given

2. ∠MON ≅ ∠POQ 2. Vertical angles are congruent.

3. ∆MON ≅ ∆POQ 3. ?

4. MO ≅ PO 4. ?

a. ASA; CPCTC c. AAS; CPCTCb. ASA; Substitution d. SAS; CPCTC

____ 23. R, S, and T are the vertices of one triangle. E, F, and D are the vertices of another triangle. m∠R = 70, m∠S = 80, m∠F = 70, m∠D = 30, RS = 4, and EF = 4. Are the two triangles congruent? If yes, explain and

tell which segment is congruent to RT.

a. yes, by AAS; ED

b. yes, by ASA; FD

c. yes, by SAS; EDd. No, the two triangles are not congruent.

Name: ________________________ ID: A

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____ 24. Supply the reasons missing from the proof shown below.

Given: AB ≅ AC, ∠BAD ≅ ∠CAD

Prove: AD bisectsBC

Statements Reasons

1. AB ≅ AC 1. Given

2. ∠BAD ≅ ∠CAD 2. Given

3. AD ≅ AD 3. Reflexive Property

4. ∆BAD ≅ ∆CAD 4. ?

5. BD ≅ CD 5. ?

6. AD bisectsBC 6. Def. of segment bisector

a. SAS; Reflexive Property c. SSS; Reflexive Propertyb. SAS; CPCTC d. ASA; CPCTC

____ 25. Find the values of x and y.

a. x = 70, y = 20 c. x = 90, y = 20b. x = 20, y = 70 d. x = 90, y = 70

Name: ________________________ ID: A

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____ 26. The octagon in the figure is equiangular and AB ≅ AC. Find m∠ACB.

a. 135 b. 45 c. 30 d. 90

____ 27. In an A-frame house, the two congruent sides extend from the ground to form a 44° angle at the peak. What angle does each side form with the ground?a. 146 b. 68 c. 73 d. 136

____ 28. What is the measure of a base angle of an isosceles triangle if the vertex angle measures 44° and the two congruent sides each measure 21 units?

a. 73° b. 146° c. 136° d. 68°

____ 29. What is the measure of the vertex angle of an isosceles triangle if one of its base angles measures 52°?a. 64° b. 128° c. 104° d. 76°

____ 30. Use the information in the figure. Find m∠D.

a. 78° b. 51° c. 129° d. 39°

Name: ________________________ ID: A

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____ 31. Find the value of x. The diagram is not to scale.

a. x = 40 b. x = 23 c. x = 13 d. none of these

____ 32. Find the value of x. The diagram is not to scale.

Given: RS ≅ ST, m∠RST = 5x − 54, m∠STU = 8x

a. 20 b. 18 c. 144 d. 23

____ 33. Two sides of an equilateral triangle have lengths 3x − 1 and −2x + 24. Which of 19 − x or 4x + 2 could be the length of the third side?a. both 19 – x and 4x + 2 c. neither 19 – x nor 4x + 2b. 4x + 2 only d. 19 – x only

____ 34. For which situation could you prove ∆1 ≅ ∆2 using the HL Theorem?

a. II only b. I and II c. II and III d. I only

Name: ________________________ ID: A

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____ 35. Is there enough information to conclude that the two triangles are congruent? If so, what is a correct congruence statement?

a. Yes; ∆ABC ≅ ∆ACD.b. Yes; ∆ACB ≅ ∆ADC.c. Yes; ∆CAB ≅ ∆DAC.d. No, the triangles cannot be proven congruent.

____ 36. What additional information will allow you to prove the triangles congruent by the HL Theorem?

a. ∠A ≅ ∠E c. AC ≅ DC

b. AC ≅ BD d. m∠BCE = 90

____ 37. RQ is perpendicular to PS at Q between P and S. ∠SPR ≅ ∠PSR. By which of the five congruence statements, HL, AAS, ASA, SAS, and SSS, can you conclude that ∆PQR ≅ ∆SQR?a. HL, AAS, ASA, SAS, and SSS c. HL, AAS, ASA, and SASb. HL, AAS, and ASA d. HL and ASA

Name: ________________________ ID: A

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____ 38. BE is the bisector of ∠ABC and CD→

is the bisector of ∠ACB. Also, ∠XBA ≅ ∠YCA. Which of AAS, SSS,

SAS, or ASA would you use to help you prove BL ≅ CM?

a. AAS b. SAS c. ASA d. SSS

____ 39. Which overlapping triangles are congruent by AAS?

a. ∆ABE ≅ ∆CDA c. ∆ADC ≅ ∆EDAb. ∆ABE ≅ ∆DEA d. ∆ADC ≈ ∆EBC

____ 40. The sides of an isosceles triangle have lengths 2x + 4, −x + 16. The base has length 5x + 1. What is the length of the base?a. 4 c. 21b. 12 d. cannot be determined

Short Answer

41. ∆PQR ≅ ∆TSR. List the six pairs of congruent corresponding parts.

Name: ________________________ ID: A

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42. For the two quadrilaterals below, ∠B ≅ ∠G, ∠BCD ≅ ∠GCD, ∠CDE ≅ ∠CDF, and ∠E ≅ ∠F. Complete this congruence statement for the two quadrilaterals.

EDCB ≅ ___?___

43. Write the missing reasons to complete the proof.

Given: AB ≅ CD, ∠A ≅ ∠D, and AF ≅ DE Prove: ∆FAC ≅ ∆EDB

Statement Reason

1. AF ≅ DE 1. Given2. ∠A ≅ ∠D 2. Given

3. AB ≅ CD 3. Given4. AB = CD 4. Definition of congruent segments5. AB + BC = CD + BC 5. ? 6. AC = BD 6. Segment Addition Postulate

7. AC = BD 7. Definition of congruent segments8. ∆FAC ≅ ∆EDB 8. ?

44. Based on the given information, can you conclude that ∆QRS ≅ ∆TUV? Explain.

Given: QR ≅ TU , QS ≅ TV , and ∠R ≅ ∠U

45. Sketch ∆PQR and ∆STU so that PQ ≅ ST, PR ≅ SU , and ∠R ≅ ∠U , but ∆PQR is NOT congruent to ∆STU .

Name: ________________________ ID: A

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46. Explain how you can use SSS, SAS, ASA, or AAS with CPCTC to prove that ∠D ≅ ∠B.

47. Explain how you can use SSS, SAS, ASA, or AAS with CPCTC to complete a proof.

Given: ON ≅ OP, ∠NOM ≅ ∠POM

Prove: NM ≅ PM

Name: ________________________ ID: A

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48. Fill in the missing reasons to complete the proof.Given: ∠VUY ≅ ∠UWT ≅ ∠X

Prove: UW ≅ UT

Statement Reason1. ∠VUY ≅ ∠X 1. Given

2. UY Ä TX 2. Converse of the Corresponding Angles Postulate3. ∠T ≅ ∠VUY 3. ? 4. ∠VUY ≅ ∠UWT 4. Given5. ∠T ≅ ∠UWT 5. Transitive Property

6. UT ≅ UW 6. ?

49. Complete the statement BF ≅ ? . Explain why it is true.

Name: ________________________ ID: A

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50. Is ∆MNP ≅ ∆ONP by HL? If so, name the legs that allow the use of HL.

51. Write the missing reasons to complete the flow proof.

Given: ∠ADB and∠CDB are right angles, ∠A ≅ ∠CProve: ∆ADB = ∆CDB

Name: ________________________ ID: A

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52. Separate and redraw ∆ABC and ∆CDA. Identify any common angles or sides.

53. Determine which triangles in the figure are congruent by AAS.

54. Name a pair of triangles in the figure and state whether they are congruent by SSS, SAS, ASA, AAS, or HL.

Given: NP ≅ OM , MN ≅ PO

Name: ________________________ ID: A

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55. Write a two-column proof to show that ∆CEF ≅ ∆AEF.Given: BC ≅ DA, ∠1 ≅ ∠2, and CF ≅ AF

56. Is there enough information to prove the two triangles congruent? If yes, write the congruence statement and name the postulate you would use. If no, write not possible and tell what other information you would need.

57. In the figure, ∠1 ≅ ∠2, ∠3 ≅ ∠4, and TK ≅ TL. Prove that ∆RTK ≅ ∆STL.

58. For ∆RST and ∆UVW , ∠R ≈ ∠U , ST ≅ VW , and ∠S ≅ ∠V . Explain how you can prove ∆RST ≅ ∆UVW by ASA.

Name: ________________________ ID: A

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59. Can you conclude the triangles are congruent? Justify your answer.

60. Complete the proof by providing the missing reasons.

Given: SR ⊥ RT, TU ⊥ US, SR ≅ TUProve: ∆TRS ≅ ∆SUT

Statement Reason

1. SR ≅ TU , SR ⊥ RT, and TU ⊥ US, 1. Given

2. ∠SRT and∠TUS are right angles 2. Definition of perpendicular segments3. ∠SRT ≅ ∠TUS 3. ?

4. ST ≅ ST 4. ? 5. ∆TRS ≅ ∆SUT 5. ?

Name: ________________________ ID: A

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Essay

61. Write a proof.

Given: NO ≅ PM , ∠1 ≅ ∠2, and OR ≅ MRProve: ∆ORQ ≅ ∆MRQ

62. Write a paragraph proof to show that ∆ABC ≅ ∆DEC.

Given: AC ≅ DC and BC ≅ CE

63. Write a two-column proof.

Given: BC ≅ EC and AC ≅ DC

Prove: BA ≅ ED

Name: ________________________ ID: A

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64. Write a two-column proof:Given: ∠BAC ≅ ∠DAC, ∠DCA ≅ ∠BCA

Prove: BC ≅ CD

Other

65. Guy wires of equal length anchor a vertical post to the flat ground. The guy wires are attached to the post at the same height. Explain why the guy wires reach the ground at the same distance from the base of the post.

66. When you open a stepladder, you use a brace on each side of the ladder to lock the legs in place. Explain why the triangles formed on each side by the legs and the ground (∆ABC and∆DEF in the diagram) are congruent.

Name: ________________________ ID: A

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67. Is there enough information to prove the two triangles congruent by AAS? If yes, write the congruence statement and explain. If no, write not possible and tell what other information you would need.

Given: ∠B ≅ ∠D

68. It appears from the name of the HL Theorem that you actually need to know that only two parts of two triangles are congruent in order to prove two triangles congruent. Is this the case?

69. Given that ∠EAC ≅ ∠ECA, what else do you need to prove that BA ≅ DC? Outline a proof that uses the needed information.

ID: A

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Geometry Ch 4 WorksheetAnswer Section

MULTIPLE CHOICE

1. ANS: B PTS: 1 REF: 4-1 Congruent FiguresTOP: 4-1 Example 1 KEY: congruent figures | corresponding parts | word problem

2. ANS: D PTS: 1 REF: 4-1 Congruent FiguresTOP: 4-1 Example 2 KEY: congruent figures | corresponding parts

3. ANS: D PTS: 1 REF: 4-1 Congruent FiguresTOP: 4-1 Example 4 KEY: congruent figures | corresponding parts | proof

4. ANS: C PTS: 1 REF: 4-1 Congruent FiguresTOP: 4-1 Example 1 KEY: congruent figures | corresponding parts | word problem

5. ANS: B PTS: 1 REF: 4-1 Congruent FiguresTOP: 4-1 Example 1 KEY: congruent figures | corresponding parts

6. ANS: D PTS: 1 REF: 4-1 Congruent FiguresTOP: 4-1 Example 1 KEY: congruent figures | corresponding parts

7. ANS: A PTS: 1 REF: 4-1 Congruent FiguresTOP: 4-1 Example 1 KEY: congruent figures | corresponding parts

8. ANS: C PTS: 1 REF: 4-1 Congruent FiguresKEY: congruent figures | corresponding parts

9. ANS: A PTS: 1 REF: 4-1 Congruent FiguresKEY: congruent figures | corresponding parts

10. ANS: A PTS: 1 REF: 4-2 Triangle Congruence by SSS and SASTOP: 4-2 Example 1 KEY: SSS | reflexive property | proof

11. ANS: D PTS: 1 REF: 4-2 Triangle Congruence by SSS and SASTOP: 4-2 Example 2 KEY: SAS | reasoning

12. ANS: C PTS: 1 REF: 4-2 Triangle Congruence by SSS and SASTOP: 4-2 Example 3 KEY: SSS | SAS | reasoning

13. ANS: A PTS: 1 REF: 4-2 Triangle Congruence by SSS and SASTOP: 4-2 Example 2 KEY: angle

14. ANS: C PTS: 1 REF: 4-3 Triangle Congruence by ASA and AASTOP: 4-3 Example 1 KEY: ASA

15. ANS: D PTS: 1 REF: 4-3 Triangle Congruence by ASA and AASTOP: 4-3 Example 4 KEY: ASA | proof

16. ANS: B PTS: 1 REF: 4-3 Triangle Congruence by ASA and AASTOP: 4-3 Example 3 KEY: ASA | reasoning

17. ANS: B PTS: 1 REF: 4-3 Triangle Congruence by ASA and AASTOP: 4-3 Example 3 KEY: ASA | AAS | reasoning

18. ANS: B PTS: 1 REF: 4-3 Triangle Congruence by ASA and AASTOP: 4-3 Example 3 KEY: ASA | SAS | reasoning

19. ANS: D PTS: 1 REF: 4-3 Triangle Congruence by ASA and AASTOP: 4-3 Example 4 KEY: ASA | reasoning

20. ANS: B PTS: 1 REF: 4-3 Triangle Congruence by ASA and AASTOP: 4-3 Example 3 KEY: ASA | AAS | SAS

ID: A

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21. ANS: D PTS: 1 REF: 4-3 Triangle Congruence by ASA and AASKEY: ASA | AAS

22. ANS: A PTS: 1 REF: 4-4 Using Congruent Triangles: CPCTCTOP: 4-4 Example 1 KEY: ASA | CPCTC | proof

23. ANS: B PTS: 1 REF: 4-4 Using Congruent Triangles: CPCTCTOP: 4-4 Example 1 KEY: ASA | CPCTC | word problem

24. ANS: B PTS: 1 REF: 4-5 Isosceles and Equilateral TrianglesTOP: 4-5 Example 1 KEY: segment bisector | isosceles triangle | proof

25. ANS: C PTS: 1 REF: 4-5 Isosceles and Equilateral TrianglesTOP: 4-5 Example 2 KEY: angle bisector | isosceles triangle

26. ANS: B PTS: 1 REF: 4-5 Isosceles and Equilateral TrianglesTOP: 4-5 Example 3 KEY: isosceles triangle | Isosceles Triangle Theorem | Polygon Angle-Sum Theorem

27. ANS: B PTS: 1 REF: 4-5 Isosceles and Equilateral TrianglesTOP: 4-5 Example 2 KEY: Isosceles Triangle Theorem | isosceles triangle | Triangle Angle-Sum Theorem | word problem | problem solving

28. ANS: D PTS: 1 REF: 4-5 Isosceles and Equilateral TrianglesTOP: 4-5 Example 2 KEY: isosceles triangle | Converse of Isosceles Triangle Theorem | Triangle Angle-Sum Theorem

29. ANS: D PTS: 1 REF: 4-5 Isosceles and Equilateral TrianglesTOP: 4-5 Example 2 KEY: isosceles triangle | Isosceles Triangle Theorem | Triangle Angle-Sum Theorem | word problem

30. ANS: D PTS: 1 REF: 4-5 Isosceles and Equilateral TrianglesTOP: 4-5 Example 2 KEY: Isosceles Triangle Theorem | Triangle Angle-Sum Theorem | isosceles triangle

31. ANS: B PTS: 1 REF: 4-5 Isosceles and Equilateral TrianglesTOP: 4-5 Example 2 KEY: Isosceles Triangle Theorem | isosceles triangle

32. ANS: B PTS: 1 REF: 4-5 Isosceles and Equilateral TrianglesTOP: 4-5 Example 2 KEY: Isosceles Triangle Theorem | isosceles triangle | problem solving | Triangle Angle-Sum Theorem

33. ANS: D PTS: 1 REF: 4-5 Isosceles and Equilateral TrianglesKEY: equilateral triangle | word problem | problem solving

34. ANS: C PTS: 1 REF: 4-6 Congruence in Right TrianglesTOP: 4-6 Example 1 KEY: HL Theorem | right triangle | reasoning

35. ANS: B PTS: 1 REF: 4-6 Congruence in Right TrianglesTOP: 4-6 Example 1 KEY: HL Theorem | right triangle | reasoning

36. ANS: C PTS: 1 REF: 4-6 Congruence in Right TrianglesTOP: 4-6 Example 1 KEY: HL Theorem | right triangle | reasoning

37. ANS: C PTS: 1 REF: 4-6 Congruence in Right TrianglesTOP: 4-6 Example 1 KEY: right triangle | HL Theorem | ASA | SAS | AAS | SSS | proof | word problem | problem solving | reasoning

38. ANS: C PTS: 1 REF: 4-7 Using Corresponding Parts of Congruent TrianglesTOP: 4-7 Example 2 KEY: corresponding parts | congruent figures | ASA | SAS | AAS | SSS | reasoning

ID: A

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39. ANS: D PTS: 1 REF: 4-7 Using Corresponding Parts of Congruent TrianglesTOP: 4-7 Example 2 KEY: congruent figures | corresponding parts | overlapping triangles | proof

40. ANS: C PTS: 1 REF: 4-5 Isosceles and Equilateral TrianglesKEY: isosceles triangle | Isosceles Triangle Theorem | word problem | problem solving

SHORT ANSWER

41. ANS:

Sides: PQ ≅ TS , QR ≅ SR, RP ≅ RTAngles: ∠P ≅ ∠T, ∠Q ≅ ∠S , ∠PRQ ≅ ∠TRS

PTS: 1 REF: 4-1 Congruent Figures TOP: 4-1 Example 1KEY: congruent figures | corresponding parts

42. ANS: FDCG

PTS: 1 REF: 4-1 Congruent Figures TOP: 4-1 Example 1KEY: congruent figures | corresponding parts

43. ANS: Step 4: Addition property of equalityStep 7: SAS

PTS: 1 REF: 4-2 Triangle Congruence by SSS and SAS KEY: SAS | proof

44. ANS: Answers may vary. Sample: Two pairs of sides are congruent, but the angle is not included. There is no SSA Congruence Theorem, so you cannot conclude ∆QRS ≅ ∆TUV with the information given.

PTS: 1 REF: 4-2 Triangle Congruence by SSS and SAS KEY: reasoning | SAS

45. ANS: Sketches may vary. Sample:

PTS: 1 REF: 4-2 Triangle Congruence by SSS and SAS KEY: congruent figures | word problem | problem solving

ID: A

4

46. ANS:

Answers may vary. Sample: Because the two triangles share the side AC, they are congruent by SAS. Then ∠D ≅ ∠B by CPCTC.

PTS: 1 REF: 4-4 Using Congruent Triangles: CPCTC TOP: 4-4 Example 2 KEY: CPCTC | SAS | writing in math | reasoning

47. ANS:

Answers may vary. Sample: Since the two triangles share the side RP, they are congruent by SAS. Then

QP ≅ SP by CPCTC.

PTS: 1 REF: 4-4 Using Congruent Triangles: CPCTC TOP: 4-4 Example 2 KEY: SAS | CPCTC | writing in math | reasoning

48. ANS: Step 3: Corresponding Angles PostulateStep 6: Converse of Isosceles Triangle Theorem

PTS: 1 REF: 4-5 Isosceles and Equilateral Triangles TOP: 4-5 Example 1 KEY: Converse of Isosceles Triangle Theorem | Corresponding Angles Postulate | Converse of the Corresponding Angles Postulate | isosceles triangle | proof

49. ANS:

Answers may vary. Sample: EF; ∆BDF ≅ ∆EDF by SAS, so BF ≅ EF by CPCTC.

PTS: 1 REF: 4-4 Using Congruent Triangles: CPCTC KEY: CPCTC | SAS

50. ANS:

No

PTS: 1 REF: 4-6 Congruence in Right Triangles TOP: 4-6 Example 1KEY: HL Theorem | right triangle

51. ANS: a. Definition of right trianglesb. Converse of Isosceles Triangle Theoremc. Reflexive propertyd. Hypotenuse-Leg Theorem

PTS: 1 REF: 4-6 Congruence in Right Triangles TOP: 4-6 Example 2KEY: flow proof | HL Theorem | Converse of Isosceles Triangle Theorem | right triangle | proof

ID: A

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52. ANS:

AC is the only common side.

PTS: 1 REF: 4-7 Using Corresponding Parts of Congruent TrianglesTOP: 4-7 Example 1 KEY: overlapping triangles | common parts

53. ANS: ∆MNQ ≅ ∆ONQ

PTS: 1 REF: 4-7 Using Corresponding Parts of Congruent TrianglesKEY: overlapping triangles | congruent figures | AAS

54. ANS: ∆MNP ≅ ∆POM by SSS

PTS: 1 REF: 4-7 Using Corresponding Parts of Congruent TrianglesTOP: 4-7 Example 2 KEY: overlapping triangles | congruent figures | SSS

55. ANS:

Statement Reason

1. BC ≅ DA 1. Given

2. ∠1 ≅ ∠2 2. Given

3. ∠CEB ≅ ∠AED 3. Vertical angles are congruent.

4. ∆CEB ≅ ∆AED 4. AAS

5. CE ≅ AE 5. CPCTC

6. CF ≅ AF 6. Given

7. EF ≅ EF

8. ∆CEF ≅ ∆AEF

7. Reflexive Property

8. SSS

PTS: 1 REF: 4-3 Triangle Congruence by ASA and AAS KEY: ASA | CPCTC | congruent figures | proof

ID: A

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56. ANS: Yes; ∆WXD ≅ ∆YXD by SAS.

PTS: 1 REF: 4-2 Triangle Congruence by SSS and SAS KEY: SAS | proof | reasoning

57. ANS: Answers may vary. Sample: ∆RTL ≅ ∆STK by SAS, so ∠R ≅ ∠S . Supplements of congruent angles are congruent, so ∠RKT ≅ ∠SLT. ∆RTK ≅ ∆STL by AAS.

PTS: 1 REF: 4-7 Using Corresponding Parts of Congruent TrianglesKEY: overlapping triangles | proof | AAS

58. ANS: Answers may vary. Sample: Two angles of one triangle are congruent to two angles of the other triangle, so their third angles are also congruent. The congruent sides are included between two pairs of congruent angles, so the triangles are congruent by ASA.

PTS: 1 REF: 4-3 Triangle Congruence by ASA and AAS TOP: 4-3 Example 4 KEY: ASA | proof | reasoning

59. ANS: Yes, the vertical angles are congruent so the triangles are congruent by ASA.

PTS: 1 REF: 4-1 Congruent Figures TOP: 4-1 Example 3KEY: SAS | ASA | congruent figures | reasoning

60. ANS: Step 3: All right angles are congruent.Step 4: Reflexive PropertyStep 5. HL Theorem

PTS: 1 REF: 4-7 Using Corresponding Parts of Congruent TrianglesTOP: 4-7 Example 2 KEY: HL Theorem | overlapping triangles | proof

ID: A

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ESSAY

61. ANS: [4] Statement Reason

1. NO ≅ PM 1. Given

2. ∠1 ≅ ∠2 2. Given

3. ∠NQO ≅ ∠PQM 3. Vertical angles are congruent.

4. ∆NQO ≅ ∆PQM 4. AAS

5. OQ ≅ MQ 5. CPCTC

6. OR ≅ MR

7. QR ≅ QR

8. ∆ORQ ≅ ∆MRQ

6. Given

7. Reflexive Property

8. SSS

[3] correct idea, some details inaccurate[2] correct idea, not well organized[1] correct idea, one or more significant steps omitted

PTS: 1 REF: 4-7 Using Corresponding Parts of Congruent TrianglesTOP: 4-7 Example 3 KEY: AAS | CPCTC | corresponding parts | congruent figures | proof | rubric-based question | extended response

62. ANS: [4] Answers may vary. Sample: You are given that AC ≅ DC and BC ≅ CE. Vertical angles

BCA and ECD are congruent, so ∆ABC ≅ ∆DEC by SAS.

[3] correct idea, some details inaccurate[2] correct idea, not well organized[1] correct idea, one or more significant steps omitted

PTS: 1 REF: 4-2 Triangle Congruence by SSS and SAS KEY: congruent figures | SAS | proof | AAS | rubric-based question | extended response | proof

ID: A

8

63. ANS: [4]

Statement Reason

1. BC ≅ EC and AC ≅ DC 1. Given

2. ∠BCA ≅ ∠ECD 2. Vertical angles are congruent.3. ∆BCA ≅ ∆ECD 3. SAS

4. BA ≅ ED 4. CPCTC

[3] correct idea, some details inaccurate[2] correct idea, not well organized[1] correct idea, one or more significant steps omitted

PTS: 1 REF: 4-4 Using Congruent Triangles: CPCTC KEY: CPCTC | congruent figures | proof | SAS | rubric-based question | extended response

64. ANS: [4]

Statement Reason1. ∠BAC ≅ ∠DAC and ∠DCA ≅ ∠BCA

1. Given

2. CA ≅ CA 2. Reflexive Property3. ∆CBA ≅ ∆CDA 3. ASA

4. BC ≅ CD 4. CPCTC

[3] correct idea, some details inaccurate[2] correct idea, not well organized[1] correct idea, one or more significant steps omitted

PTS: 1 REF: 4-4 Using Congruent Triangles: CPCTC KEY: ASA | CPCTC | congruent figures | corresponding parts | rubric-based question | extended response | proof

OTHER

65. ANS: Answers may vary. Sample: Because the post is vertical, both triangles are right triangles. In each triangle, the guy wire is the hypotenuse, and the vertical post is a leg. The triangles are congruent by the HL theorem. The wires reach the ground at the same distance from the base by CPCTC.

PTS: 1 REF: 4-6 Congruence in Right Triangles TOP: 4-6 Example 1KEY: CPCTC | HL Theorem | right triangle | word problem | problem solving | writing in math

66. ANS: Answers may vary. Sample: Each leg on one side of the ladder is the same length as the corresponding leg on the other side. The locking braces hold the legs apart by the same angle measure. The triangles are congruent by SAS.

PTS: 1 REF: 4-2 Triangle Congruence by SSS and SAS KEY: SAS | word problem | problem solving | writing in math

ID: A

9

67. ANS: Not possible; you have one pair of congruent angles (∠B ≅ ∠D) and one pair of congruent sides

(AC = AC), but you would need to know that one more pair of angles are congruent, either ∠BCA ≅ ∠CAD or ∠BAC ≅ ∠DCA, to prove the triangles congruent by AAS.

PTS: 1 REF: 4-3 Triangle Congruence by ASA and AAS TOP: 4-3 Example 3 KEY: multi-part question | AAS | proof | writing in math

68. ANS: Answers may vary. Sample: No; to use the HL Theorem, you also need to know that you have two right triangles. The right angles are a third pair of corresponding congruent parts.

PTS: 1 REF: 4-6 Congruence in Right Triangles KEY: HL Theorem | congruent figures | corresponding parts | reasoning

69. ANS:

Answers may vary. Sample: BA ≅ DC if ∆BAC ≅ ∆DCA. These two triangles are congruent if you know that ∠BAC ≅ ∠DCA (to get SAS) or ∠B ≅ ∠D (to get AAS).

PTS: 1 REF: 4-7 Using Corresponding Parts of Congruent TrianglesTOP: 4-7 Example 2 KEY: multi-part question | SAS | ASA | overlapping triangles | reasoning