Congruent Opposite Sides of Quadrilaterals are Parallel

In this post, we are going to discuss one basic concepts of quadrilaterals. That is, if we have a quadrilateral, and the opposite sides are congruent, then these opposite sides are parallel.  This is the same if the two opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

Consider the quadrilateral below. The lengths of AB and CD are equal. Also, The lengths of AD and BC are also equal. So, we are going to show that these pairs of sides are parallel.

parallelogram

 

Theorem: If the two pairs of the opposite sides of a quadrilateral are congruent, then they are parallel.

Given: AB = CD  and AD = BC

What we want to show: AB \parallel CD and AD \parallel BC.

Proof

We draw diagonal AC.

From the given BC= AD (Side) and AB = CD (Side).

Also, AC is common to both triangles ABC and CDA (Side)

 

parallelogram proof

Therefore, by the Side-Side-Side or SSS congruence theorem, triangles ABC and CDA are congruent.

In effect, \angle BCA \cong \angle ACB.

So, BC \parallel AD because they are corresponding angles of the two triangles.

Using the same argument, it can also be shown that AB \parallel AD (left as an exercise).

Therefore, if the two pairs of the opposite sides of a quadrilateral are congruent, then they are parallel.

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