1. 3.5 Using Properties of Parallel Lines Geometry Mrs. Spitz Fall 2005

2. Standard/Objectives: Standard 3: Students will learn and apply geometric concepts. Objectives: Use properties of parallel lines in real-life situations, such as building a CD rack. Construct parallel lines using a straight edge and a compass. To understand how light bends when it passes through glass or water.

3. Assignment: Quiz after this section. Pg. 159 – Do constructions on this page “Copying an Angle” and “Parallel Lines.” Place in your binder under computer/lab work. In your homework, you are asked to do constructions on pg. 161 #25-28. Place these in your binder. Pgs. 160-162 #1-24, 33-36.

4. Ex. 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. m n 1 k 2 3

5. Proof: Statements: 1. m ║n 2.  1 ≅  2 3. n ║k 4.  2 ≅  3 5.  1 ≅  3 6. m║k Reasons: 1. Given 2. Corresponding Angles Postulate 3. Given 4. Corresponding Angles Postulate 5. Transitive POC 6. Corresponding Angles Converse

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

7. 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. m n p If m  p and n  p, then m ║n.

8. Ex. 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. k1k1 k2k2 k3k3 k4k4

9. Solution You are given that k 1 ║ k 2 and k 2 ║ k 3. By transitivity of parallel lines, k 1 ║ k 3. Since k 1 ║ k 3 and k 3 ║ k 4, it follows that k 1 ║ k 4. So the top step is parallel to the floor.

10. Ex. 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.

11. Proof: Given: m  1 = 45 °, m  2 = 45 °, m  3 = 45 °, m  = 45 ° Prove: BA║CD 1 2 3 4 B C A D

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

13. Constructions: Do the two constructions on page 159. Label and place in your binder. Do Constructions on page 161 #25- 28. Label and place in your binder.

14. Reminder: Quiz 2 will be the next time we meet. To look at a similar quiz, check out page 164.