Overview
This lecture covers two-column proofs involving angle congruence and supplementary angles, demonstrating how to logically prove relationships between angles using key geometric properties.
Two-Column Proof: Vertical Angles
- Given: Angle 1 is congruent to Angle 2.
- Statement: Angle 2 and Angle 3 are vertical angles.
- Reason: Vertical angles are congruent.
- Conclusion: Angle 1 is congruent to Angle 3 by the transitive property.
Two-Column Proof: Supplements of Congruent Angles
- Given: Angle 1 supplementary to Angle 2; Angle 3 supplementary to Angle 4; Angle 1 is congruent to Angle 4.
- If Angle 1 = 100°, then Angle 4 = 100° (since they are congruent).
- Angle 2 and Angle 3 each measure 80° (supplements to 100°).
- Conclusion: Angle 2 is congruent to Angle 3.
- Reason: Supplements of congruent angles are congruent.
Two-Column Proof: Angle Addition and Subtraction
- Given: Angle BAC is congruent to Angle BCA; Angle 2 is congruent to Angle 4.
- Definition: Congruent angles have equal measures.
- Angle BAC = Angle 1 + Angle 2; Angle BCA = Angle 3 + Angle 4 (Angle Addition Postulate).
- Since angle measures are equal, set Angle 1 + Angle 2 = Angle 3 + Angle 4 (Substitution Property).
- Since Angle 2 = Angle 4, subtract these from both sides to get Angle 1 = Angle 3 (Subtraction Property).
- Conclusion: Angle 1 is congruent to Angle 3 (Definition of Congruent Angles).
Key Terms & Definitions
- Congruent Angles — Angles that have equal measures.
- Vertical Angles — Angles formed opposite each other when two lines cross; always congruent.
- Supplementary Angles — Two angles whose measures sum to 180 degrees.
- Transitive Property — If angle A ≅ angle B and angle B ≅ angle C, then angle A ≅ angle C.
- Angle Addition Postulate — The measure of a larger angle is the sum of its non-overlapping parts.
- Substitution Property — Replacing one quantity with an equal value in an equation.
- Subtraction Property — Subtracting equal amounts from both sides of an equation produces equality.
Action Items / Next Steps
- Practice writing two-column proofs for angle congruence and supplementary relationships.
- Review definitions and properties of angles for upcoming assignments.