Right Angle: An angle whose measure is 90. Straight Angle: An - - PowerPoint PPT Presentation

Connect Proofs to Section 2-4: Special Pairs of Angles

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

Given: 1 and 2 are complementary Prove: ABC is a right angle. A B C 1 2

Statements Reasons

• 1. 1 and 2 are complementary 1. Given
• 2. m1 + m2 = 90
• 2. Definition of

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

• 5. 3 and 4 are supplementary.
• 1. Given
• 4. m3 + m4 = 180
• 2. Definition of a straight

angle

• 3. m3 + m4 = mDEF
• 3. Angle Addition Postulate
• 2. mDEF= 180
• 4. Substitution
• 1. mDEF is a straight angle.
• 5. Definition of

supplementary angles

Vertical Angle Theorem:

Vertical Angles are Congruent.

Hypothesis: Two angles are vertical angles. Conclusion: The angles are congruent. Conditional: If two angles are vertical angles, then the angles are congruent.

Given: Prove:

Vertical Angle Theorem Proof

Given: 1 and 2 are vertical angles. Prove: 1 @ 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!!

1 3 2 4

Vertical Angle Theorem Proof

Given: Diagram Below Prove: 1 @ 2 1 3 2 4

Reasons Statements

• 1. m1 + m3 = 180

m3 + m2 = 180

• 1. Angle Addition Postulate
• 2. m1 + m3 = m3 + m2
• 2. Substitution

**. m3 = m3 **. Reflexive Property

• 4. m1 = m2
• 4. Subtraction Property

Proof Example Given: 2 @ 3;

Prove: 1 @ 4

1 3 2 4

Reasons Statements

• 1. 2 @ 3
• 1. Given
• 2. 2 @ 1
• 3. 1 @ 3
• 4. 3 @ 4
• 5. 4 @ 1
• 2. Vertical Angles are Congruent
• 4. Vertical Angles are Congruent
• 3. Substitution
• 5. Substitution

You can also say “Vertical Angle Theorem”

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

Proof Example Given: 1 and 2 are supplementary; 3 and 4 are supplementary; 2 @ 4 Prove: 1 @ 3

1 3 2 4

• 1. 1 and 2 are supplementary

3 and 4 are supplementary

• 1. Given
• 2. m1 + m2 = 180

m3 + m4 = 180

• 2. Definition of Supplementary

Angles

• 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

Reasons Statements