Warm-Up

1. Warm-Up What is the converse of the Corresponding Angles Postulate? If two parallel lines are cut by a transversal, then pairs of corresponding angles are congruent. Is this converse necessarily true?

2. 3.3 Prove Lines are Parallel Objectives: • To use angle pair relationships to prove that two lines are parallel

3. Copying an Angle Draw angle A on your paper. How could you copy that angle to another part of your paper using only a compass and a straightedge?

4. Copying an Angle • Draw angle A.

5. Copying an Angle • Draw a ray with endpoint A’.

6. Copying an Angle • Put point of compass on A and draw an arc that intersects both sides of the angle. Label these points B and C.

7. Copying an Angle • Put point of compass on A’ and use the compass setting from Step 3 to draw a similar arc on the ray. Label point B’ where the arc intersects the ray.

8. Copying an Angle • Put point of compass on B and pencil on C. Make a small arc.

9. Copying an Angle • Put point of compass on B’ and use the compass setting from Step 5 to draw an arc that intersects the arc from Step 4. Label the new point C’.

10. Copying an Angle • Draw ray A’C’.

11. Copying an Angle Click on the button to watch a video of the construction.

12. Constructing Parallel Lines Now let’s apply the construction for copying an angle to create parallel lines by making congruent corresponding angles.

13. Constructing Parallel Lines • Draw line l and point P not on l.

14. Constructing Parallel Lines • Draw a transversal through point P intersecting line l.

15. Constructing Parallel Lines • Copy the angle formed by the transversal and line l at point P.

16. Constructing Parallel Lines Click on the image to watch a video of the construction.

17. Proving Lines Parallel Converse of Corresponding Angles Postulate If two lines are cut by a transversal so that corresponding angles are congruent, then the lines are parallel. Converse of Alternate Interior Angles Theorem If two lines are cut by a transversal so that alternate interior angles are congruent, then the lines are parallel.

18. Proving Lines Parallel Converse of Alternate Exterior Angles Theorem If two lines are cut by a transversal so that alternate exterior angles are congruent, then the lines are parallel. Converse of Consecutive Interior Angles Theorem If two lines are cut by a transversal so that consecutive interior angles are supplementary, then the lines are parallel.

19. Example 1 Can you prove that lines a and b are parallel? Explain why or why not. No, not enough information Yes, alt. ext angles are congruent Yes, corresponding angles are congruent

20. Example 2 Find the value of x that makes m||n. x=24

21. Example 3 Prove the Converse of the Alternate Interior Angles Theorem. Given: Prove: This proof has the angles numbered differently, but you get the idea

22. Example 4 Given: 1 and 3 are supplementary Prove: Do this is your notebook. You can do it. I BELIEVE in you!!

23. Example 5 Find the values of x and y so that l||m. x=15 y=10

24. Oh, My, That’s Obvious! Transitive Property of Parallel Lines If two lines are parallel to the same line, then they are parallel to each other.