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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.
TRANSITIVE If A B and B C, then A C
SYMMETRIC If A B, then B A
REFLEX IVE For any angle A, A A
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.
GIVEN A B, PROVE A C
AB
C
B C
Transitive Property of Angle Congruence
GIVEN A B,
B C
PROVE A C
Statements Reasons
1
2
3
4
m A = m B Definition of congruent angles
5 A C Definition of congruent angles
A B, GivenB C
m B = m C Definition of congruent angles
m A = m C Transitive property of equality
Using the Transitive Property
This two-column proof uses the Transitive Property.
Statements Reasons
2
3
4
m 1 = m 3 Definition of congruent angles
GIVEN m 3 = 40°, 1 2, 2 3
PROVE m 1 = 40°
1
m 1 = 40° Substitution property of equality
1 3 Transitive property of Congruence
Givenm 3 = 40°, 1 2,2 3
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 2 are right angles
PROVE 1 2
Proving Theorem 2.3
Statements Reasons
1
2
3
4
m 1 = 90°, m 2 = 90° Definition of right angles
m 1 = m 2 Transitive property of equality
1 2 Definition of congruent angles
GIVEN 1 and 2 are right angles
PROVE 1 2
1 and 2 are right angles Given
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.
1 233
If m 1 + m 2 = 180°m 2 + m 3 = 180°
and1
then
1 3
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.
45
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
If m 4 + m 5 = 90°
m 5 + m 6 = 90°
andthen
4 6
566
4
Proving Theorem 2.4
Statements Reasons
1
2
GIVEN 1 and 2 are supplements
PROVE 2 3
3 and 4 are supplements
1 4
1 and 2 are supplements Given
3 and 4 are supplements
1 4
m 1 + m 2 = 180° Definition of supplementary angles
m 3 + m 4 = 180°
Proving Theorem 2.4
Statements Reasons
3
GIVEN 1 and 2 are supplements
PROVE 2 3
3 and 4 are supplements
1 4
4
5 m 1 + m 2 = Substitution property of equality
m 3 + m 1
m 1 + m 2 = Transitive property of equalitym 3 + m 4
m 1 = m 4 Definition of congruent angles
Proving Theorem 2.4
Statements Reasons
GIVEN 1 and 2 are supplements
PROVE 2 3
3 and 4 are supplements
1 4
6
7
m 2 = m 3 Subtraction property of equality
2 3 Definition of congruent angles
POSTULATE
POSTULATE 12 Linear Pair Postulate
If two angles for m a linear pair, then they are supplementary.
m 1 + m 2 = 180°
PROPERTIES OF SPECIAL PAIRS OF ANGLES
Proving Theorem 2.6
THEOREM
THEOREM 2.6 Vertical Angles Theorem
Vertical angles are congruent
1 3, 2 4
Proving Theorem 2.6
PROVE 5 7
GIVEN 5 and 6 are a linear pair,6 and 7 are a linear pair
1
2
3
Statements Reasons
5 and 6 are a linear pair, Given6 and 7 are a linear pair
5 and 6 are supplementary, Linear Pair Postulate6 and 7 are supplementary
5 7 Congruent Supplements Theorem