Post on 27-Jul-2020
Name: ________________________ Class: ___________________ Date: __________ ID: A
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Geometry CH 6 Test Review
1. In the figure shown, QX ⊥ XP,RP ⊥ XP,QT ≅ RS ,and XS ≅ TP. Determine whether ∆XQT is congruent to
∆PRS . Explain your reasoning.
2. Using the figure shown, determine whether ∆TAR is congruent to ∆PAR. Explain your reasoning.
3. Triangle CLG and ∆AER are isosceles triangles with CL ≅ AE, HG ≅ ZR, and ∠L ≅ ∠E. Prove that
CH ≅ AZ. Explain your reasoning.
4. Using the figure shown, determine whether ∆RAC is congruent to ∆YTC. Explain your reasoning.
Name: ________________________ ID: A
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Congruence Theorems SSS, SAS, ASA, and AAS
Define each Theorem:
a. Side-Side-Side (SSS) Congruence Theorem
b. Side-Angle-Side (SAS) Congruence Theorem
c. Angle-Side-Angle (ASA) Congruence Theorem
d. Angle-Angle-Side (AAS) Congruence Theorem
____ 5. .
____ 6. .
____ 7. .
____ 8. .
Name: ________________________ ID: A
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Congruence Theorems SSS, SAS, ASA, and AAS
For each figure, determine whether there is enough information to conclude that the indicated
triangles are congruent. If so, state the theorem you used.
EXAMPLE
Given: AC Ä BD, AB Ä CD
Is ABD ≅ DCA?
Yes. There is enough information to conclude that ABD ≅ DCA by ASA.
9.
Given: EI ≅ GH , EH ≅ GI
Is EHI ≅ GIH?
10.
Given: TQ ≅ TP
Is TRQ ≅ TRP?
Name: ________________________ ID: A
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11.
Given: TU ≅ RU , NU ≅ PU
Is TUN ≅ RUP?
12.
Given: ∠A ≅ ∠X, ∠UYA ≅ ∠WZX, AZ ≅ XY
Is AUY ≅ XWZ?
13.
Given: HJ bisects ∠H, HI ≅ HK
Is HLK ≅ HLI?
Name: ________________________ ID: A
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14.
Given: ∠A ≅ ∠E
Is ACG ≅ EGC?
15.
Given: FL ≅ IJ , FJ ≅ IL
Is FJL ≅ ILJ?
16.
Given: ∠NMO ≅ ∠POM, MN ≅ OP
Is MNQ ≅ OPQ?
Name: ________________________ ID: A
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Right Triangle Congruence Theorems: HL, LL, HA, and LA
Vocabulary: Choose the diagram that models each right triangle congruence theorem. EXPLAIN
EACH THEOREM IN YOUR OWN WORDS.
a. Explain the theorem used:
b. Explain the theorem used:
c. Explain the theorem used:
d. Explain the theorem used:
____ 17. Hypotenuse-Leg (HL) Congruence Theorem
____ 18. Leg-Leg (LL) Congruence Theorem
____ 19. Hypotenuse-Angle (HA) Congruence Theorem
____ 20. Leg-Angle (LA) Congruence Theorem
Name: ________________________ ID: A
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Right Triangle Congruence Theorems: HL, LL, HA, and LA
For each figure, determine if there is enough information to prove the two triangles are congruent. If
so, name the congruence theorem used.
EXAMPLE
Given: GF bisects ∠RGS, and ∠R and ∠S are right angles.
Is FRG ≅ FSG?
Yes. There is enough information to conclude that FRG ≅ FSG by HA.
21.
Given: DV ⊥ TU
Is DVT ≅ DVU ?
22.
Given: NM ≅ EM , DM ≅ OM , and ∠NMD and ∠EMO are right angles.
Is NMD ≅ EMO?
Name: ________________________ ID: A
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23.
Given: GO ≅ MI, and ∠E and ∠K are right angles.
Is GEO ≅ MKI?
24.
Given: HM ≅ VM , and ∠H and ∠V are right angles.
Is GHM ≅ UVM?
Name: ________________________ ID: A
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Right Triangle Congruence Theorems: HL, LL, HA, and LA
Create a two-column proof to prove each statement.
Given:WZ bisects VY , WV ⊥ VY , and YZ ⊥ VY
Prove: WVX ≅ ZYX
Statements Reasons
1. WV ⊥ VY and YZ ⊥ VY 1. Given
2. ∠WVX and ∠ZYX are right angles. 2. Definition of perpendicular angles
3. WVX and ZYX are right triangles. 3. Definition of right triangles
4. WZ bisects VY . 4. Given
5. VX ≅ YX 5. Definition of segment bisector
6. ∠WXV ≅ ∠ZXY 6. Vertical Angle Theorem
7. WVX ≅ ZYX 7. LA Congruence Theorem
_____________________________________________________________________
25. Given: Point D is the midpoint of EC, ADB is an isosceles triangle with base AB, and ∠E and ∠C are right
angles.
Prove: AED ≅ BCD
Statements Reasons
1.________________________________ 1.________________________________
2.________________________________ 2.________________________________
3.________________________________ 3.________________________________
4.________________________________ 4.________________________________
5.________________________________ 5.________________________________
6.________________________________ 6.________________________________
7.________________________________ 7.________________________________
8.________________________________ 8.________________________________
9.________________________________ 9.________________________________
*** You may have more space than what you need ***
Name: ________________________ ID: A
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26. Given: SU ⊥ UP, TP ⊥ UP, and UR ≅ PR
Prove: SUR ≅ TPR
Statements Reasons
1.________________________________ 1.________________________________
2.________________________________ 2.________________________________
3.________________________________ 3.________________________________
4.________________________________ 4.________________________________
5.________________________________ 5.________________________________
6.________________________________ 6.________________________________
7.________________________________ 7.________________________________
8.________________________________ 8.________________________________
9.________________________________ 9.________________________________
27. Given: Rectangle MNWX and ∠NMW ≅ ∠XWM
Prove: MNW ≅ WXM
Statements Reasons
1.________________________________ 1.________________________________
2.________________________________ 2.________________________________
3.________________________________ 3.________________________________
4.________________________________ 4.________________________________
5.________________________________ 5.________________________________
6.________________________________ 6.________________________________
7.________________________________ 7.________________________________
8.________________________________ 8.________________________________
9.________________________________ 9.________________________________
Name: ________________________ ID: A
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CPCTC: Corresponding Parts of Congruent Triangles are Congruent
Problem Set :Create a two-column proof to prove each statement.
EXAMPLE
Given: RS is the ⊥ bisector of PQ.
Prove: ∠SPT ≅ ∠SQT
Statements Reasons
1. RS is the ⊥ bisector of PQ. 1. Given
2. RS ⊥ PQ 2. Definition of perpendicular bisector
3. ∠PTS and ∠QTS are right angles. 3. Definition of perpendicular lines
4. PTS and QTS are right triangles. 4. Definition of right triangles
5. RS bisects PQ. 5. Definition of perpendicular bisector
6. PT ≅ QT 6. Definition of bisect
7. TS ≅ TS 7. Reflexive Property of ≅
8. PTS ≅ QTS 8. Leg-Leg Congruence Theorem
9. ∠SPT ≅ SQT 9. CPCTC
_____________________________________________________________________
28. Given: ∠JHK ≅ ∠LHK, ∠JKH ≅ ∠LKH
Prove: JK ≅ LK
Name: ________________________ ID: A
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Statements Reasons
1.________________________________ 1.________________________________
2.________________________________ 2.________________________________
3.________________________________ 3.________________________________
4.________________________________ 4.________________________________
5.________________________________ 5.________________________________
6.________________________________ 6.________________________________
7.________________________________ 7.________________________________
8.________________________________ 8.________________________________
9.________________________________ 9.________________________________
29. Given: AC⊥ DB, AC bisects DB
Prove: AD ≅ AB
Statements Reasons
1.________________________________ 1.________________________________
2.________________________________ 2.________________________________
3.________________________________ 3.________________________________
4.________________________________ 4.________________________________
5.________________________________ 5.________________________________
6.________________________________ 6.________________________________
7.________________________________ 7.________________________________
8.________________________________ 8.________________________________
9.________________________________ 9.________________________________
Name: ________________________ ID: A
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30. Given: AT ≅ AQ, AC bisects ∠TAQ
Prove: AC bisects TQ
Statements Reasons
1.________________________________ 1.________________________________
2.________________________________ 2.________________________________
3.________________________________ 3.________________________________
4.________________________________ 4.________________________________
5.________________________________ 5.________________________________
6.________________________________ 6.________________________________
7.________________________________ 7.________________________________
8.________________________________ 8.________________________________
9.________________________________ 9.________________________________
Name: ________________________ ID: A
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CPCTC: Corresponding Parts of Congruent Triangles are Congruent
Use the given information to answer each question.
EXAMPLE
Samantha is hiking through the forest and she comes upon a canyon. She wants to know how wide the
canyon is. She measures the distance between points A and B to be 35 feet. Then she measures the distance
between points B and C to be 35 feet. Finally, she measures the distance between points C and D to be 80
feet. How wide is the canyon? Explain.
The canyon is 80 feet wide.
The triangles are congruent by the Leg-Angle Congruence Theorem. Corresponding parts of congruent
triangles are congruent, so CD = AE.
_____________________________________________________________________
31. Explain why m∠NMO = 20°.
32. Calculate MR given that the perimeter of HMR is 60 centimeters.
Name: ________________________ ID: A
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33. Greta has a summer home on Lake Winnie. Using the diagram, how wide is Lake Winnie near Greta’s
summer home?
34. Jill is building a livestock pen in the shape of a triangle. She is using one side of a barn for one of the sides of
her pen and has already placed posts in the ground at points A, B, and C, as shown in the diagram. If she
places fence posts every 10 feet, how many more posts does she need? (Note: There will be no other posts
placed along the barn wall.)
35. Given rectangle ACDE, calculate the measure of ∠CDB.
Name: ________________________ ID: A
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Isosceles Triangle Theorems: Isosceles Triangle Base Theorem, Vertex Angle Theorem, Perpendicular
Bisector Theorem, Altitude to Congruent Sides Theorem, and Angle Bisector to Congruent Sides
Theorem
Complete each two-column proof.
Given: Isosceles ABC with AB ≅ CB, BD ⊥ AC, DE ⊥ AB, and DF ⊥ CB
Prove: ED ≅ FD
Statements Reasons
1. AB ≅ CB 1. Given
2. BD ⊥ AC, DE ⊥ AB, DF ⊥ CB 2. Given
3. ∠AED and ∠CFD are right angles. 3. Definition of perpendicular lines
4. AED and CFD are right triangles. 4. Definition of right triangles
5. ∠A ≅ ∠C 5. Base Angle Theorem
6. AD ≅ CD 6. Isosceles Triangle Base Theorem
7. AED ≅ CFD 7. HA Congruence Theorem
8. ED ≅ FD 8. CPCTC
_____________________________________________________________________
.
Name: ________________________ ID: A
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36. Given: Isosceles IAE with IA ≅ IE ,AG ⊥ IE, EK⊥IA
Prove: IGA ≅ IKE
Statements Reasons
1.________________________________ 1.________________________________
2.________________________________ 2.________________________________
3.________________________________ 3.________________________________
4.________________________________ 4.________________________________
5.________________________________ 5.________________________________
6.________________________________ 6.________________________________
7.________________________________ 7.________________________________
8.________________________________ 8.________________________________
9.________________________________ 9.________________________________
Direct Proof vs. Indirect Proof: Inverse, Contrapositive, Direct Proof, and Indirect Proof
Vocabulary: Define each term in your own words.
37. inverse
.
38. contrapositive
.
39. direct proof
.
40. indirect proof (or proof by contradiction)
.
41. Hinge Theorem
Name: ________________________ ID: A
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42. Hinge Converse Theorem
.
Direct Proof vs. Indirect Proof: Inverse, Contrapositive, Direct Proof, and Indirect Proof
Problem Set: Write the converse of each conditional statement. Then determine whether the converse
is true.
EXAMPLE
If two lines do not intersect and are not parallel, then they are skew lines.
The converse of the conditional would be:
If two lines are skew lines, then they do not intersect and are not parallel.
The converse is true.
_____________________________________________________________________
43. If two lines are coplanar and do not intersect, then they are parallel lines.
.
44. If a triangle has two sides with equal lengths, then it is an isosceles triangle.
.
45. If the lengths of the sides of a triangle are 3 cm, 4 cm, and 5 cm, then the triangle is a right triangle.
.
46. If the corresponding angles of two triangles are congruent, then the triangles are similar.
Name: ________________________ ID: A
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Direct Proof vs. Indirect Proof: Inverse, Contrapositive, Direct Proof, and Indirect Proof
Write the inverse of each conditional statement. Then determine whether the inverse is true.
EXAMPLE
If a triangle is an equilateral triangle, then it is an isosceles triangle.
The inverse of the conditional would be:
If a triangle is not an equilateral triangle, then it is not an isosceles triangle.
The inverse is not true.
_____________________________________________________________________
47. If a triangle is a right triangle, then the sum of the measures of its acute angles is 90°.
.
48. If a polygon is a triangle, then the sum of its exterior angles is 360°.
.
49. If two angles are complementary, then the sum of their measures is 90°
.
50. If a polygon is a trapezoid, then it is a quadrilateral.
Name: ________________________ ID: A
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Direct Proof vs. Indirect Proof: Inverse, Contrapositive, Direct Proof, and Indirect Proof
Write the contrapositive of each conditional statement. Then determine whether the contrapositive is
true.
EXAMPLE
If one of the acute angles of a right triangle measures 45°, then it is an isosceles right triangle.
The contrapositive of the conditional would be:
* If a triangle is not an isosceles right triangle, then it is not a right triangle with an acute angle that measures
45°.
* The contrapositive is true.
_____________________________________________________________________
51. If one of the acute angles of a right triangle measures 30°, then it is a 30° − 60° − 90° triangle.
.
52. If a quadrilateral is an isosceles trapezoid, then it has two pairs of congruent base angles.
.
53. If two angles are supplementary, then the sum of their measures is 180°.
.
54. If the diameter of a circle is 12 inches, then the radius of the circle is 6 inches.
Name: ________________________ ID: A
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Use the Hinge Theorem or its converse to write an inequality for each pair of triangles.
SP >>>> GQ
55.
56.
Name: ________________________ ID: A
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Use the given information to answer each question.
EXAMPLE
Samantha and Devon are going to meet at the library before school to study for a test. Samantha leaves her
house and walks 5 blocks due north and then 7 blocks northeast to reach the library. Devon walks 7 blocks
northwest from his house and then 5 blocks due west to reach the library. Use the diagram to determine
whose house is a greater distance “as the crow flies” from the library. Explain your reasoning.
Because the included angle from Devon’s house is larger than the included angle from Samantha’s
house, Devon’s house is a greater distance from the library than Samantha’s house.
_____________________________________________________________________
57. A local resident donated two triangular plots of land, a blue spruce tree, and a ginkgo tree to the park district.
The members of the park district board decided that one of the trees would be planted at point A and the other
at point B. Because the blue spruce would grow to be much larger than the ginkgo tree they decided that the
blue spruce should be planted in the plot with the larger angle. Use the diagram to explain why the board
members decided to plant the blue spruce at point B.
Name: ________________________ ID: A
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Direct Proof vs. Indirect Proof: Inverse, Contrapositive, Direct Proof, and Indirect Proof
Create an indirect proof to prove each statement.
EXAMPLE
Given: WY bisects ∠XYZ and XW ≅/ ZW
Prove: XY ≅/ ZY
Statements Reasons
1. XY ≅ ZY 1. Assumption
2. WY bisects ∠XYZ. 2. Given
3. ∠XYW ≅ ∠ZYW 3. Definition of angle bisector
4. YW ≅ YW 4. Reflexive Property of ≅≅≅≅
5. XYW ≅ ZYW 5. SAS Congruence Theorem
6. XW ≅ ZW 6. CPCTC
7.XW ≅/ ZW 7. Given
8. XY ≅ ZY is false. 8. Step 7 contradicts Step 6.The assumption
is false.
9. XY ≅/ ZY is true. 9. Proof by contradiction
_____________________________________________________________________
.
Name: ________________________ ID: A
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58. Given: m∠EBX ≠ m∠EBZ
Prove:EB is not an altitude of EZX.
Statements Reasons
1.________________________________ 1.________________________________
2.________________________________ 2.________________________________
3.________________________________ 3.________________________________
4.________________________________ 4.________________________________
5.________________________________ 5.________________________________
6.________________________________ 6.________________________________
7.________________________________ 7.________________________________
8.________________________________ 8.________________________________
9.________________________________ 9.________________________________
59. Given: ∠OMP ≅ ∠MOP and NP does not bisect ∠ONM.
Prove: NM ≅/ NO
Statements Reasons
1.________________________________ 1.________________________________
2.________________________________ 2.________________________________
3.________________________________ 3.________________________________
4.________________________________ 4.________________________________
5.________________________________ 5.________________________________
6.________________________________ 6.________________________________
7.________________________________ 7.________________________________
8.________________________________ 8.________________________________
9.________________________________ 9.________________________________
ID: A
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Geometry CH 6 Test Review
Answer Section
1. ANS:
Both ∆XQT and ∆PRS are right triangles. If XS ≅ TP, then XT ≅ PS . Therefore, ∆XQT ≅ ∆PRS by the HL
Congruence Theorem.
PTS: 1 REF: Ch6.3 TOP: Pre Test
2. ANS:
In the figure, TR ≅ PR. Side AR is shared by both triangles. All right angles are congruent, so
∠TRA ≅ ∠PRA. Therefore, ∆TAR ≅ ∆PAR by SAS Congruence Theorem.
PTS: 1 REF: Ch6.2 TOP: Pre Test
3. ANS:
Because CL ≅ AE and the triangles are isosceles, the triangles are congruent by SAS. So, CG ≅ AR by
CPCTC. Because ∠CHL and ∠AZE are right angles, ∠CHG and ∠AZR are right angles. By the HL
Congruence Theorem, ∆CHG and ∆AZR are congruent. Therefore, CH ≅ AZ.
PTS: 1 REF: Ch6.5 TOP: Pre Test
4. ANS:
In the figure, AR ≅ TY and AC = TC. Angle RCA and angle YCT are vertical angles, so ∠RCA ≅ ∠YCT. So,
two pairs of sides and a pair of non-included angles are known. Because the SAS Congruence Theorem must
use the included angles, there is not enough information to determine whether the triangles are congruent.
PTS: 1 REF: Ch6.2 TOP: Mid Ch Test
5. ANS: C PTS: 1 REF: Ch6.2 TOP: Skills Practice
6. ANS: B PTS: 1 REF: Ch6.2 TOP: Skills Practice
7. ANS: D PTS: 1 REF: Ch6.2 TOP: Skills Practice
8. ANS: A PTS: 1 REF: Ch6.2 TOP: Skills Practice
9. ANS:
Yes. There is enough information to conclude that EHI ≅ GIH by SSS.
PTS: 1 REF: Ch6.2 TOP: Skills Practice
10. ANS:
No. TRQ might not be congruent to TRP because you do not know that QR ≅ PR or that ∠QTR ≅ ∠PTR.
There is not enough information.
PTS: 1 REF: Ch6.2 TOP: Skills Practice
11. ANS:
Yes. There is enough information to conclude that TUN ≅ RUP by SAS.
PTS: 1 REF: Ch6.2 TOP: Skills Practice
ID: A
2
12. ANS:
Yes. Since it can be shown that AY ≅ XZ, there is enough information to conclude that AUY ≅ XWZ by
ASA.
PTS: 1 REF: Ch6.2 TOP: Skills Practice
13. ANS:
Yes. There is enough information to conclude that HLK ≅ HLI by SAS.
PTS: 1 REF: Ch6.2 TOP: Skills Practice
14. ANS:
No. ACG might not be congruent to EGC, because you do not know that ∠ACG ≅ ∠EGC or that
∠AGC ≅ ∠ECG. There is not enough information.
PTS: 1 REF: Ch6.2 TOP: Skills Practice
15. ANS:
Yes. There is enough information to conclude that FJL ≅ ILJ by SSS.
PTS: 1 REF: Ch6.2 TOP: Skills Practice
16. ANS:
Yes. There is enough information to conclude that MNQ ≅ OPQ by AAS.
PTS: 1 REF: Ch6.2 TOP: Skills Practice
17. ANS: B PTS: 1 REF: Ch6.3 TOP: Skills Practice
18. ANS: D PTS: 1 REF: Ch6.3 TOP: Skills Practice
19. ANS: A PTS: 1 REF: Ch6.3 TOP: Skills Practice
20. ANS: C PTS: 1 REF: Ch6.3 TOP: Skills Practice
21. ANS:
No. DVT might not be congruent to DVU . There is not enough information.
PTS: 1 REF: Ch6.3 TOP: Skills Practice
22. ANS:
Yes. There is enough information to conclude that NMD ≅ EMO by LL.
PTS: 1 REF: Ch6.3 TOP: Skills Practice
23. ANS:
No. GEO might not be congruent to MKI. There is not enough information.
PTS: 1 REF: Ch6.3 TOP: Skills Practice
24. ANS:
Yes. There is enough information to conclude that GHM ≅ UVM by LA.
PTS: 1 REF: Ch6.3 TOP: Skills Practice
ID: A
3
25. ANS:
Statements Reasons
1. ∠E and ∠C are right angles. 1. Given
2. AED and BCD are right triangles. 2. Definition of right triangles
3. Point D is the midpoint of EC. 3. Given
4. ED ≅ CD 4. Definition of midpoint
5. ADB is an isosceles triangle with base AB. 5. Given
6. AD ≅ BD 6. Definition of isosceles triangle
7. AED ≅ BCD 7. HL Congruence Theorem
PTS: 1 REF: Ch6.3 TOP: Skills Practice
26. ANS:
Statements Reasons
1. SU ⊥ UP and TP ⊥ UP 1. Given
2. ∠U and ∠P are right angles. 2. Definition of perpendicular lines
3. SUR and TPR are right triangles. 3. Definition of right triangles
4. UR ≅ PR 4. Given
5. ∠PRT and ∠URS are vertical angles. 5. Definition of vertical angles
6. ∠PRT ≅ ∠URS 6. Vertical Angle Theorem
7. SUR ≅ TPR 7. LA Congruence Theorem
PTS: 1 REF: Ch6.3 TOP: Skills Practice
27. ANS:
Statements Reasons
1. MNWX is a rectangle. 1. Given
2. ∠N and ∠X are right angles. 2. Definition of rectangle
3. MNW and WXM are right triangles. 3. Definition of right triangles
4. MW = WM 4. Reflexive Property of Equality
5. MW ≅ WM 5. Definition of congruent segments
6. ∠NMW ≅ ∠XWM 6. Given
7. MNW ≅ WXM 7. HA Congruence Theorem
PTS: 1 REF: Ch6.3 TOP: Skills Practice
ID: A
4
28. ANS:
Statements Reasons
1. ∠JHK ≅ ∠LHK 1. Given
2. ∠JKH ≅ ∠LKH 2. Given
3. HK ≅ HK 3. Reflexive Property of ≅
4. HJK ≅ HLK 4. ASA Congruence Theorem
5. JK ≅ LK 5. CPCTC
PTS: 1 REF: Ch6.4 TOP: Skills Practice
29. ANS:
Statements Reasons
1. AC ⊥ DB 1. Given
2. ∠DEA is a right angle. 2. Definition of perpendicular lines
3. ∠BEA is a right angle. 3. Definition of perpendicular lines
4. DEA is a right triangle. 4. Definition of right triangle
5. BEA is a right triangle. 5. Definition of right triangle
6. AC bisects DB. 6. Given
7. DE ≅ BE 7. Definition of bisect
8. AE ≅ AE 8. Reflexive Property of ≅
9. DEA ≅ BEA 9. Leg-Leg Congruence Theorem
10. AD ≅ AB 10. CPCTC
PTS: 1 REF: Ch6.4 TOP: Skills Practice
30. ANS:
Statements Reasons
1. AT ≅ AQ 1. Given
2. ∠T ≅ ∠Q 2. Base Angle Theorem
3. AC bisects ∠TAQ. 3. Given
4. ∠TAC ≅ ∠QAC 4. Definition of bisect
5. TAC ≅ QAC 5. ASA Congruence Theorem
6. TC ≅ QC 6. CPCTC
7. AC bisects TQ. 7. Definition of bisect
PTS: 1 REF: Ch6.4 TOP: Skills Practice
ID: A
5
31. ANS:
Using QMN and the Base Angle Theorem, m∠MNO = 60°. Using PMO and the Base Angle Theorem,
m∠POM = 80°. Since ∠POM and ∠MON are supplementary, m∠MON = 100°. Since the sum of the
measures of the angles in a triangle is 180°, m∠NMO = 20°.
PTS: 1 REF: Ch6.4 TOP: Skills Practice
32. ANS:
MR = 20 cm. Using the Base Angle Converse Theorem, MR = HR. Solve the perimeter equation
x + x + 20 = 60, where x = MR and x = HR. So, x = 20.
PTS: 1 REF: Ch6.4 TOP: Skills Practice
33. ANS:
The two triangles in the diagram are congruent by the Hypotenuse-Leg Congruence Theorem. Using CPCTC,
Lake Winnie is 48 meters wide.
PTS: 1 REF: Ch6.4 TOP: Skills Practice
34. ANS:
Eight posts are needed to complete the fence. Using the Base Angle Converse Theorem, AC = 50 ft. She will
need four more posts for side AC and four more posts for side BC.
PTS: 1 REF: Ch6.4 TOP: Skills Practice
35. ANS:
The measure of ∠CDB = 60°. Using the Base Angle Theorem, m∠BDE = 30°. Since ACDE is a rectangle,
m∠CDE = 90°. So m∠CDB = m∠CDE − m∠BDE, or 90° − 30°.
PTS: 1 REF: Ch6.4 TOP: Skills Practice
36. ANS:
Statements Reasons
1. IA ≅ IE 1. Given
2. AG ⊥ IE,EK ⊥ IA 2. Given
3. ∠IGA and ∠IKE are right angles. 3. Definition of perpendicular lines
4. IGA and IKE are right triangles. 4. Definition of right triangle
5. AG ≅ EK 5. Isos. Triangle Altitude to Congruent Sides
Theorem
6. IGA ≅ IKE 6. HL Congruence Theorem
PTS: 1 REF: Ch6.5 TOP: Skills Practice
37. ANS:
The inverse of the conditional statement “If p, then q”, is the statement “If not p, then not q.”
PTS: 1 REF: Ch6.6 TOP: Skills Practice
38. ANS:
The contrapositive of the conditional statement “If p, then q”, is the statement “If not q, then not p.”
PTS: 1 REF: Ch6.6 TOP: Skills Practice
ID: A
6
39. ANS:
A direct proof is a proof that begins with the given information and works to the desired conclusion directly
through the use of givens, definitions, properties, postulates, and theorems.
PTS: 1 REF: Ch6.6 TOP: Skills Practice
40. ANS:
An indirect proof, or proof by contradiction, is a proof that uses the contrapositve. If you prove the
contrapositive true, then the statement is true.
PTS: 1 REF: Ch6.6 TOP: Skills Practice
41. ANS:
If two sides of one triangle are congruent to two sides of another triangle and the included angle of the first
pair is larger than the included angle of the second pair, then the third side of the first triangle is longer than
the third side of the second triangle.
PTS: 1 REF: Ch6.6 TOP: Skills Practice
42. ANS:
If two sides of one triangle are congruent to two sides of another triangle and the third side of the first
triangle is longer than the third side of the second triangle, then the included angle of the first pair of sides is
larger than the included angle of the second pair of sides.
PTS: 1 REF: Ch6.6 TOP: Skills Practice
43. ANS:
The converse of the conditional would be:
If two lines are parallel lines, then they are coplanar and do not intersect.
The converse is true.
PTS: 1 REF: Ch6.6 TOP: Skills Practice
44. ANS:
The converse of the conditional would be:
If a triangle is an isosceles triangle, then it has two sides with equal lengths.
The converse is true.
PTS: 1 REF: Ch6.6 TOP: Skills Practice
45. ANS:
The converse of the conditional would be:
If a triangle is a right triangle, then the lengths of its sides are 3 cm, 4 cm, and 5 cm.
The converse is not true.
PTS: 1 REF: Ch6.6 TOP: Skills Practice
ID: A
7
46. ANS:
The converse of the conditional would be:
If two triangles are similar, then the corresponding angles of the two triangles are congruent.
The converse is true.
PTS: 1 REF: Ch6.6 TOP: Skills Practice
47. ANS:
The inverse of the conditional would be:
If a triangle is not a right triangle, then the sum of the measures of its acute angles is not 90°.
The inverse is true.
PTS: 1 REF: Ch6.6 TOP: Skills Practice
48. ANS:
The inverse of the conditional would be:
If a polygon is not a triangle, then the sum of its exterior angles is not 360°.
The inverse is not true.
PTS: 1 REF: Ch6.6 TOP: Skills Practice
49. ANS:
The inverse of the conditional would be:
If two angles are not complementary, then the sum of their measures is not 90°.
The inverse is true.
PTS: 1 REF: Ch6.6 TOP: Skills Practice
50. ANS:
The inverse of the conditional would be:
If a polygon is not a trapezoid, then it is not a quadrilateral.
The inverse is not true.
PTS: 1 REF: Ch6.6 TOP: Skills Practice
51. ANS:
The contrapositive of the conditional would be:
If a triangle is not a 30° − 60° − 90° triangle, then it is not a right triangle with an acute angle that measures
30°.
The contrapositive is true.
PTS: 1 REF: Ch6.6 TOP: Skills Practice
52. ANS:
The contrapositive of the conditional would be:
If a quadrilateral does not have two pairs of congruent base angles, then it is not an isosceles trapezoid.
The contrapositive is true.
PTS: 1 REF: Ch6.6 TOP: Skills Practice
ID: A
8
53. ANS:
The contrapositive of the conditional would be:
If the sum of the measures of two angles is not 180°, then the angles are not supplementary.
The contrapositive is true.
PTS: 1 REF: Ch6.6 TOP: Skills Practice
54. ANS:
The contrapositive of the conditional would be:
If the radius of a circle is not 6 inches, then the diameter of the circle is not 12 inches.
The contrapositive is true.
PTS: 1 REF: Ch6.6 TOP: Skills Practice
55. ANS:
AD > RQ
PTS: 1 REF: Ch6.6 TOP: Skills Practice
56. ANS:
m∠Z > m∠I
PTS: 1 REF: Ch6.6 TOP: Skills Practice
57. ANS:
Because both triangular plots have a pair of sides measuring 2000 yards and 3000 yards, the Hinge Converse
Theorem guarantees that the angle opposite the longer of the two remaining sides is the larger angle.
Therefore, the blue spruce should be planted at point B which is opposite the 4500 yard side.
PTS: 1 REF: Ch6.6 TOP: Skills Practice
58. ANS:
Statements Reasons
1. EB is an altitude of EZX. 1. Assumption
2. ∠EBX and ∠EBZ are right angles. 2. Definition of altitude
3. ∠EBX ≅ ∠EBZ 3. Right Angles Congruence Theorem
4. m∠EBX = m∠EBZ 4. Definition of congruent angles
5. m∠EBX ≠ m∠EBZ 5. Given
6. EB is an altitude of EZX is false. 6. Step 5 contradicts Step 4.The assumption is
false.
7. EB is not an altitude of EZX is true. 7. Proof by contradiction
PTS: 1 REF: Ch6.6 TOP: Skills Practice
ID: A
9
59. ANS:
Statements Reasons
1. NM ≅ NO 1. Assumption
2. ∠OMP ≅ ∠MOP 2. Given
3. NP does not bisect ∠ONM . 3. Given
4.MP ≅ OP 4. Isosceles Triangle Base Angle
Converse Theorem
5.PN ≅ PN 5. Reflexive Property of ≅
6. ONP ≅ MNP 6. SSS Congruence Theorem
7.∠ONP ≅ ∠MNP 7. CPCTC
8.NP bisects ∠ONM . 8. Definition of angle bisector
9.NM ≅ NO is false. 9. Step 8 contradicts Step 3.
The assumption is false.
10.NM ≅/ NO is true. 10. Proof by contradiction
PTS: 1 REF: Ch6.6 TOP: Skills Practice