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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
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
What you know
REASONS
Postulated, definitions,
theorems, properties, etc.
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3Subtraction prop. of Equality (SPE)
m1 = m2
Simplify
1  2
Vertical Angle Theorem
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 + 3
Vertical Angles Theorem
4x = 2x + 104
Addition Property of Equality
2x = 104
Subtraction Property of Equality
x = 52
Division Property of Equality
The vertical angles, as we found, measure 107º
107
107
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
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
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
m1 + m2 = 180
m3 + m2 = 180
Def of Sup. s
m1 + m2 = m3 + m2
Substitution
m1 + m2 - m2 = m3 + m2 - m2Subtraction prop. of Equality (SPE)
m1 = m3
Simplify
1  3
Congruence Supplement Theorem
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
1 and 2 are complementary
3 and 2 are complementary
m1 + m2 = 90
m3 + m2 = 90
m1 + m2 = m3 + m2
REASON
Given
Def of Comp s
Substitution
m1 + m2 - m2 = m3 + m2 - m2SPE
1  3
Simplify
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
m1 + m2 = 180
m3 + m2 = 180
Def of Sup. s
m1 + m2 = m3 + m2
Substitution
m1 + m2 - m2 = m3 + m2 - m2Subtraction prop. of Equality (SPE)
m1 = m3
Simplify
1  3
Congruence Supplement Theorem