Proof That The Opposite Sides of a Parallelogram Are Congruent

A parallelogram is a quadrilateral whose opposite sides are parallel. In the figure below, PQRS is a parallelogram. PQ is parallel to RS and PS is parallel to QR.

parallelogram-1

 

In this post, aside from being parallel, we will also prove that the opposite sides of a parallelogram are congruent.

Theorem

The opposite sides of a parallelogram are congruent.

Given

Parallelogram PQRS.

What To Show

\overline{PQ} \cong \overline{SR} and \overline{PS} \cong \overline{QR}.

Proof

Draw diagonal \overline{QS}.

parallelogram

 

Since \overline{QS} is a transversal to parallel lines \overline{PQ} and \overline {SR}, \angle PSQ \cong \angle SQR because they are alternate interior angles. (A)

Now, segment is congruent to itself (reflexive proeprty) so, \overline{QS} \cong \overline{QS}. (S)

Since \overline{PS}  is parallel to  \overline {QR}\angle PQS \cong \angle QSR because they are alternate interior angles. (A)

By ASA congruence, \triangle PQS \cong \triangle QSR.

Since corresponding parts of congruent triangles are congruent, \overline{PQ} \cong \overline{SR} and \overline{PS} \cong \overline{QR}.

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