1. Section 2-5: Proving Angles Congruent SPI 32E: solve problems involving complementary, supplementary, congruent, vertical or adjacent angles given angle measures expressed algebraically.
Objectives:
Prove and apply theorems about angles

2. Using Deductive Reasoning to show a Conjecture is True
Deductive Reasoning (logical)
Reason from a given statement to produce a conclusion
Real-world examples:
Doctors diagnose a patients illness
Carpenters to determine what materials are needed for a job.
Proof
Set of steps you take to show a conjecture is true
Theorem
The statement that you prove to be true
Format of a Proof to derive a Theorem (Side by Side)
Given: What you know
Prove: What you will show to be true, based on known information.
STATEMENT REASONS
What you know Postulated, definitions,
theorems, properties, etc.

3. Prove the Vertical Angles Theorem
Theorem 2-1: Vertical Angles Theorem
Vertical angles are congruent.
Given: 1 and 2 are vertical angles
Prove: 1  2
STATEMENT REASON
1 and 2 are vertical angles
Def. of vertical angles
m1 + m3 = 180
Angle Addition Postulate
m2 + m3 = 180
Angle Addition Postulate
m1 + m3 = m2 + m3
Substitution
m1 + m3 - m3 = m2 + m3 - m3
Subtraction prop. of Equality (SPE)
m1 = m2
Simplify
1  2
Vertical Angle Theorem

4. Use Theorem 2-1 (Vertical Angle Theorem)
to solve problems since it is proven.
Find the value of x.
The angles with labeled measures are vertical angles because their sides are opposite rays. Apply the Vertical Angles Theorem to find x.
Problem Reason
4x – 101 = 2x Vertical Angles Theorem
4x = 2x Addition Property of Equality
2x = Subtraction Property of Equality
x = Division Property of Equality

5. The vertical angles, as we found, measure 107º
What is the measure of the other pair of vertical angles?
73º
HELP!!
How do you know?
(What Def, postulate, theorem….?)
Def: Adjacent angles are supplementary
and vertical angles are congruent

6. Prove the Congruence Supplements Theorem
(Vertical Angle Thm is a special case of this Thm)
Theorem 2-2: Congruence Supplement Theorem
If two angles are supplements of the same angle (or of
congruent angles), then the two angles are congruent.
What do you know based on the definition of:
Supplementary Angles?
Two angles whose measures have a sum of 180
Congruent Angles?
Two angles have the same measure

7. Prove Theorem 2-2: Congruence Supplement Theorem
If two angles are supplements of the same angle (or of
congruent angles), then the two angles are congruent.
Given: 1 and 2 are supplementary
3 and 2 are supplementary
Prove: 1  3
STATEMENT REASON
1 and 2 are supplementary
3 and 2 are supplementary
Given
Def of Sup. s
m1 + m2 = 180
m3 + m2 = 180
m1 + m2 = m3 + m2
Substitution
m1 + m2 - m2 = m3 + m2 - m2
Subtraction prop. of Equality (SPE)
m1 = m3
Simplify
1  3
Congruence Supplement Theorem

8. Prove the Congruence Complements Theorem
Do Now
Prove the Congruence Complements Theorem
Theorem 2-3
If two angles are complements of
the same angle (or of congruent angles),
then the two angles are congruent.
Given: 1 and 2 are complementary
3 and 2 are complementary
Prove: 1  3
STATEMENT REASON
1 and 2 are complementary
3 and 2 are complementary
Given
m =
+ m2 = 90 m
Def of Comp s
m3
Substitution
m1 + m2 = m3 + m2
m1 + m2 - m2 = m3 + m2 - m2
SPE
Simplify
1  3

9. Prove Theorem 2-2: Congruence Supplement Theorem
If two angles are supplements of the same angle (or of
congruent angles), then the two angles are congruent.
Given: 1 and 2 are supplementary
3 and 2 are supplementary
Prove: 1  3
STATEMENT REASON
1 and 2 are supplementary
3 and 2 are supplementary
Given
Def of Sup. s
m1 + m2 = 180
m3 + m2 = 180
m1 + m2 = m3 + m2
Substitution
m1 + m2 - m2 = m3 + m2 - m2
Subtraction prop. of Equality (SPE)
m1 = m3
Simplify
1  3
Congruence Supplement Theorem