Notes 2.4

2.4 Special Pairs of Angles Objective- Find the measures of complementary, supplementary and vertical angles.

Complementary Angles • 2 angles whose sum is 90º

30° x 90 - x

60° 01/23/13

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Definition of Complementary • Given: ∠1 and ∠2 are complements • Your statement will say…

– m ∠1 + m ∠2 = 90

• Your reason will be…

– Def of comp.

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Supplementary Angles โ ข 2 angles whose sum is 180ยบ

180 - x

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x

4

Definition of Supplementary • Given: ∠3 and ∠4 are supplements • Your statement will say…

– m ∠3 + m ∠4 = 180

• Your reason will be…

– Def of supp.

• Caution: AAP vs. Def. of Supp 01/23/13

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Example 1 • A supplement of an angle is 3 times as large as the complement. Find the measure of the angle.

• Let x = angle measure • Then 180 – x = supplement • And 90 – x = complement • 180 – x = 3(90 – x) • x = 45 01/23/13

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Vertical Angles • The sides of one angle are opposite rays to the sides of the other angle

1 4

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2 3

7

VAC • Theorem: Vertical angles are congruent.

1 7

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2 6

3 5

4

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Proof!

• Given: ∠1 and ∠2 are vertical angles.

• Prove: ∠1 = ∠2 1 2

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Statements

Reason

1. ∠1 and ∠2 are vertical 2. m∠1 + m ∠3 = 180 m∠2 + m∠3 = 180 3. m∠1 + m ∠3 = m∠2 + m∠3 4. m∠3 = m∠3 5. m∠1 = m∠2

1. Given 2. AAP 3. Subst.

4. Reflexive 5. Subtraction

1 2

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Example 2 • Solve for x.

3x + 8

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6x - 22

11

Example 3 • Find each angle measure.

x + 20 4x - 15 01/23/13

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Example 4 • Find the value of x.

3x

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6x

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Homework…

• Page 52-53 • 1-22, 28, 29

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