Warm-UpWarm-UpWhat 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 Parallel3.3 Prove Lines are ParallelObjectives:

1.To use angle pair relationships to prove that two lines are parallel

Copying an AngleCopying an AngleDraw 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 AngleCopying an Angle1. Draw angle A.

Copying an AngleCopying an Angle2. Draw a ray with endpoint A’.

Copying an AngleCopying an Angle3. 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 AngleCopying an Angle4. 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’ wherethe arc intersects the ray.

Copying an AngleCopying an Angle5. Put point of compass on B and pencil on

C. Make a small arc.

Copying an AngleCopying an Angle6. 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 AngleCopying an Angle7. Draw ray A’C’.

Copying an AngleCopying an AngleClick on the

button to watch a video of the construction.

Constructing Parallel LinesConstructing Parallel LinesNow let’s apply

the construction for copying an angle to create parallel lines by making congruent corresponding angles.

Constructing Parallel LinesConstructing Parallel Lines1. Draw line l and

point P not on l.

Constructing Parallel LinesConstructing Parallel Lines2. Draw a

transversal through point P intersecting line l.

Constructing Parallel LinesConstructing Parallel Lines3. Copy the angle

formed by the transversal and line l at point P.

Constructing Parallel LinesConstructing Parallel LinesClick on the

image to watch a video of the construction.

Proving Lines ParallelProving Lines ParallelConverse of Corresponding Converse of Corresponding

Angles PostulateAngles PostulateIf two lines are cut by a transversal

so that corresponding angles are congruent, then the lines are parallel.

Converse of Alternate Interior Converse of Alternate Interior Angles TheoremAngles Theorem

If two lines are cut by a transversal so that alternate interior angles are congruent, then the lines are parallel.

Proving Lines ParallelProving Lines ParallelConverse of Alternate Exterior Converse of Alternate Exterior

Angles TheoremAngles TheoremIf two lines are cut by a transversal

so that alternate exterior angles are congruent, then the lines are parallel.

Converse of Consecutive Converse of Consecutive Interior Angles TheoremInterior Angles Theorem

If two lines are cut by a transversal so that consecutive interior angles are supplementary, then the lines are parallel.

Example 1Example 1Can you prove that lines a and b are

parallel? Explain why or why not.

Yes, alt. ext angles are congruent

Yes, corresponding angles are congruent

No, not enough information

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

x=24

Example 3Example 3Prove the Converse of the Alternate Interior

Angles Theorem.Given:Prove:

3 6 l m

This proof has the angles numbered differently, but you get the idea

Example 4Example 4Given: 1 and 3 are supplementary

Prove:

2 3 RA TP

Do this is your notebook. You can do it. I BELIEVE in you!!

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

o

n

m

l

10y+2

2x-6

5x+3

15y+6x=15y=10

Oh, My, That’s Obvious!Oh, My, That’s Obvious!Transitive Property of Transitive Property of

Parallel LinesParallel LinesIf two lines are parallel

to the same line, then they are parallel to each other.