Proof

Parallelogram Opposite Sides Theorem

The opposite sides in a parallelogram are congruent.

If $P Q R S$ is a parallelogram, then the following statement holds true.

$\overline{P Q} ≅ \overline{S R}$ and $\overline{Q R} ≅ \overline{P S}$

This can be proven using the ASA Congruence Theorem. In the parallelogram $P Q R S,$ the diagonal $\overline{Q S}$ is drawn.

According to the definition of a parallelogram, the opposite sides are parallel, $P Q ∥ S R$ and $Q R ∥ P S .$

Then, by the Alternate Interior Angles Theorem, the following angles are congruent:

∠ P Q S ≅ ∠ Q S R and ∠ P S Q ≅ ∠ R Q S .

Since the triangles have two congruent angles and share one side,

\overline{Q S},

the ASA Congruence Theorem applies. It states that two triangles are congruent if two angles and their including side are congruent.

△ P Q S ≅ △ R S Q

In congruent triangles, all corresponding parts are congruent. Therefore, the opposite sides in the parallelogram are congruent as well.

This can be summarized by a two-column proof.

Statement	Reason
$P Q R S$ is a parallelogram.	Given
Draw the diagonal $\overline{Q S} .$	Diagonal to create triangles.
$P Q ∥ S R,$ $Q R ∥ P S$	Parallel lines
$∠ P Q S ≅ ∠ Q S R,$ $∠ P S Q ≅ ∠ R Q S$	Alternative Angle Theorem
$△ P Q S ≅ △ R S Q$	ASA Congruence Theorem
$P Q ≅ S R,$ $Q R ≅ P S$	Parallel lines in a parallelogram are congruent.

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