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Converse of Parallel Lines Theorem - Concept

Teacher/Instructor Brian McCall
Brian McCall

Univ. of Wisconsin
J.D. Univ. of Wisconsin Law school

Brian was a geometry teacher through the Teach for America program and started the geometry program at his school

If two lines are intersected by a transversal, then alternate interior angles, alternate exterior angles, and corresponding angles are congruent. The converse of the theorem is true as well. If two corresponding angles are congruent, then the two lines cut by the transversal must be parallel. Similarly, if two alternate interior or alternate exterior angles are congruent, the lines are parallel.

So the question is, if we have two lines that might be parallel and they're intersected by a transversal, can we do the converse of the parallel lines theorem? Which says, if we have alternate interior angles or alternate exterior angles, or corresponding angles that are congruent, is that enough to say that these two lines are parallel? And as we read right here, yes it is. If two lines and a transversal form alternate interior angles, notice I abbreviated it, so if these alternate interior angles are congruent, that is enough to say that these two lines must be parallel.
If you have alternate exterior angles. That is these two angles right here that are alternate exterior, if those two are congruent, you don't even need to know about these interior ones. That's enough to say that they're parallel.
And finally, corresponding angles. If you have one pair of corresponding angles that are congruent you can say these two lines must be parallel. So the converse of the parallel lines there is true.

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