2-4 Special Pairs of Angles & Proofs.

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Presentation transcript:

2-4 Special Pairs of Angles & Proofs

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.)

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.

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.

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:

Vertical Angle Theorem Proof Prove: Ð1 @ 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!!

Given: Ð1 and Ð2 are vertical angles. Vertical Angle Theorem Proof Prove: Ð1 @ 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 Ð1 @ Ð2 4. Subtraction 4. mÐ1 = mÐ2 4. Subtraction Property 5. Ð1 @ Ð2 5. Definition of @ Angles.

Given: Ð2 @ Ð3; Prove: Ð1 @ Ð4 1. Ð2 @ Ð3 1. Given 2. Ð2 @ Ð1 EXAMPLE 3 1 3 2 4 Given: Ð2 @ Ð3; Prove: Ð1 @ Ð4 Statements Reasons 1. Ð2 @ Ð3 1. Given You can also say “Vertical Angle Theorem” 2. Ð2 @ Ð1 2. Vertical Angles are Congruent 3. Ð1 @ Ð3 3. Substitution You can also say “Vertical Angle Theorem” 4. Ð3 @ Ð4 4. Vertical Angles are Congruent 5. Ð4 @ Ð1 5. Substitution

YOU CANNOT UNDER ANY CIRCUMSTANCES USE THE REASON “DEFINITION OF VERTICAL ANGLES” IN A PROOF!!

Ð1 and Ð2 are supplementary; Ð3 and Ð4 are supplementary; Ð2 @ Ð4 Given: Ð1 and Ð2 are supplementary; Ð3 and Ð4 are supplementary; Ð2 @ Ð4 Prove: Ð1 @ 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. Ð2 @ Ð4 or mÐ2 = mÐ4 4. Given 5. mÐ1 = mÐ3 or Ð1 @ Ð3 5. Subtraction Property