# 2.7 Prove Angle Pair Relationships - PowerPoint PPT Presentation

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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.

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2.7 Prove Angle Pair Relationships

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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.

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

and

and

Given: form a linear pair.

Prove:

### Example 1:

Proof:

Statements Reasons

1.

1. Given

2.

2. Linear pairs are supplementary.

3.

3. Definition of supplementary angles

4.

4. Subtraction Property

5.

5. Substitution

6.

6. Definition of congruent angles

### 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 AXYare congruent.

Proof:

Statements Reasons

1.

1. Given

linear pairs.

2.

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

3.

3.Given

4.

4.

If 1 and 2are vertical angles and m1 andm2 find m1 and m2.

1

2

Vertical Angles Theorem

m1

m2

Definition of congruent angles

### Example 2:

Substitution

Divide each side by 3.

### Example 2:

Answer: m1 = 37 and m2 = 37

If and are vertical angles and

and

find and

form a linear pair and

If

and

find

### Example 3:

Supplement Theorem

Subtraction Property

If

are complementary angles and .

and

find