Honors Math 2
Vertical Angles Theorem (Section 6.05)
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Date:
Theorem: When two lines intersect, the angles formed that
are opposite each other are congruent. (Vertical Angles Theorem)
A
D
P
B
C
Proof:
1.
2.
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3.
4.
Statement
∠CPA and ∠APD are a linear pair
∠BPD and ∠APD are a linear pair
∠CPA is supplementary to ∠APD
∠BPD
is supplementary to ∠APD
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m∠CPA
+ m∠APD = 180°
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m∠BPD + m∠APD
€ = 180°
∠CPA + ∠APD ≅ ∠BPD + ∠APD
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Reason
1. Assumed from diagram
2. Definition of linear pair
3. Definition of supplementary
4. Transitive Property
5. ∠APD ≅ ∠APD
5. Reflexive Property
6. ∠CPA ≅ ∠DPB
6. Subtraction Property
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N
L
Now that we have proven this fact about vertical angles,
we can use it to prove other things.
M
Given: LM ≅ RM, NM ≅ QM
R
Prove: ∠L ≅ ∠R
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Q
Statement
1. LM ≅ RM, NM ≅ QM
1. Given
2. ∠LMN ≅ ∠RMQ
2. Vertical Angles Theorem
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3. ΔLMN ≅ ΔRMQ
3. SAS Congruence Postulate
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4. ∠L ≅ ∠R
4. CPCTC
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Reason
C
Let’s try some other proofs.
Given: AC ≅ BC, AD ≅ BD
Prove: CD bisects ∠ACB
A
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Statement
€ 1. AC ≅ BC,
€ AD ≅ BD
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B
D
Reason
1. Given
2. 𝐶𝐷 ≅ 𝐶𝐷
2. Reflexive Property
3. ∆𝐴𝐶𝐷 ≅ ∆𝐵𝐶𝐷
3. SSS Congruence Postulate
4. ∠𝐴𝐶𝐷 ≅ ∠𝐵𝐶𝐷
4. CPCTC
5. CD bisects ∠ACB
5. Definition of Bisect
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V
Given: WX ≅ VW , YZ ≅ VW
Prove: WY ≅ XZ
W
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Statement
€ 1. WX ≅ VW , YZ ≅ VW
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X
Y
Reason
1. Given
2. 𝑊𝑋 ≅ 𝑌𝑍
2. Transitive Property
3. 𝑋𝑌 ≅ 𝑋𝑌
3. Reflexive Property
4. 𝑊𝑌 ≅ 𝑋𝑍
4. Addition Property
Z
Homework
1) State the postulate, definition, or theorem that justifies the statement about the diagram.
A.
B.
Q
1 2
C
B
P
A
E
3 4
T
R
F
1. If E is the midpoint of AF , then AE ≅ EF
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2. If AE ≅ EF , then E is the midpoint of AF
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3. If ∠BEC ≅ ∠CEF , then EC bisects ∠BEF
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4. If EC is the bisector of ∠𝐵𝐸𝐹, then
∠BEC ≅ ∠CEF
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5. ∠AEB + ∠BEC = ∠AEC
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6. AE + EF = AF
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S
7. If PR ⊥ QS then ∠4 is a right angle.
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8. If ∠PSR is a right angle, then PS ⊥ SR
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9. If ∠3 is a right angle, then ∠3 = 90
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10. If ∠PSR = 90 , then ∠PSR is a right angle.
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11. If ∠3 and ∠4 are supplementary angles, then
∠3 + ∠4 = 180∘
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2) On last night’s homework, you did the following proof:
Here are a few related theorems that are useful shortcuts to problems like this.
a. If angles are supplementary to the same angle, then they are congruent.
b. If angles are supplementary to congruent angles, then they are congruent.
c. If angles are complementary to the same angle, then they are congruent.
d. If angles are complementary to congruent angles, then they are congruent.
Choose one of these theorems and write a proof (on a separate sheet of paper). Note that you
will need to set up the proof on your own. Draw a picture, decide on the given information, and
decide the statement that you are trying to prove.
3) On last night’s homework, you did the problem below. Rewrite the proof using one of the
theorems above.
Given: ∠1 ≅ ∠4
Prove: ∠2 ≅ ∠3
4) Prove the following theorem (on a separate piece of paper):
Angles that are both supplementary and congruent are right angles.
5) Do the following problem in the textbook: p. 461 #6