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

Consecutive Interior Angles Converse Theorem

Last updated: Oct 5, 2019 by Ido Sarig · This website generates income via ads and uses cookies · Terms of use · Privacy policy

In today's lesson, we will show a simple method for proving the Consecutive Interior Angles Converse Theorem.

The Consecutive Interior Angles Theorem states that the consecutive interior angles on the same side of a transversal line intersecting two parallel lines are supplementary (That is, their sum adds up to 180).

Here we will prove its converse of that theorem. We will show that if the consecutive interior angles on the same side of a transversal line intersecting two lines are supplementary, then the two lines are parallel.

Problem

m∠5 + m∠4 = 180°, prove that AB||CD

Consecutive Interior Angles Theorem in Geometry

Strategy

This is a converse theorem. A converse of a theorem is a statement formed by interchanging what is given in a theorem and what is to be proved.

So it makes sense to start by looking at what we did in the original theorem. Then, we will try to replicate that.

In proving the original theorem, we relied on the fact that a linear pair of angles are supplementary. Let's do that here, too: m∠1 + m∠4 = 180° as a linear pair, m∠5 + m∠4 = 180° is given, so m∠5=m∠1, and by the converse of the corresponding angles theorem, the lines are parallel.

And this is how you prove the Consecutive Interior Angles Converse Theorem!

Proof

(1) m∠5 + m∠4 = 180° //From the problem statement
(2) m∠1 + m∠4 = 180° // Linear pair of angles are supplementary
(3) m∠5=m∠1 // (1), (2) , transitive property of equality
(4) AB||CD // Converse of the corresponding angles theorem

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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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