1. 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, A  A
SYMMETRIC
If A  B, then B  A
TRANSITIVE
If A  B and B  C, then A  C

2. 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
B C A C B

3. m A = m B Definition of congruent angles
Transitive Property of Angle Congruence
GIVEN
A B,
B C
PROVE
A C
Statements
Reasons
A  B, Given
B  C 1 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

4. 1 3 Transitive property of Congruence
Using the Transitive Property
This two-column proof uses the Transitive Property.
GIVEN
m 3 = 40°, 1 2, 2 3
PROVE
m 1 = 40°
Statements
Reasons
Given
m 3 = 40°, 1 2,
2 3 1 2
1 3 Transitive property of Congruence 3
m 1 = m 3 Definition of congruent angles 4
m 1 = 40° Substitution property of equality

5. 1 and 2 are right angles 1 2 You can prove Theorem 2.3 as shown.
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

6. 1 and 2 are right angles Given
Proving Theorem 2.3
GIVEN
1 and 2 are right angles
PROVE
1 2
Statements
Reasons
1 and 2 are right angles Given 1 2
m 1 = 90°, m 2 = 90° Definition of right angles 3
m 1 = m 2 Transitive property of equality 4
1  Definition of congruent angles

7. 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

8. If m 1 + m 2 = 180° m 2 + m 3 = 180° and then 1  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. 3 1 2 3 1
If m 1 + m 2 = 180°
m 2 + m 3 = 180°
and
then
1  3

9. 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 6 4

10. If m 4 + m 5 = 90° m 5 + m 6 = 90° and then 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. 6 5 4 6 4
If m 4 + m 5 = 90°
m 5 + m 6 = 90°
and
then
4  6

11. 1 and 2 are supplements Given
Proving Theorem 2.4
GIVEN
1 and 2 are supplements
3 and 4 are supplements
1 4
PROVE
2 3
Statements
Reasons 1
1 and 2 are supplements Given
3 and 4 are supplements
1  4 2
m 1 + m 2 = 180° Definition of supplementary angles
m 3 + m 4 = 180°

12. m 1 + m 2 = Transitive property of equality m 3 + m 4
Proving Theorem 2.4
GIVEN
1 and 2 are supplements
3 and 4 are supplements
1 4
PROVE
2 3
Statements
Reasons 3
m 1 + m 2 = Transitive property of equality
m 3 + m 4 4
m 1 = m 4 Definition of congruent angles 5
m 1 + m 2 = Substitution property of equality
m 3 + m 1

13. m 2 = m 3 Subtraction property of equality
Proving Theorem 2.4
GIVEN
1 and 2 are supplements
3 and 4 are supplements
1 4
PROVE
2 3
Statements
Reasons 6
m 2 = m 3 Subtraction property of equality
2 3 Definition of congruent angles 7

14. m 1 + m 2 = 180° PROPERTIES OF SPECIAL PAIRS OF ANGLES POSTULATE
POSTULATE 12 Linear Pair Postulate
If two angles for m a linear pair, then they are supplementary.
m 1 + m 2 = 180°

15. 1 3, 2 4 THEOREM Vertical angles are congruent Proving Theorem 2.6
THEOREM 2.6 Vertical Angles Theorem
Vertical angles are congruent
1 3, 2 4

16. 5 and 6 are a linear pair, Given 6 and 7 are a linear pair
Proving Theorem 2.6
GIVEN
5 and 6 are a linear pair,
6 and 7 are a linear pair
PROVE
5 7
Statements
Reasons 1
5 and 6 are a linear pair, Given
6 and 7 are a linear pair 2
5 and 6 are supplementary, Linear Pair Postulate
6 and 7 are supplementary 3
5 7 Congruent Supplements Theorem