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CONGRUENCE OF ANGLES
THEOREM
THEOREM 2.2 Properties of Angle Congruence
Angle congruence is r ef lex ive, sy mme tric, and transitive.
Here are some examples.
REFLEX IVE
For any angle A,
SYMMETRIC If
A 
B, then
TRANSITIVE If
A 
B and
B 
C, then
A 
A
B 
A
A 
C
Transitive Property of Angle Congruence
Prove the Transitive Property of Congruence for angles.
SOLUTION
To prove the Transitive Property of Congruence for angles,
begin by drawing three congruent angles. Label the vertices as
A, B, and C.
A
B
GIVEN
B,
C
PROVE
A
C
B
A
C
Transitive Property of Angle Congruence
A
B
GIVEN
B,
C
Statements
A 
B 
1
PROVE
A
C
Reasons
B,
C
Given
2
m
A=m
B
Definition of congruent angles
3
m
B=m
C
Definition of congruent angles
4
m
A=m
C
Transitive property of equality
5
A 
C
Definition of congruent angles
Using the Transitive Property
This two-column proof uses the Transitive Property.
GIVEN
m
3 = 40°,
PROVE
m
1 = 40°
1
2,
Statements
1
2
m
1
3
3
m
1=m
4
m
1 = 40°
3
Reasons
3 = 40°,
2
3
1
2
2,
Given
Transitive property of Congruence
3
Definition of congruent angles
Substitution property of equality
Proving Theorem 2.3
THEOREM
THEOREM 2.3 Right Angle Congruence Theorem
All right angles are congruent.
You can prove Theorem 2.3 as shown.
GIVEN
1 and
PROVE
1
2 are right angles
2
Proving Theorem 2.3
GIVEN
1 and
PROVE
1
2 are right angles
2
Statements
1 and
1
Reasons
2 are right angles
2
m
1 = 90°, m
3
m
1=m
4
1
2
2
2 = 90°
Given
Definition of right angles
Transitive property of equality
Definition of congruent angles
PROPERTIES OF SPECIAL PAIRS OF ANGLES
THEOREMS
THEOREM 2.4 Congruent Supplements Theorem
If two angles are supplementary to the same angle (or to
congruent angles) then they are congruent.
1
2
3
PROPERTIES OF SPECIAL PAIRS OF ANGLES
THEOREMS
THEOREM 2.4 Congruent Supplements Theorem
If two angles are supplementary to the same angle (or to
congruent angles) then they are congruent.
1
2
3
If m 1 + m 2 = 180° and
m 2 + m 3 = 180° then
1
3
1
PROPERTIES OF SPECIAL PAIRS OF ANGLES
THEOREMS
THEOREM 2.5 Congruent Complements Theorem
If two angles are complementary to the same angle (or to
congruent angles) then the two angles are congruent.
5
4
6
PROPERTIES OF SPECIAL PAIRS OF ANGLES
THEOREMS
THEOREM 2.5 Congruent Complements Theorem
If two angles are complementary to the same angle (or to
congruent angles) then the two angles are congruent.
4
5
4
If m 4 + m 5 = 90° and
m 5 + m 6 = 90° then
4
6
6
Proving Theorem 2.4
GIVEN
1 and 2 are supplements
3 and 4 are supplements
1
4
PROVE
2
3
Statements
1 and 2 are supplements
3 and 4 are supplements
1 4
1
2
Reasons
m
m
1+m
3+m
2 = 180°
4 = 180°
Given
Definition of supplementary angles
Proving Theorem 2.4
GIVEN
1 and 2 are supplements
3 and 4 are supplements
1
4
PROVE
2
Statements
3
Reasons
3
m
m
1+m
3+m
2=
4
Transitive property of equality
4
m
1=m
4
Definition of congruent angles
5
m
m
1+m
3+m
2=
1
Substitution property of equality
Proving Theorem 2.4
GIVEN
1 and 2 are supplements
3 and 4 are supplements
1
4
PROVE
2
Statements
6
7
m
2=m
2
3
3
Reasons
3
Subtraction property of equality
Definition of congruent angles
PROPERTIES OF SPECIAL PAIRS OF ANGLES
POSTULATE
POSTULATE 12 Linear Pair Postulate
If two angles form a linear pair, then they are supplementary.
m
1+m
2 = 180°
Proving Theorem 2.6
THEOREM
THEOREM 2.6 Vertical Angles Theorem
Vertical angles are congruent
1
3,
2
4
Proving Theorem 2.6
GIVEN
PROVE
5 and
6 and
5
6 are a linear pair,
7 are a linear pair
7
Statements
Reasons
1
5 and
6 and
6 are a linear pair,
7 are a linear pair
Given
2
5 and
6 and
6 are supplementary,
7 are supplementary
Linear Pair Postulate
3
5
7
Congruent Supplements Theorem