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Home » Lines and Angles » Parallel Lines » Converse Alternate Interior Angles Theorem

Converse Alternate Interior Angles Theorem

Last updated: Oct 29, 2021 by Ido Sarig · This website generates income via ads and uses cookies · Terms of use · Privacy policy

In today's geometry lesson, we'll prove the converse of the Alternate Interior Angles Theorem.

We have shown that when two parallel lines are intersected by a transversal line, the interior alternating angles and exterior alternating angles are congruent (that is, they have the same measure of the angle.)

We will now show that the opposite is also true. If the interior alternating angles or exterior alternating angles created by the transversal line are congruent, then the two lines intersected by the transversal line are parallel.

Problem

two parallel line segments

Given m∠5 = m∠3 or m∠1 = m∠7 , prove that AB||CD

Strategy

To prove two lines are parallel, we can use the converse of the Corresponding Angles Theorem - if we find a pair of corresponding angles that are congruent, then the two lines are parallel.

So let's do exactly what we did when we proved the Alternate Interior Angles Theorem, but in reverse - going from congruent alternate angles to showing congruent corresponding angles.

Proof

Here's how you prove the converse of the Alternate Interior Angles Theorem:

(1) m∠5 = m∠3 //given

(2) m∠1 = m∠3 //vertical, or opposite angles

(3) m∠1 = m∠5 //using (1) and (2) and transitive property of equality, both equal m∠3

(4) ∠1 ≅ ∠5       //(3), the definition of congruent angles

(5) AB||CD        //converse of the Corresponding Angles Theorem

And similarly, for the exterior pair:

(1) m∠1 = m∠7 //given

(2) m∠5 = m∠7 //vertical, or opposite angles

(3) m∠1 = m∠5 //using (1) and (2) and transitive property of equality, both equal m∠7

(4) ∠1 ≅ ∠5       //(3), the definition of congruent angles

(5) AB||CD        //converse of the Corresponding Angles Theorem

« Prove the Pythagorean Theorem Using Triangle Similarity
Proving that a Quadrilateral is a Parallelogram »

About the Author

Ido Sarig is a high-tech executive with a BSc degree in Computer Engineering. His goal is to help you develop a better way to approach and solve geometry problems. You can contact him at [email protected]

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Welcome to Geometry Help! I'm Ido Sarig, a high-tech executive with a BSc degree in Computer Engineering and an MBA degree in Management of Technology. I'm here to tell you that geometry doesn't have to be so hard! My goal with this website is to help you develop a better way to approach and solve geometry problems, even if spatial awareness is not your strongest quality. Read More…

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