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

Consecutive Interior Angles Theorem

Last updated: Mar 27, 2021 by Ido Sarig · This website generates income via ads and uses cookies · Terms of use · Privacy policy

The Theorem

The Consecutive Interior Angles Theorem states that the two interior angles formed by a transversal line intersecting two parallel lines are supplementary (i.e: they sum up to 180°).

Consecutive Interior Angles Theorem

The problem

AB||CD, prove m∠5 + m∠4 = 180°, and that m∠3 + m∠6 = 180°

So how do we go about this? We already know that the two angles that are next to each other and which form a straight line are “Supplementary angles” and their sum is 180°. So we will try to use that here, too, since here we also need to prove that the sum of two angles is  180°.

So let’s proceed to the proof, using what we already know about angles that are next to each other and which form a straight line.

Proof

Here's how you prove the Consecutive Interior Angles Theorem:

(1)    AB||CD                                 //given
(2)    ∠1 ≅ ∠5                                //from the axiom of parallel lines – corresponding angles
(3)    m∠1 = m∠5                        //definition of congruent angles 
(4)    m∠1 + m∠4 = 180°           //straight line measures 180° 
(5)    m∠5 + m∠4 = 180°           //using (3) and (4), and performing algebraic substitution, replacing  m∠1 with the equivalent m∠5

And we can repeat this proof for the second pair of interior angles. We'll replace m∠1 with  m∠2, m∠5 with m∠6 and m∠4 with m∠3

And we have proven the theorem.

« Vertical Angles Theorem
Alternate Interior Angles Theorem »

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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About

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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