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Home » Quadrangles » Parallelograms » Parallelograms: Consecutive Angles are Supplementary

Parallelograms: Consecutive Angles are Supplementary

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

One of the basic properties of parallelograms is that any pair of consecutive angles are supplementary. This property will be very useful in many problems involving parallelograms. We'll prove this property using one of the theorems about parallel lines - the Consecutive Interior Angles Theorem.

Problem

ABCD is a parallelogram. Show that the pairs of consecutive angles are supplementary.

properties of parallelograms

Strategy

The definition of a parallelogram is that both pairs of opposing sides are parallel.

Therefore, it's a simple use of the properties of parallel lines to show that the consecutive angles are supplementary.

We have already proven that for the general case of parallel lines, a transversal line creates interior angles that sum up to 180°.

But, a parallelogram is simply two pairs of parallel lines. So, let's apply the above theorem to each pair of sides. In other words, the two opposing sides will be used as the parallel lines. And, we'll use one of the other sides as the transversal line. In conclusion, doing this for each one of the pairs of sides gives the required proof.

Proof

Here's how to prove that parallelograms consecutive angles are supplementary:

(1) AB||CD                                    //Given, definition of a parallelogram
(2) m∠ABC + m∠DCB = 180°     // consecutive interior angles between 2 parallel lines
(3) AD||BC                                   //Given, see (1)
(4) m∠BAD + m∠CDA = 180°     // consecutive interior angles between 2 parallel lines

« Parallelograms: The Two Pairs of Opposite Angles are Congruent
Calculating the Height of Tall Objects »

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…

Geometry Topics

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    • Triangle Inequalities

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