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.

0. Statement	0. Reason
1. $P Q R S$ is a parallelogram.	1. Given
2. Draw the diagonal $\overline{Q S} .$	2. Diagonal to create triangles.
3. $P Q ∥ S R,$ $Q R ∥ P S$	3. Parallel lines
4. $∠ P Q S ≅ ∠ Q S R,$ $∠ P S Q ≅ ∠ R Q S$	4. Alternative Angle Theorem
5. $△ P Q S ≅ △ R S Q$	5. ASA Congruence Theorem
6. $P Q ≅ S R,$ $Q R ≅ P S$	6. Parallel lines in a parallelogram are congruent.