CK-12 Geometry - Second Edition

http://www.ck12.org Chapter 2. Reasoning and Proof

Proof of the Right Angle Theorem

Given:^6 Aand^6 Bare right angles

Prove:^6 A∼=^6 B

TABLE2.20:

Statement Reason

1.^6 Aand^6 Bare right angles Given
2.m^6 A= 90 ◦andm^6 B= 90 ◦ Definition of right angles
3.m^6 A=m^6 B Transitive PoE
4.^6 A∼=^6 B ∼=angles have = measures

This theorem may seem redundant, but anytime right angles are mentioned, you need to use this theorem to say the
angles are congruent.

Same Angle Supplements Theorem: If two angles are supplementary to the same angle then the angles are
congruent.

So, ifm^6 A+m^6 B= 180 ◦andm^6 C+m^6 B= 180 ◦, thenm^6 A=m^6 C. Using numbers to illustrate, we could say
that if^6 Ais supplementary to an angle measuring 56◦, thenm^6 A= 124 ◦.^6 Cis also supplementary to 56◦, so it too
is 124◦. Therefore,m^6 A=m^6 C. This example, however, does not constitute a proof.

Proof of the Same Angles Supplements Theorem

Given:^6 Aand^6 Bare supplementary angles.^6 Band^6 Care supplementary angles.

Prove:^6 A∼=^6 C

TABLE2.21:

Statement Reason

1.^6 Aand^6 Bare supplementary^6 Band^6 C are
supplementary

Given

2.m^6 A+m^6 B= 180 ◦

m^6 B+m^6 C= 180 ◦ Definition of supplementary angles

3.m^6 A+m^6 B=m^6 B+m^6 C Substitution PoE

4.m^6 A=m^6 C Subtraction PoE

5.^6 A∼=^6 C ∼=angles have = measures

Example 1:Given that^61 ∼=^6 4 and^6 Cand^6 Fare right angles, show which angles are congruent.