Download Unit 5 Test Review

Name: ______________________
Class: _________________
Date: _________
ID: A
Unit 5 Test Review
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____
1. Justify the last two steps of the proof.
Given: RS ≅ UT and RT ≅ US
Prove: ΔRST ≅ ΔUTS
Proof:
1. RS ≅ UT
2. RT ≅ US
3. ST ≅ TS
4. ΔRST ≅ ΔUTS
1. Given
2. Given
3. ?
4.
?
a. Symmetric Property of ≅; SSS b. Reflexive Property of ≅; SAS c. Reflexive Property of ≅; SSS
d. Symmetric Property of ≅; SAS
____
2. What other information do you need in order to prove the triangles congruent using the SAS Congruence
Postulate?
a. ∠BAC ≅ ∠DAC b. AC ⊥ BD c. ∠CBA ≅ ∠CDA d. AC ≅ BD
1
____
3. Which two triangles are congruent by ASA?
AF bisects EC, and ∠AED ≅ ∠FCD.
a. ΔABD and ΔCBD b. ΔADE and ΔFDC c. ΔAED and ΔADB d. none
____
4. Which pair of triangles is congruent by ASA?
a.
b.
d.
2
c.
____
5. What is the missing reason in the two-column proof?
⎯⎯
⎯
→
⎯⎯
⎯
→
Given: MO bisects ∠PMN and OM bisects ∠PON
Prove: ΔPMO ≅ ΔNMO
Statements
Reasons
1. MO bisects ∠PMN
2. ∠PMO ≅ ∠NMO
3. MO ≅ MO
1. Given
2. Definition of angle bisector
3. Reflexive property
4. OM bisects ∠PON
5. ∠POM ≅ ∠NOM
6. ΔPMO ≅ ΔNMO
4. Given
5. Definition of angle bisector
6. ?
⎯⎯
⎯
→
⎯⎯
⎯
→
a. ASA Postulate b. SSS Postulate c. AAS Theorem d. SAS Postulate
____
6. Can you use the SAS Postulate, the AAS Theorem, or both to prove the triangles congruent?
a. either SAS or AAS b. SAS only c. AAS only d. neither
____
7. Name the theorem or postulate that lets you immediately conclude ΔABD ≅ ΔCBD.
a. AAS b. SAS c. ASA d. none of these
3
____
8. Based on the given information, what can you conclude, and why?
Given: ∠H ≅ ∠L, HJ ≅ JL
a. ΔHIJ ≅ ΔLKJ by ASA b. ΔHIJ ≅ ΔJLK by SAS c. ΔHIJ ≅ ΔJLK by ASA d. ΔHIJ ≅ ΔLKJ by
SAS
____
9. R, S, and T are the vertices of one triangle. E, F, and D are the vertices of another triangle. m∠R = 60,
m∠S = 80, m∠F = 60, m∠D = 40, RS = 4, and EF = 4. Are the two triangles congruent? If yes, explain
and tell which segment is congruent to RT.
a. yes, by ASA; FD b. yes, by AAS; ED c. yes, by SAS; ED d. No, the two triangles are not
congruent.
____ 10. Supply the missing reasons to complete the proof.
Given: ∠Q ≅ ∠T and QR ≅ TR
Prove: PR ≅ SR
Statement
1. ∠Q ≅ ∠T and
Reasons
1. Given
QR ≅ TR
2. ∠PRQ ≅ ∠SRT
2. Vertical angles are congruent.
3. ΔPRQ ≅ ΔSRT
3.
?
4. PR ≅ SR
4.
?
a. ASA; Substitution b. SAS; Corresp. parts of ≅ Δ are ≅. c. AAS; Corresp. parts of ≅ Δ are ≅. d.
ASA; Corresp. parts of ≅ Δ are ≅.
4
____ 11. For which situation could you immediately prove Δ1 ≅ Δ2 using the HL Theorem?
a. I only b. II only c. III only d. II and III
____ 12. ABCD is a parallelogram. If m∠CDA = 66, then m∠BCD =
? . The diagram is not to scale.
a. 66 b. 124 c. 114 d. 132
____ 13. For the parallelogram, if m∠2 = 5x − 28 and m∠4 = 3x − 10, find m∠3. The diagram is not to scale.
a. 9 b. 17 c. 173 d. 163
____ 14. ABCD is a parallelogram. If m∠DAB = 115, then m∠BCD =
a. 125 b. 65 c. 75 d. 115
5
? . The diagram is not to scale.
____ 15. What is the missing reason in the proof?
Given: ABCD with diagonal BD
Prove: ABD ≅ CDB
Statements
1. AD Ä BC
2. ∠ADB ≅ ∠CBD
3. AB Ä CD
4. ∠ABD ≅ ∠CDB
5. DB ≅ DB
6. ΔABD ≅ ΔCDB
Reasons
1. Definition of parallelogram
2. Alternate Interior Angles Theorem
3. ?
4. Alternate Interior Angles Theorem
5. Reflexive Property of Congruence
6. ASA
a. Reflexive Property of Congruence b. Definition of parallelogram c.
Alternate Interior Angles Theorem d. ASA
____ 16. WXYZ is a parallelogram. Name an angle congruent to ∠WZY.
a. ∠ZXY b. ∠XWZ c. ∠ZXW d. ∠WXY
6
____ 17. In parallelogram DEFG, DH = x + 3, HF = 3y, GH = 4x – 5, and HE = 2y + 3. Find the values of x and y.
The diagram is not to scale.
a. x = 6, y = 3 b. x = 2, y = 3
c. x = 3, y = 2
d. x = 3, y = 6
____ 18. Find AM in the parallelogram if PN =10 and AO = 5. The diagram is not to scale.
a. 5 b. 10 c. 5 d. 10
____ 19. LMNO is a parallelogram. If NM = x + 15 and OL = 3x + 5, find the value of x and then find NM and OL.
a. x = 7, NM = 20, OL = 22 b. x = 5, NM = 20, OL = 20 c. x = 7, NM = 22, OL = 22 d. x = 5, NM
= 22, OL = 20
7
____ 20. Find values of x and y for which ABCD must be a parallelogram. The diagram is not to
scale.
a. x = 10, y = 38
b. x = 10, y = 21 c. x = 10, y = 7
d. x = 7, y = 10
____ 21. Based on the information in the diagram, can you prove that the figure is a parallelogram? Explain.
a. Yes; both pairs of opposite sides are congruent. b. Yes; opposite angles are congruent. c. No;
you cannot prove that the quadrilateral is a parallelogram. d. Yes; two opposite sides are both parallel
and congruent.
____ 22. Which statement can you use to conclude that quadrilateral XYZW is a parallelogram?
a. XN ≅ NY and ZN ≅ NW b. XY ≅ YZ and YZ ≅ XW c. XY ≅ WZ and YZ ≅ XW d.
XY ≅ WZ and YZ ≅ ZW
8
____ 23. Based on the information given, can you determine that the quadrilateral must be a parallelogram?
Explain.
Given: XY ≅ WZ and XW ≅ YZ
a. Yes; opposite sides are congruent. b. Yes; diagonals of a parallelogram bisect each other. c. Yes;
two opposite sides are both parallel and congruent. d. No; you cannot determine that the quadrilateral
is a parallelogram.
____ 24. Which description does NOT guarantee that a quadrilateral is a square?
a. is both a rectangle and a rhombus b. is a parallelogram with perpendicular diagonals c.
has all sides congruent and all angles congruent d. has all right angles and has all sides congruent
____ 25. Classify the figure in as many ways as possible.
a. rectangle, square, quadrilateral, parallelogram, rhombus b. rectangle, square, parallelogram c.
rhombus, quadrilateral, square d. square, rectangle, quadrilateral
____ 26. DEFG is a rectangle. DF = 5x – 5 and EG = x + 11. Find the value of x and the length of each diagonal.
a. x = 4, DF = 13, EG = 13 b. x = 4, DF = 15, EG = 18 c. x = 4, DF = 15, EG = 15 d. x = 2, DF
= 13, EG = 13
9
____ 27. In rectangle KLMN, KM = 6x + 16 and LN = 49. Find the value of x.
a. 5.5 b. 4.5 c. 33 d. 6.5
____ 28. In rectangle PQRS, PR = 18x – 24 and QS = x + 146. Find the value of x and the length of each diagonal.
a. x = 10, PR = 156, QS = 156 b. x = 10, PR = 78, QS = 78 c. x = 5, PR = 151, QS = 151 d. x =
11, PR = 174, QS = 174
____ 29. In quadrilateral ABCD, m∠ACD = 2x + 4 and m∠ACB = 5x − 11. For what value of x is ABCD a rhombus?
a. 4 b. 5 c. 6 d. 7
____ 30. In quadrilateral ABCD, AE = x + 10 and BE = 3x − 18. For what value of x is ABCD a rectangle?
a. 24 b. 14 c. 18 d. 16
10
____ 31. Lucinda wants to build a square sandbox, but she has no way of measuring angles. Explain how she can
make sure that the sandbox is square by only measuring length.
a. Arrange four equal-length sides so the diagonals bisect each other. b. Arrange four equal-length
sides so the diagonals are equal lengths also. c. Make each diagonal the same length as four equal-length
sides. d. Not possible; Lucinda has to be able to measure a right angle.
____ 32. ∠J and ∠M are base angles of isosceles trapezoid JKLM. If m∠J = 20x + 9, and
m∠M = 14x + 15, find m∠K.
a. 151 b. 1 c. 29 d. 75.5
____ 33. Parallelogram ABCD has the angle measures shown. Can you conclude that it is a rhombus, a rectangle, or
a square? Explain.
a. Parallelogram ABCD is a rhombus, because the diagonal bisects two angles. b. Parallelogram ABCD
is a square, because all four angles have the same measure. c. Parallelogram ABCD is a rectangle,
because the diagonal creates congruent angles. d. There is not enough information.
Short Answer
34. Are the triangles congruent? Justify your answer.
35. Based on the given information, can you conclude that ΔQRS ≅ ΔTUV? Explain.
Given: QR ≅ TU, QS ≅ TV, and ∠R ≅ ∠U
11
36. 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.
37. Complete the proof by providing the missing reasons.
Given: CB ⊥ BD, DE ⊥ EC, CB ≅ DE
Prove: ΔDBC ≅ ΔCED
Statement
Reason
1. CB ≅ DE, CB ⊥ BD, and DE ⊥ EC
1. Given
2. ∠CBD and ∠DEC are right angles
3. ∠CBD ≅ ∠DEC
2. Definition of perpendicular segments
3. ?
4. ?
5. ?
4. CD ≅ CD
5. ΔDBC ≅ ΔCED
12
38. For what values of x and y must this quadrilateral be a parallelogram? Find the lengths of the sides. The
diagram is not to scale.
Essay: Write a proof.
39. Given: BC ≅ DA, ∠1 ≅ ∠2, and CF ≅ AF
Prove: ΔCFE ≅ ΔAFE
40. Given: AC ≅ DC and BC ≅ CE
Prove: ΔABC ≅ ΔDEC.
41. Given: BC ≅ EC and AC ≅ DC
Prove: BA ≅ ED
13
42. Given: ∠BAC ≅ ∠DAC, ∠DCA ≅ ∠BCA
Prove: BC ≅ CD
43. Given:
Prove:
STUV
44. Given: SV Ä TU and ΔSVX ≅ ΔUTX
SXT ≅
UXV
Prove: VUTS is a parallelogram
Other
45. ABCD is a rhombus. Explain why ΔABC ≅ ΔCDA.
14
ID: A
Unit 5 Test Review
Answer Section
MULTIPLE CHOICE
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
C
B
B
B
A
C
A
A
A
D
C
C
D
D
B
D
C
A
B
C
B
C
A
B
A
C
A
A
B
B
B
A
A
SHORT ANSWER
34. Yes, the diagonal segment is congruent to itself, so the triangles are congruent by SAS.
35. 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 ΔQRS ≅ ΔTUV with the information given.
36. Yes; ΔPQS ≅ ΔRQS by SAS.
1
ID: A
37. Step 3: All right angles are congruent.
Step 4: Reflexive Property
Step 5. HL Theorem
38. x = 7, y = 3; 18, 28
ESSAY
39.
[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
6. Given
7. EF ≅ EF
7. Reflexive Property
8. ΔCFE ≅ ΔAFE
8. SSS
[3]
[2]
[1]
correct idea, some details inaccurate
correct idea, not well organized
correct idea, one or more significant steps omitted
[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.
correct idea, some details inaccurate
correct idea, not well organized
correct idea, one or more significant steps omitted
40.
[3]
[2]
[1]
41. [4]
Statement
1.
2.
3.
4.
[3]
[2]
[1]
BC ≅ EC and AC ≅ DC
∠BCA ≅ ∠ECD
ΔBCA ≅ ΔECD
BA ≅ ED
Reason
1. Given
2. Vertical angles are congruent.
3. SAS
4. Corresp. parts of ≅ Δ are ≅.
correct idea, some details inaccurate
correct idea, not well organized
correct idea, one or more significant steps omitted
2
ID: A
42. [4]
Statement
1. ∠BAC ≅ ∠DAC and
∠DCA ≅ ∠BCA
2. CA ≅ CA
3. ΔCBA ≅ ΔCDA
4. BC ≅ CD
Reason
1. Given
2. Reflexive Property
3. ASA
4. Corresp. parts of ≅ Δ are ≅.
[3] correct idea, some details inaccurate
[2] correct idea, not well organized
[1] correct idea, one or more significant steps omitted
43.
[4]
STUV is a
.
Given
VU ≅ TS
VU ≅ TS
Opposite sides are ≅.
Opposite sides are ≅.
VU Ä ST
Definition of a
∠UVT ≅ ∠STV
.
∠VUS ≅ ∠UST
Alternate interior angles are ≅.
SXT ≅
Alternate interior angles are ≅.
UXV
[3]
[2]
[1]
ASA
correct idea, some details inaccurate
correct idea, not well organized
correct idea, one or more significant steps omitted
[4]
Because ΔSVX ≅ ΔUTX, SV ≅ TU because corresponding parts of congruent
[3]
[2]
[1]
triangles are congruent. It is given that SV Ä TU. Therefore quadrilateral VUTS is a
parallelogram because if one pair of opposite sides of a quadrilateral is both
congruent and parallel, then the quadrilateral is a parallelogram.
correct idea, some details inaccurate
correct idea, not well organized
correct idea, one or more significant steps omitted
44.
3
ID: A
OTHER
45. AB ≅ CD and BC ≅ DA by the definition of rhombus. AC ≅ AC by the Reflexive Property, so
ΔABC ≅ ΔCDA by SSS.
4