3.5 Using Properties of Parallel Lines

Example 1: Proving Two Lines are Parallel Goal 1: Using Parallel Lines in Real Life Example 1: Proving Two Lines are Parallel Lines m, n and k represent three of the oars of a team of rowers. Given: m║n and n║k. Prove: m║k. 1 m 2 n 3 k

Proof Statements: m║n 1 ≅ 2 n║k 2 ≅ 3 1 ≅ 3 m║k Reasons: Given Corresponding Angles Postulate Transitive POC Corresponding Angles Converse

Parallel/Perpendicular lines Theorems Theorem 3.11: If two lines are parallel to the same line, then they are parallel to each other. r q p If p║q and q║r, then p║r.

Parallel/Perpendicular lines Theorems Theorem 3.12: In a plane, if two lines are perpendicular to the same line, then they are parallel to each other. If mp and np, then m║n. m n p

Example 2: Why Steps are Parallel In the diagram at the right, each step is parallel to the step immediately below it and the bottom step is parallel to the floor. Explain why the top step is parallel to the floor. k1 k2 k3 k4

Example 2: Why Steps are Parallel (cont.) Solution You are given that k1║ k2 and k2║ k3. By transitivity of parallel lines, k1║ k3. Since k1║ k3 and k3║ k4, it follows that k1║ k4. So the top step is parallel to the floor. k1 k2 k3 k4

Example 3: Building a CD Rack You are building a CD rack. You cut the sides, bottom, and top so that each corner is composed of two 45° angles. Prove that the top and bottom front edges of the CD rack are parallel.

Proof Given: m1 = 45°, m2 = 45°, m3 = 45°, m = 45° Prove: BA║CD B

BCCD BABC BA║CD m3=45°, m4=45° m1=45°, m2=45° mABC = 90° mBCD = m3+ m4 mABC = m1+ m2 Angle Addition Postulate Given Angle Addition Postulate Given mABC = 90° mBCD = 90° Substitution Property Substitution Property ABC is a right angle. BCD is a right angle. Definition of Right Angle Definition of Right Angle BCCD BABC Definition of  lines Definition of  lines BA║CD In a plane, 2 lines to the same line are ║.