2.6 Proving Statements about Angles

Properties of Angle Congruence

Reflexive For any angle, A <A <A.

Symmetric If <A <B, then <B <A.

Transitive If <A <B and <B <C, then <A <C.

Right Angle Congruence Theorem

• All right angles are congruent.

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.

.

.

A

B C

X

Y Z

Congruent Supplements Theorem

• If two angles are supplementary to the same angle, then they are congruent– If m<1 + m<2 = 180° and m<2 + m<3 = 180°,

then m<1 = m<3 or 31

Congruent Complements Theorem

• If two angles are complementary to the same angle, then the two angles are congruent.– If m<4 + m<5 = 90° and m<5 + m<6 = 90°,

then m<4 = m<6 or 64

Linear Pair Postulate

• If two angles form a linear pair, then they are supplementary.

1 2

m<1 + m<2 = 180°

Example:

• < 1 and < 2 are a linear pair.If m<1 = 78°, then find m<2.

Vertical Angles Theorem

• Vertical angles are congruent.

1

23

4

42,31

Example

<1 and <2 are complementary angles.<1 and <3 are vertical angles.If m<3 = 49°, find m<2.

Proving the Right Angle Congruence Theorem

Given: Angle 1 and angle 2 are right anglesProve: 1 2

4. 1 2

1. Givensrightand '21.1 902901.2 and

21.3 mm

2. Def. of right ’s

3. Trans. POE

4. Def. of ’s

Statements Reasons

Proving the Vertical Angles Theorem

56

7

Given: 5 and 6 are a linear pair. 6 and 7 are a linear pair.

3. 5 7

1. Given

3. Supplements Theorem

Prove: 5 7

1. 5 and 6 are a linear pair. 6 and 7 are a linear pair.

2. 5 and 6 are supplementary.6 and 7 are supplementary.

2. Linear Pair Postulate

Statements Reasons

Solve for x.

Give a reason for each step of the proof. Choose from the list of reasons given.

Given: 6 7 Prove: 5 8

Plan for Proof: First show that 5 6 and 7 8. Then use transitivity to show that 5 8.)

1. Given

4. Vertical ’s Theorem

2. Vertical ’s Theorem

Statements Reasons

1. 6 7

4. 5 6

2. 7 8

3. 6 8 3. Trans. POC

5. 5 8 5. Trans. POC