Name: __________________________________________ Period: ___________________

1

Chapter 5 Practice

Multiple Choice

1. Classify ∆DBC by its angle measures, given m∠DAB = 60°, m∠ABD = 75°, and m∠BDC = 25°.

a. obtuse triangle c. right triangle

b. acute triangle d. equiangular triangle

2. Classify ∆ABC by its side lengths.

a. equilateral b. isosceles c. scalene d. obtuse

3. What is the measure of the vertex angle of an isosceles triangle if one of its base angles measures 42°?

a. 69° b. 84° c. 138° d. 96°

4. One of the acute angles in a right triangle has a measure of 34.6°. What is the measure of the other acute angle?

a. 145.4° c. 55.4°

b. 34.6° d. 90°

5. If ∆MNO ≅ ∆PQR, which of the following can you NOT conclude as being true?

a. MN ≅ PR c. NO ≅ QR

b. ∠M ≅ ∠P d. ∠N ≅ ∠Q

2

6. ∆ABC is an isosceles triangle. The base AB = 8x + 4. BC = 7x + 3 and CA = 6x + 6. Find AB.

a. AB = 3 b. AB = 24 c. AB = 28 d. AB = 76

7. ∠ABC ≅ ?

a. ∠PMN b. ∠NPM c. ∠NMP d. ∠MNP

8. Find the value of x.

a. x = 6 b. x = 4 c. x = 2 d. x = 8

9. Find the measure of each numbered angle.

a. m∠1 = 54°, m∠2 = 117°, m∠3 = 63° c. m∠1 = 54°, m∠2 = 63°, m∠3 = 63°

b. m∠1 = 117°, m∠2 = 63°, m∠3 = 63° d. m∠1 = 54°, m∠2 = 63°, m∠3 = 117°

3

10. Given that ∆ABC ≅ ∆DEC and m∠E = 23º, find m∠ACB.

a. m∠ACB = 77º c. m∠ACB = 23º

b. m∠ACB = 67º d. m∠ACB = 113º

11. Find m∠K .

a. m∠K = 63° c. m∠K = 79°

b. m∠K = 55° d. m∠K = 39°

12. For these triangles, select the triangle congruence statement and the postulate or theorem that supports it.

a. ∆ABC ≅ ∆JLK, SSA c. ∆ABC ≅ ∆JLK, SAS

b. ∆ABC ≅ ∆JKL, HL d. ∆ABC ≅ ∆JKL, AAS

4

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

a. x = 21 b. x = 60 c. x = 15 d. none of these

14. Refer to the figure below with GJ congruent to JI . Which of the following statements is true?

a. ∆GHJ ≅ ∆IHJ by SAS c. ∆GJH ≅ ∆IJH by SSS

b. ∆GHJ ≅ ∆IHJ by HL d. ∆GIJ ≅ ∆JHG by SSS

15. What must be true in order for ∆ABC ≅ ∆EDC by the SAS Congruence Postulate?

a. ∠B ≅ ∠D b. ∠A ≅ ∠E c. AC ≅ CE d. AB ≅ DE

16. Find x (Note: Not drawn to scale)

a. 79 b. 101 c. 120 d. 128

5

17. Daphne folded a triangular sheet of paper into the shape shown. Find m∠ECD , given m∠CAB = 61°,

m∠ABC = 22°, and m∠BCD = 42°.

a. m∠ECD = 41° b. m∠ECD = 97° c. m∠ECD = 22° d. m∠ECD = 61°

18. Can you prove ∆FDG ≅ ∆FDB? Explain.

a. yes, by ASA b. yes, by AAA c. yes, by SAS d. no

19. What additional information will allow you to prove

the triangles congruent by the HL Theorem?

a. ∠A ≅ ∠E b. m∠BCE = 90 c. AC ≅ DC d. AC ≅ BD

6

20. Refer to the figure below. m∠A = .

a. 67° b. 35° c. 141° d. 74°

21. Name the theorem or postulate that lets you conclude ∆ABD ≅ ∆CBD.

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

22. Find CA.

a. CA = 10 b. CA = 12 c. CA = 14 d. Not enough

information.

7

23. Find m∠BDC.

a. 68° c. 56°

b. 84° d. 112°

24. Find the value of x.

a. 13 c. 26

b. 60 d. 13 3

25. Find m∠KMN .

a. 20° c. 101°

b. 79° d. 5°

8

26. Find the value of x.

a. 10 b. 4 c. 5 d. 6

27. Write a congruence statement for the triangles.

a. PRQ ≅ TUV b. PRQ ≅ UTV c. QRP ≅ TUV d. QRP ≅ UTV

28. If BCDE is congruent to OPQR, then DE is congruent to ? .

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

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

a. either ASA or AAS c. AAS only

b. ASA only d. neither

9

30. In the diagram, CDE ≅ GHI . Find the value of y.

a. y = 51 c. y = 42

b. y = 54 d. y = 64

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

Given: ∠H ≅ ∠L, HJ ≅ JL

a. ∆HIJ ≅ ∆LKJ by ASA c. ∆HIJ ≅ ∆JLK by ASA

b. ∆HIJ ≅ ∆JLK by SAS d. ∆HIJ ≅ ∆LKJ by SAS

32. Find the values of x and y.

a. x = 90, y = 47 c. x = 47, y = 43

b. x = 43, y = 47 d. x = 90, y = 43

10

33. Use the information in the figure. Find m∠D.

a. 32° b. 122° c. 64° d. 58°

34. Find m∠M .

a. 35° c. 70°

b. 55° d. 20°

35. Given ∆QRS ≅ ∆TUV, QS = 5v + 2, and TV = 6v − 8, TU = 6v + 5, find the length of QS.

a. 52 b. 53 c. 10 d. 68

36. Given ∆ABC ≅ ∆PQR, m∠A = 3v + 3, m∠B = 3v + 4, and m∠Q = 8v − 6, find m∠B.

a. 22 b. 11 c. 10 d. 25

11

37. Name the angle included by the sides PN and NM .

a. ∠N b. ∠P c. ∠M d. none of these

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

a. BC ≅ EF c. AC ≅ DF

b. ∠B ≅ ∠E d. AB ≅ EF

39. What is the measure of a base angle of an isosceles triangle if the vertex angle measures 38° and the two

congruent sides each measure 21 units?

a. 71° b. 142° c. 152° d. 76°

12

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

Given: RS ≅ ST , m∠RST = 3x − 48, m∠STU = 9x

a. 19 b. 105 c. 21 d. 24

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

Given: AC→

bisects ∠DAB and CA→

bisects ∠DCB

Prove: ∆DAC ≅ ∆ABC

Statements Reasons

1. AC→

bisects ∠DAB 1. Given

2. ∠DAC ≅ ∠BAC 2. Definition of angle bisector

3. AC ≅ AC 3. Reflexive property

4. CA→

bisects ∠DCB 4. Given

5. ∠DAC ≅ ∠BCA 5. Definition of angle bisector

6. ∆DAC ≅ ∆BAC 6. ?

a. ASA Postulate c. SAS Postulate

b. SSS Postulate d. AAS Theorem

42. Two sides of an equilateral triangle have lengths 2x − 2 and 3x − 6. Which of 10 − x or 6x + 5 could be the length

of the third side?

a. neither 10 – x nor 6x + 5 c. both 10 – x and 6x + 5

b. 10 – x only d. 6x + 5 only

13

43. Given: P is the midpoint of TQ and RS .

Prove: ∆TPR ≅ ∆QPS

Statements Reasons

1. P is the midpoint of TQ and RS . 1. Given

2. TP ≅ QP, RP ≅ SP 2. [1]

3. [2] 3. Vertical Angles Theorem

4. ∆TPR ≅ ∆QPS 4. [3]

a. [1]. Definition of midpoint

[2] ∠TPR ≅ ∠QPS

[3] SAS

c. [1] Definition of midpoint

[2] ∠PRT ≅ ∠PSQ

[3] SAS

b. [1] Definition of midpoint

[2] RT ≅ SQ

[3] SSS

d. [1] Definition of midpoint

[2] ∠TPR ≅ ∠QPS

[3] SSS

44. Given: ∠Q ≅ ∠T and QR ≅ TR Prove: PR ≅ SR

Statement Reasons

1.∠Q ≅ ∠T and

QR ≅ TR

1. Given

2. ∠PRQ ≅ ∠SRT 2. Vertical angles are congruent.

3. ∆PRQ ≅ ∆SRT 3. ?

4. PR ≅ SR 4. ?

a. 3. ASA; 4. Substitution c. 3. AAS; 4. CPCTC

b. 3. SAS; 4. CPCTC d. 3. ASA; 4. CPCTC

14

45. Given: AB ≅ AC, ∠BAD ≅ ∠CAD Prove: AD bisects BC

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 bisects BC 6. Def. of segment bisector

a. (4)ASA; (5)CPCTC c. (4)SSS; (5) Reflexive Property

b. (4)SAS; (5)Reflexive Property d. (4)SAS; (5)CPCTC

Short Answer

46. Find the measure of each acute angle.

15

47. Fill in the missing reasons to complete the proof.

Given: ∠VUY ≅ ∠UWT ≅ ∠X

Prove: UW ≅ UT

Statement Reason

1. ∠VUY ≅ ∠X 1. Given

2. UY Ä TX 2. Converse of the Corresponding Angles Postulate

3. ∠T ≅ ∠VUY 3. ?

4. ∠VUY ≅ ∠UWT 4. Given

5. ∠T ≅ ∠UWT 5. Transitive Property

6. UT ≅ UW 6. ?

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

16

Essay

49. Write a two-column proof.

Given: BC ≅ EC and AC ≅ DC

Prove: BA ≅ ED

50. Write a two-column proof:

Given: ∠BAC ≅ ∠DAC, ∠DCA ≅ ∠BCA

Prove: BC ≅ CD

ID: A

1

Chapter 5 Practice

Answer Section

MULTIPLE CHOICE

1. A

2. A

3. D

4. C

5. A

6. C

7. D

8. A

9. C

10. B

11. A

12. B

13. A

14. A

15. C

16. A

17. A

18. A

19. C

20. A

21. A

22. C

23. D

24. A

25. C

26. B

27. D

28. D

29. A

30. C

31. A

32. D

33. A

34. D

35. A

36. C

37. A

38. B

39. A

40. A

41. A

42. B

43. A

ID: A

2

44. D

45. D

SHORT ANSWER

46. 26°, 64°

47. Step 3: Corresponding Angles Postulate

Step 6: Converse of Isosceles Triangle Theorem

48. Yes, the diagonal segment is congruent to itself, so the triangles are congruent by SAS.

ESSAY

49. [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

50. [4]

Statement Reason

1. ∠BAC ≅ ∠DAC and

∠DCA ≅ ∠BCA1. Given

2. CA ≅ CA 2. Reflexive Property

3. ∆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