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?

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

Investigation 1 Use the following Investigation to help you test the converse of the Corresponding Angles Postulate. You will need patty papers.

Investigation 1 Draw 2 intersecting lines on a patty paper and copy these lines onto a second patty paper. Slide the top copy so that the two transversals stay lined up.

Investigation 1 Trace the lines and the angles from the bottom original onto the top copy. When you do this, you are constructing sets of congruent corresponding, alternate interior, and alternate exterior angles.

Investigation 1 Do the lines appear to be parallel? You can test this with the right angle of a patty paper to see if the distance between the two lines remains the same. Let’s test this another way, but first let’s learn how to do a basic compass and straightedge construction.

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?

Copying an Angle Draw angle A.

Copying an Angle Draw a ray with endpoint A’.

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.

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.

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

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’.

Copying an Angle Draw ray A’C’.

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

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

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

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

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

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

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.

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.

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

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

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

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

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

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.

Assignment P. 165-169: 2-8 even, 9, 19-22, 27, 28, 31, 38, 39, 40, 42, 44, 46, 47 Challenge Problems