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2-4 Special Pairs of Angles & Proofs

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**REVIEW: VOCABULARY from Section 2-4**

Right Angle: An angle whose measure is 90. Straight Angle: An angle whose measure is 180. Complementary Angles: Two angles whose measures sum to 90. Supplementary Angles: Two angles whose measures sum to 180. Vertical Angles: The two non-adjacent angles that are created by a pair of intersecting lines. (They are across from one another.)

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**EXAMPLE 1 Given: Ð1 and Ð2 are complementary**

Prove: ÐABC is a right angle. A 1 2 B C Statements Reasons 1. Ð1 and Ð2 are complementary 1. Given 2. Definition of Complementary Angles 2. mÐ1 + mÐ2 = 90 3. mÐ1 + mÐ2 = mÐABC 3. Angle Addition Postulate 4. mÐABC = 90 4. Substitution 5. ÐABC is a right angle. 5. Definition of a right angle.

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**Given: ÐDEF is a straight angle. Prove: Ð3 and Ð4 are supplementary 3 **

EXAMPLE 2 Given: ÐDEF is a straight angle. Prove: Ð3 and Ð4 are supplementary 3 4 D E F Statements Reasons 1. mÐDEF is a straight angle. 1. Given 2. Definition of a straight angle 2. mÐDEF= 180 3. mÐ3 + mÐ4 = mÐDEF 3. Angle Addition Postulate 4. mÐ3 + mÐ4 = 180 4. Substitution 5. Definition of supplementary angles 5. Ð3 and Ð4 are supplementary.

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**Given: Prove: Vertical Angle Theorem: Vertical Angles are Congruent.**

Conditional: If two angles are vertical angles, then the angles are congruent. Given: Hypothesis: Two angles are vertical angles. Prove: Conclusion: The angles are congruent. Aside: Would the converse of this theorem work? If two angles are congruent, then the angles are vertical angles. FALSE Counterexample:

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**Vertical Angle Theorem Proof**

Prove: Ð2 Given: Ð1 and Ð2 are vertical angles. 1 3 4 2 NOTE: You cannot use the reason “Vertical Angle Theorem” or “Vertical Angles are Congruent” in this proof. That is what we are trying to prove!!

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**Given: Ð1 and Ð2 are vertical angles.**

Vertical Angle Theorem Proof Prove: Ð2 Given: Ð1 and Ð2 are vertical angles. 1 3 4 2 Statements Reasons 1. Ð1 and Ð2 are vertical Ðs. 1. Given 2. mÐ1 + mÐ3 = 180 mÐ3 + mÐ2 = 180 2. Angle Addition Postulate 3. mÐ1 + mÐ3 = mÐ3 + mÐ2 3. Substitution **. mÐ3 = mÐ3 **. Reflexive Property 4. mÐ1 = mÐ2 and Ð2 4. Subtraction 4. mÐ1 = mÐ2 4. Subtraction Property 5. Ð2 5. Definition Angles.

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**Given: Ð2 @ Ð3; Prove: Ð1 @ Ð4 1. Ð2 @ Ð3 1. Given 2. Ð2 @ Ð1**

EXAMPLE 3 1 3 2 4 Given: Ð3; Prove: Ð4 Statements Reasons 1. Ð3 1. Given You can also say “Vertical Angle Theorem” 2. Ð1 2. Vertical Angles are Congruent 3. Ð3 3. Substitution You can also say “Vertical Angle Theorem” 4. Ð4 4. Vertical Angles are Congruent 5. Ð1 5. Substitution

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**YOU CANNOT UNDER ANY CIRCUMSTANCES USE THE REASON “DEFINITION OF VERTICAL ANGLES”**

IN A PROOF!!

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**Ð1 and Ð2 are supplementary; Ð3 and Ð4 are supplementary; Ð2 @ Ð4 **

Given: Ð1 and Ð2 are supplementary; Ð3 and Ð4 are supplementary; Ð4 Prove: Ð3 1 2 4 3 Statements Reasons 1. Ð1 and Ð2 are supplementary Ð3 and Ð4 are supplementary 1. Given 2. mÐ1 + mÐ2 = 180 mÐ3 + mÐ4 = 180 2. Definition of Supplementary Angles 3. mÐ1 + mÐ2 = mÐ3 + mÐ4 3. Substitution 4. Ð4 or mÐ2 = mÐ4 4. Given 5. mÐ1 = mÐ3 or Ð3 5. Subtraction Property

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