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