Connect Proofs to Section 2-4: Special Pairs of Angles 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.)
Given: 1 and 2 are complementary A Prove: ABC is a right angle. 1 2 B C Statements Reasons 1. 1 and 2 are complementary 1. Given 2. Definition of 2. m 1 + m 2 = 90 Complementary Angles 3. m 1 + m 2 = m ABC 3. Angle Addition Postulate 4. m ABC = 90 4. Substitution 5. ABC is a right angle. 5. Definition of a right angle.
Given: DEF is a straight angle. Prove: 3 and 4 are supplementary 3 4 D E F Statements Reasons 1. Given 1. m DEF is a straight angle. 2. m DEF= 180 2. Definition of a straight angle 3. m 3 + m 4 = m DEF 3. Angle Addition Postulate 4. Substitution 4. m 3 + m 4 = 180 5. Definition of 5. 3 and 4 are supplementary. supplementary angles
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. Conclusion: The angles are congruent. Prove:
Vertical Angle Theorem Proof Given: 1 and 2 are vertical angles. Prove: 1 @ 2 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!!
Vertical Angle Theorem Proof Prove: 1 @ 2 Given: Diagram Below 1 3 4 2 Statements Reasons 1. m 1 + m 3 = 180 1. Angle Addition Postulate m 3 + m 2 = 180 2. m 1 + m 3 = m 3 + m 2 2. Substitution **. m 3 = m 3 **. Reflexive Property 4. Subtraction Property 4. m 1 = m 2
Proof Example Given: 2 @ 3; Prove: 1 @ 4 YOU CANNOT UNDER 1 4 ANY CIRCUMSTANCES 2 3 Statements Reasons USE THE REASON 1. 2 @ 3 1. Given “DEFINITION OF 2. Vertical Angles are Congruent 2. 2 @ 1 VERTICAL ANGLES” 3. 1 @ 3 3. Substitution IN A PROOF!! You can also say 4. 3 @ 4 4. Vertical Angles are Congruent “Vertical Angle Theorem” 5. 4 @ 1 5. Substitution
Given: Prove: 1 @ 3 Proof Example 1 and 2 are supplementary; 1 2 3 and 4 are supplementary; 2 @ 4 4 3 Statements Reasons 1. 1 and 2 are supplementary 1. Given 3 and 4 are supplementary 2. m 1 + m 2 = 180 2. Definition of Supplementary Angles m 3 + m 4 = 180 3. m 1 + m 2 = m 3 + m 4 3. Substitution 4. 2 @ 4 or m 2 = m 4 4. Given 5. m 1 = m 3 or 1 @ 3 5. Subtraction Property
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