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Home » Quadrangles » Parallelograms » Opposite Sides of a Parallelogram are Equal

Opposite Sides of a Parallelogram are Equal

Last updated: May 1, 2024 by Ido Sarig · This website generates income via ads and uses cookies · Terms of use · Privacy policy

A parallelogram is defined as a quadrilateral where the two opposite sides are parallel. We will show that in that case, they are also equal to each other.

Problem

ABCD is a parallelogram, AD||BC and AB||DC.  Show that AD = BC and that AB = DC.

parallelogram with diagonal

Strategy

Once again, since we are trying to show line segments are equal, we will use congruent triangles. Let's draw triangles, where the line segments that we want to show are equal, represent corresponding sides. We will do it by drawing one of the diagonals of the parallelogram, as above.

If we can show that ΔABD and  ΔCDB are congruent, we'll have what we need.

So, what can we use to show these two triangles are congruent?

We know this is a parallelogram so the two opposite sides are parallel, and the diagonal acts as a transversal line, intersecting both pairs of parallel lines - hinting we should use the Alternate Interior Angles Theorem.

Proof

Here's how you prove that opposite sides of a parallelogram are equal:

(1) ABCD is a parallelogram  //Given
(2) AD || BC                                 //From the definition of a parallelogram
(3) ∠ADB ≅ ∠CBD                      //Alternate Interior Angles Theorem
(4) AB || DC                                 //From the definition of a parallelogram
(5) ∠ABD ≅ ∠CDB                      //Alternate Interior Angles Theorem
(6) BD= BD                                 // Common side, reflexive property of equality
(7) ΔABD ≅ ΔCDB                      // (3), (6), (5) Angle-Side-Angle postulate
(8) AD=BC                                // Corresponding sides in congruent triangles (CPCTC)
(9) AB=DC                                // Corresponding sides in congruent triangles (CPCTC)

« Right Triangles: Median to the Hypotenuse is Equal to Half the Hypotenuse
Parallelograms: The Two Pairs of Opposite Angles are Congruent »

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…

Geometry Topics

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