Chapter 4 – Triangles and Congruence
4.1
Answer Key
Triangle Sum Theorem
Answers 1.
m ∠1 = 41°
2.
m ∠1 = 86°
3.
m ∠1 = 61°
4.
m ∠1 = 51°
5.
m ∠1 = 13°
6.
m ∠1 = 60°
7.
m ∠1 = 70°
8.
84°
9.
57°
10.
21°
11.
x = 14°
12.
x = 9°
13.
x = 22°
14.
x = 17°
15.
x = 12°
CK12 Geometry Concepts
1
Answer Key
Chapter 4 – Triangles and Congruence
4.2
Exterior Angles Theorems
Answers 1.
m ∠1 = 118°
2.
m ∠1 = 68°
3.
m ∠1 = 116°
4.
m ∠1 = 161°
5.
m ∠1 = 141°
6.
m ∠1 = 135°
7.
180°
8.
360°
9.
360°
10.
x = 30°
11.
x = 25°
12.
x = 7°
13.
Answers will vary. You should draw from the “Triangle TearUp” and the proof of the Triangle Sum Theorem in the previous concept (Triangle Sum Theorem).
14.
The exterior angles are all linear pairs with the interior angles of a triangle.
15.
Suppose the three exterior angles are a , b , and c , for x , y , and z , respectively. Then we know that
because they are all linear
pairs (#14). Adding these three equations together, we have: . Substitute
(Triangle Sum Theorem), and
simplify:
CK12 Geometry Concepts
2
Chapter 4 – Triangles and Congruence
Answer Key
CK12 Geometry Concepts
3
Chapter 4 – Triangles and Congruence
4.3
Answer Key
Congruent Triangles
Answers 1.
Yes
2.
Yes
3.
Yes
4.
No
5.
No
6.
Yes
7.
No
8.
Yes
9.
No
10.
Yes
11.
No
12.
Yes
13.
Yes
14.
No
15.
No
CK12 Geometry Concepts
4
Chapter 4 – Triangles and Congruence
4.4
Answer Key
Congruence Statements
Answers 1.
Δ FGH ≅ Δ KLM
2.
so we cannot say that the triangles are congruent. No, we do not know if AC ≅BD ,
3.
Δ ABE ≅ Δ DCE
4.
No, we only know that one pair of sides and one pair of angles are congruent. This is not enough information to determine if the triangles are congruent.
5.
Line up the congruent sides: Δ BCD ≅ Δ ZYX
6.
Corresponding Parts of Congruent Triangles are Congruent or CPCTC.
7.
∠ A
8.
9.
10.
11.
∠ I
12.
∠ W
13.
14.
15.
∠ E
CK12 Geometry Concepts
5
Chapter 4 – Triangles and Congruence
4.5
Answer Key
Third Angle Theorem
Answers 1.
88°
2.
42°
3.
50°
4.
42°
5.
47°
6.
43°
7.
47°
8.
37°
9.
Lines are not marked parallel, so we cannot assume that m ∠ HIJ = 108°.
10.
35°
11.
Lines are not marked parallel, so we cannot assume that m ∠ IHJ = 37°.
12.
55°
13.
63°
14.
62°
15.
63°
CK12 Geometry Concepts
6
Answer Key
Chapter 4 – Triangles and Congruence
4.6
SSS Triangle Congruence
Answers 1.
Yes, ΔD EF ≅ Δ HJI , SSS
2.
No, one triangle is SSS and the other is SAS.
3.
Yes, ΔA BC ≅ Δ FED , SSS
4.
Yes, ΔA TD ≅ Δ ETD , SSS
5.
6.
7. Statement 1. B is the midpoint of 2.
3.
Reason ,
4. ΔABD ≅ ΔABC
Given Definition of a Midpoint Reflexive PoC SSS
8.
Triangle #1: (8, 0), (0, 3), and (5, 9) Side #1 = √73 , Side #2 = √90 , Side #3 = √61 Triangle #2: (2, 5), (10, 2), and (5, 4) Side #1 = √90 , Side #2 = √73 , Side #3 = √61 The triangles are congruent by SSS.
CK12 Geometry Concepts
7
Chapter 4 – Triangles and Congruence 9.
Answer Key
Triangle #1: (7, 2), (1, 6), and (4, 5) Side #1 = √80 , Side #2 = √130 , Side #3 = √10 Triangle #2: (1, 10), (5, 2), and (8, 1) Side #1 = √80 , Side #2 = √130 , Side #3 = √10 The triangles are congruent by SSS.
10.
Δ ABC : AB = √18 , AC = √58 , BC = √52 , Δ DEF : DF = √17 , DE = √58 , EF = √137 The triangles are not congruent because not all the corresponding sides are congruent.
11.
Δ ABC : AB = √37 , AC = √45 , BC = √34 , Δ DEF : DE = √37 , DF = √45 , EF = √34 The triangles are congruent by SSS.
12.
Δ ABC : AB = √18 , AC = √2 , BC = 4, Δ DEF : DE = √13 , DF = √37 , EF = √18 Triangles are not congruent.
13.
Δ ABC : AB = √26 , AC = √10 , BC = √20 , Δ DEF : DE = √26 , DF = √10 , EF = √20 The triangles are congruent by SSS.
14.
Answers will vary. We do not know the angle between the two sides or the third side, so those dimensions are not fixed, leading to several possible triangles.
15.
If the two triangles are similar (same shape, not necessarily same size), we would only need to know that one pair of corresponding sides is congruent.
CK12 Geometry Concepts
8
Answer Key
Chapter 4 – Triangles and Congruence
4.7
SAS Triangle Congruence
Answers 1.
No, these are both SSA, which is not a congruence postulate.
2.
Yes, ΔA BC ≅ Δ YXZ , SAS
3.
No, these are both SSA, which is not a congruent postulate
4.
∠ C ≅ ∠ G
5.
∠ C ≅ ∠ K
6.
7. Statement 1. B is a midpoint of 2.
,
Reason
Given Definition of a midpoint
3. ∠ ABD and ∠ ABC are right angles
⊥ lines create 4 right angles
4. ∠ ABD ≅ ∠ ABC
All right angles are ≅
5.
6. Δ ABD ≅ Δ ABC
Reflexive PoC SAS
CK12 Geometry Concepts
9
Chapter 4 – Triangles and Congruence
Answer Key
CK12 Geometry Concepts
10
Answer Key
Chapter 4 – Triangles and Congruence 8. Statement 1.
Reason
is an angle bisector of ∠ DAC ,
2. ∠ DAB ≅ ∠ BAC 3.
Given Definition of an Angle Bisector Reflexive PoC
4. Δ ABD ≅ Δ ABC
SAS
9. Statement 1. B is the midpoint of
Reason and
,
Given
∠ ABE is a right angle 2.
Definition of a Midpoint
3. m ∠ ABE = 90°
Definition of a Right Angle
4. m ∠ ABE = m ∠ DBC
Vertical Angle Theorem
5. Δ ABE ≅ Δ CBD
SAS
10.
Statement 1.
is the angle bisector of ∠ ADC ,
2. ∠ ADB ≅ ∠ BDC 3.
CK12 Geometry Concepts
Reason
Given Definition of an Angle Bisector Reflexive PoC
11
Answer Key
Chapter 4 – Triangles and Congruence 4. Δ ABD ≅ Δ CBD
SAS
CK12 Geometry Concepts
12
Answer Key
Chapter 4 – Triangles and Congruence
For questions 1115, answers will vary. Below is one possibility per question. 11.
12.
13.
14.
15.
CK12 Geometry Concepts
13
Answer Key
Chapter 4 – Triangles and Congruence
4.8
ASA and AAS Triangle Congruence
Answers 1.
Yes, AAS, ΔA BC ≅ FDE
2.
Yes, ASA, ΔA BC ≅ Δ IHG
3.
No, the triangles have congruent parts that would be SSA. This is not a congruence theorem.
4.
∠ DBC ≅ ∠ DBA because they are both right angles and created by perpendicular lines.
5.
∠ CDB ≅ ∠ ADB
6.
DB ≅DB from the Reflexive Property. We have enough to say that the triangles are congruent by ASA.
7.
Statement 1.
,
Reason Given
is the angle bisector of ∠ CDA 2. ∠ DBC and ∠ ADB are right angles
Definition of perpendicular
3. ∠ DBC ≅ ∠ ADB
All right angles are ≅
4. ∠ CDB ≅ ∠ ADB
Definition of an angle bisector
5.
CK12 Geometry Concepts
Reflexive PoC
14
Answer Key
Chapter 4 – Triangles and Congruence 6. Δ CDB ≅ Δ ADB
ASA
CK12 Geometry Concepts
15
Answer Key
Chapter 4 – Triangles and Congruence 8.
∠ C ≅ ∠ A by CPCTC (corresponding parts of congruent triangles are congruent)
9.
∠ L ≅ ∠ O and ∠ P ≅ ∠ N by the Alternate Interior Angles Theorem.
10.
∠ LMP ≅ ∠ NMO by the Vertical Angles Theorem.
11.
There is more than one correct proof, this is one possible answer. Statement 1.
,
Reason Given
2. ∠ L ≅ ∠ O , ∠ P ≅ ∠ N
Alternate Interior Angles Theorem
3. Δ LMP ≅ ∠ OMN
ASA
12.
CPCTC
13.
Start with the proof from #11 and continue. Statement 1.
,
Reason Given
2. ∠ L ≅ ∠ O , ∠ P ≅ ∠ N
Alternate Interior Angles
3. Δ LMP ≅ ∠ OMN
ASA
4.
CPCTC
5. M is the midpoint of
.
Definition of a midpoint
14.
∠ A ≅ ∠ N
15.
∠ C ≅ ∠ M
16.
CK12 Geometry Concepts
16
Chapter 4 – Triangles and Congruence 17.
Answer Key
LM ≅MO or LP ≅N O
CK12 Geometry Concepts
17
Chapter 4 – Triangles and Congruence
4.9
Answer Key
HL Triangle Congruence
Answers 1.
2.
3. 4.
Given/Definition of Perpendicular Lines
5.
6.
7.
HL Congruence
8.
No
9.
Yes, by SAS
10.
No
11.
Yes, by HL
12.
No
13.
No
14.
Yes, by SSS
15.
No
CK12 Geometry Concepts
18
Answer Key
Chapter 4 – Triangles and Congruence
4.10
Isosceles Triangles
Answers 1.
x = 13°
2.
y = 16°
3.
x = 1
4.
y = 3
5.
x = 4°, y = 11°
6.
True
7.
False, only in an isosceles right triangle.
8.
False, only in the case of an equilateral triangle.
9.
True
10. Statement 1. Isosceles Δ CIS , with base angles ∠ C and ∠S
Reason Given
is the angle bisector of ∠CIS
2. ∠ C ≅ ∠ S
Base Angles Theorem
3. ∠ CIO ≅ ∠ SIO
Definition of an Angle Bisector
4.
≅
Reflexive PoC
5. Δ CIO ≅ Δ SIO 6.
ASA CPCTC
7. ∠ IOC ≅ ∠ IOS
CPCTC
8. ∠ IOC and ∠ IOS are supplementary
Linear Pair Postulate
9. m ∠ IOC = m ∠ IOS = 90°
Congruent Supplements Theorem
CK12 Geometry Concepts
19
Answer Key
Chapter 4 – Triangles and Congruence 10.
is the perpendicular bisector of
Definition of a ⊥ bisector (Steps 6 and 9)
CK12 Geometry Concepts
20
Answer Key
Chapter 4 – Triangles and Congruence 11.
Statement
Reason
1. Isosceles Δ ICS with ∠ C and ∠ S , perpendicular bisector of
is the
2. ∠ C ≅ ∠ S 3.
Given
Base Angle Theorem
Definition of a ⊥ bisector
4. m ∠IOC = m ∠IOS = 90°
Definition of a ⊥ bisector
5. Δ CIO ≅ Δ SIO
ASA
6. ∠ CIO ≅ ∠ SIO
CPCTC
7.
is the angle bisector of ∠ CIS
Definition of an Angle Bisector
12.
Side #1 = √18 , Side #2 = √18 , Side #3 = 6 This is an isosceles triangle because Side #1 and Side #2 are equal.
13.
Side #1 = √17 , Side #2 = √40 , Side #3 = √9 No sides are equal, so this is a scalene triangle.
14.
Side #1 = √72 , Side #2 = √234 , Side #3 = √162 No sides are equal, so this is a scalene triangle.
15.
Side #1 = √104 , Side #2 = √208 , Side #3 = √104 This is an isosceles triangle because Side #1 and Side #3 are equal.
CK12 Geometry Concepts
21
Chapter 4 – Triangles and Congruence 16.
Answer Key
Side #1 = 8 , Side #2 = √65 , Side #3 = √65 This is an isosceles triangle because Side #2 and Side #3 are equal.
CK12 Geometry Concepts
22
Chapter 4 – Triangles and Congruence
4.11
Answer Key
Equilateral Triangles
Answers 1.
x = 60°
2.
y = 68°
3.
x = 1.5
4.
y = 17
5.
z = 17
6.
n = 25°
7.
8.
x = 3, y = 2
9.
x = 2, y = 5
10.
z = 4
11.
a = −1 , 6
12.
m = −4 °, 15°
13.
x = 4, −1
14. 15.
x = 25°, y = 19°
CK12 Geometry Concepts
23
