2.5. Proofs about Angle Pairs and Segments http://www.ck12.org
Solution:By the Right Angle Theorem,^6 C∼=^6 F. Also,^62 ∼=^6 3 by the Same Angles Supplements Theorem.^61
and^6 2 are a linear pair, so they add up to 180◦.^6 3 and^6 4 are also a linear pair and add up to 180◦. Because^61 ∼=^6 4,
we can substitute^6 1 in for^6 4 and then^6 2 and^6 3 are supplementary to the same angle, making them congruent.
This is an example of aparagraph proof. Instead of organizing the proof in two columns, you explain everything
in sentences.
Same Angle Complements Theorem: If two angles are complementary to the same angle then the angles are
congruent.
So, ifm^6 A+m^6 B= 90 ◦andm^6 C+m^6 B= 90 ◦, thenm^6 A=m^6 C. Using numbers, we could say that if^6 Ais
supplementary to an angle measuring 56◦, thenm^6 A= 34 ◦.^6 Cis also supplementary to 56◦, so it too is 34◦.
Therefore,m^6 A=m^6 C.
The proof of the Same Angles Complements Theorem is in the Review Questions. Use the proof of the Same Angles
Supplements Theorem to help you.
Vertical Angles Theorem
Recall the Vertical Angles Theorem from Chapter 1. We will do a formal proof here.
Given: Lineskandmintersect.
Prove:^61 ∼=^6 3 and^62 ∼=^64
TABLE2.22:
Statement Reason- Lineskandmintersect Given
2.^6 1 and^6 2 are a linear pair
(^6) 2 and (^6) 3 are a linear pair
(^6) 3 and (^6) 4 are a linear pair
Definition of a Linear Pair
3.^6 1 and^6 2 are supplementary
(^6) 2 and (^6) 3 are supplementary
(^6) 3 and (^6) 4 are supplementary
Linear Pair Postulate
4.m^61 +m^62 = 180 ◦
m^62 +m^63 = 180 ◦