G Ch4 Review

Name: ______________________ Class: _________________ Date: _________ ID: A

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Geometry: Chapter 4 Review

1. (1 point) If BCDE is congruent to OPQR, then DE is congruent to ? .

2. (1 point) ∠ABC ≅ ?

3. (1 point) The two triangles are congruent as suggested by their appearance. Find the value of c. The diagrams are not to scale.

4. (1 point) Use the information given in the diagram. Tell why MN ≅ PO and ∠NOM ≅ ∠PMO.

Name: ______________________ ID: A

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5. (1 point) For the two quadrilaterals below, ∠I ≅ ∠M, ∠IJK ≅ ∠MJK, ∠LKJ ≅ ∠NKJ, and ∠L ≅ ∠N. Complete this congruence statement for the two quadrilaterals.

LKJI ≅ ___?___

6. (1 point) Justify the last two steps of the proof.Given: RS ≅ UT and RT ≅ USProve: ΔRST ≅ ΔUTS

Proof:1. RS ≅ UT 1. Given2. RT ≅ US 2. Given3. ST ≅ TS 3. ?

4. ΔRST ≅ ΔUTS 4. ?

7. (1 point) Name the angle included by the sides PN and NM.

Name: ______________________ ID: A

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8. (1 point) State whether ΔABC and ΔAED are congruent. Justify your answer.

9. (1 point) Write a paragraph proof to show that ΔABC ≅ ΔDEC.Given: AC ≅ DC and BC ≅ CE

10. (1 point) Which triangles are congruent by ASA?

Name: ______________________ ID: A

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11. (1 point) What is the missing reason in the two-column proof?Given: MO

→%%%

bisects ∠PMN and OM→%%%

bisects ∠PONProve: ΔPMO ≅ ΔNMO

Statements Reasons1. MO

→%%%

bisects ∠PMN 1. Given2. ∠PMO ≅ ∠NMO 2. Definition of angle bisector3. MO ≅ MO 3. Reflexive property

4. OM→%%%

bisects ∠PON 4. Given5. ∠POM ≅ ∠NOM 5. Definition of angle bisector6. ΔPMO ≅ ΔNMO 6. ?

12. (1 point) Can you use the SAS Postulate, the AAS Theorem, or both to prove the triangles congruent?

13. (1 point) Based on the given information, what can you conclude, and why?Given: ∠H ≅ ∠L, HJ ≅ JL

Name: ______________________ ID: A

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14. (1 point) Write a two-column proof to show that ΔOQR ≅ ΔMQR.Given: NO ≅ PM, ∠1 ≅ ∠2, and OR ≅ MR

15. (1 point) Supply the missing reasons to complete the proof.Given: ∠Q ≅ ∠T and QR ≅ TR

Prove: PR ≅ SR

Statement Reasons

1. ∠Q ≅ ∠T and

QR ≅ TR1. Given

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

3. ΔPRQ ≅ ΔSRT 3. ?

4. PR ≅ SR 4. ?

Name: ______________________ ID: A

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16. (1 point) Write a two-column proof.Given: BC ≅ EC and AC ≅ DCProve: BA ≅ ED

17. (1 point) Write a two-column proof:Given: ∠BAC ≅ ∠DAC, ∠DCA ≅ ∠BCAProve: BC ≅ CD

18. (1 point) What is the value of x?

Name: ______________________ ID: A

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19. (1 point) What is the value of x?

20. (1 point) The legs of an isosceles triangle have lengths 2x + 4 and x + 8. The base has length 5x − 2. What is the length of the base?

21. (1 point) In an A-frame house, the two congruent sides extend from the ground to form a 34° angle at the peak. What angle does each side form with the ground?

22. (1 point) Find the value of x. The diagram is not to scale.

23. (1 point) Is ΔPQS ≅ ΔRQS by HL? If so, name the legs that allow the use of HL.

Name: ______________________ ID: A

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24. (1 point) Complete the proof by providing the missing reasons.

Given: SD ⊥ HT; SH ≅ STProve: ΔSHD = ΔSTD

Statement Reason1. SD ⊥ HT 1. Given2. ∠SDH and∠SDT are right ∠s 2. ? 3. SH ≅ ST 3. ? 4. ? 4. Reflexive Property5. ΔSHD = ΔSTD 5. ?

25. (1 point) 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.

26. (1 point) What common side do ΔDHB andΔDEB share?

Name: ______________________ ID: A

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27. (1 point) What common angle do ΔCDG andΔFCE share?

28. (1 point) 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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29. (1 point) Complete the proof by providing the missing reasons.

Given: CB ⊥ BD, DE ⊥ EC, CB ≅ DEProve: ΔDBC ≅ ΔCED

Statement Reason

1. CB ≅ DE, CB ⊥ BD, and DE ⊥ EC 1. Given

2. ∠CBD and ∠DEC are right angles 2. Definition of perpendicular segments3. ∠CBD ≅ ∠DEC 3. ? 4. CD ≅ CD 4. ? 5. ΔDBC ≅ ΔCED 5. ?

30. (1 point) Write a proof.

Given: BC ≅ DA, ∠1 ≅ ∠2, and CF ≅ AFProve: ΔCFE ≅ ΔAFE

ID: A

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Geometry: Chapter 4 ReviewAnswer Section

1. ANS: QR

PTS: 1 REF: 4-1 Congruent Figures 2. ANS:

∠MNP


3


Given, Given


NKJM


Reflexive Property of ≅; SSS

PTS: 1 REF: 4-2 Triangle Congruence by SSS and SAS 7. ANS:

∠N


yes, by either SSS or SAS


[4] Answers may vary. Sample: You are given that AC ≅ DC and BC ≅ CE. Vertical angles BCA and ECD are congruent, so ΔABC ≅ ΔDEC by SAS.

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


ΔVTU andΔABC

PTS: 1 REF: 4-3 Triangle Congruence by ASA and AAS

ID: A

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11. ANS: ASA Postulate

PTS: 1 REF: 4-3 Triangle Congruence by ASA and AAS 12. ANS:

AAS only


ΔHIJ ≅ ΔLKJ by ASA


Statement Reason

1. NO ≅ PM 1. Given

2. ∠1 ≅ ∠2 2. Given

3. ∠OQN ≅ ∠MQP 3. Vertical angles are congruent.

4. ΔOQN ≅ ΔMQP 4. AAS

5. OQ ≅ MQ 5. Corresp. parts of ≅ Δ ≅

6. OR ≅ MR 6. Given

7. QR ≅ QR

8. ΔOQR ≅ ΔMQR

7. Reflexive Property

8. SSS


ASA; Corresp. parts of ≅ Δ are ≅.

PTS: 1 REF: 4-4 Using Corresponding Parts of Congruent Triangles

ID: A

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16. ANS: [4]

Statement Reason1. BC ≅ EC and AC ≅ DC 1. Given2. ∠BCA ≅ ∠ECD 2. Vertical angles are congruent.3. ΔBCA ≅ ΔECD 3. SAS4. BA ≅ ED 4. Corresp. parts of ≅ Δ are ≅.


PTS: 1 REF: 4-4 Using Corresponding Parts of Congruent Triangles 17. ANS:

[4] Statement Reason1. ∠BAC ≅ ∠DAC and ∠DCA ≅ ∠BCA 1. Given

2. CA ≅ CA 2. Reflexive Property3. ΔCBA ≅ ΔCDA 3. ASA4. BC ≅ CD 4. Corresp. parts of ≅ Δ are ≅.


PTS: 1 REF: 4-4 Using Corresponding Parts of Congruent Triangles 18. ANS:

71°

PTS: 1 REF: 4-5 Isosceles and Equilateral Triangles 19. ANS:

68°


18


73

PTS: 1 REF: 4-5 Isosceles and Equilateral Triangles

ID: A

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22. ANS: x = 23


Yes, QS (in each triangle)

PTS: 1 REF: 4-6 Congruence in Right Triangles 24. ANS:

2. Definition of ⊥ lines3. Given4. SD ≅ SD5. HL Theorem


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 the definition of a congruent triangle.


BD

PTS: 1 REF: 4-7 Congruence in Overlapping Triangles 27. ANS:

∠C


ΔMNP ≅ ΔPOM by SSS


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

PTS: 1 REF: 4-7 Congruence in Overlapping Triangles

ID: A

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30. ANS: [4] Statement Reason

1. BC ≅ DA 1. Given

2. ∠1 ≅ ∠2 2. Given

3. ∠BEC ≅ ∠DEA 3. Vertical angles are congruent.

4. ΔBEC ≅ ΔDEA 4. AAS

5. CE ≅ AE 5. Corresp. parts of ≅ Δ ≅

6. CF ≅ AF

7. EF ≅ EF

8. ΔCFE ≅ ΔAFE

6. Given

7. Reflexive Property

8. SSS


PTS: 1 REF: 4-7 Congruence in Overlapping Triangles

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G Ch4 Review