2.7 Prove Angle Pair Relationships

Objectives

Write proofs involving supplementary and complementary angles

Write proofs involving congruent and right angles

Theorems & Postulates

Theorem 2.3 (Right Angles Theorem) All right angles are congruent.

Theorem 2.4 ( Supplement Theorem) If 2 angles are supplementary to the same angle (or congruent angles), then they are congruent.

Theorem 2.5 ( Complement Theorem )If 2 angles are complementary to the same angle (or congruent angles), then they are congruent.

Theorems & Postulates

Postulate 12 ( Linear Pair Postulate)If 2 angles form a linear pair, then they are supplementary

Theorem 2.6 (Vertical Angles Theorem) Vertical angles are congruent.

Given: form a linear pair.

Prove:

In the figure, form a linear pair, and Prove that are congruent.

andand

Example 1:

1. Given1.

3. Definition of supplementary angles

3.

4. Subtraction Property4.

5. Substitution5.

6. Definition of congruent angles

6.

Proof:

Statements Reasons

2. Linear pairs are supplementary.

2.

Example 1:

1 & 4 linear pair;

In the figure, NYR and RYA form a linear pair,AXY and AXZ form a linear pair, and RYA and AXZ are congruent. Prove that RYN and AXY are congruent.

Your Turn:

Proof:

Statements Reasons

1. Given

2. If two s form a linear pair, then they are suppl. s.

3. Given

4.

1.

2.

3.

4.

linear pairs.

Your Turn:

Substitution

Add 2d to each side.

Add 32 to each side.

Divide each side by 3.

If 1 and 2 are vertical angles and m1 and m2 find m1 and m2.

Vertical Angles Theorem1 2

Definition of congruent anglesm1 m2

Example 2:

Answer: m1 = 37 and m2 = 37

Example 2:

Answer: mA = 52; mZ = 52

find andIf and are vertical angles and and

Your Turn:

Supplement Theorem

Subtraction Property

Answer: 14

form a linear pair and find If and

Example 3:

Answer: 28

are complementary angles and .

andIffind

Your Turn:

Assignment

Geometry: Pg. 127 – 131

#3 - 29, 38, 42, 46