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Parallelogram Opposite Sides Theorem

Proof

Parallelogram Opposite Sides Theorem

The opposite sides in a parallelogram are congruent.

That is, if is a parallelogram, then

This can be proven using the ASA Congruence Theorem.

In the parallelogram the diagonal is drawn.

According to the definition of a parallelogram, the opposite sides are parallel, and

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

Since the triangles have two congruent angles and share one side, the ASA Congruence Theorem applies. It states that two triangles are congruent if two angles and their including side are congruent. 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
is a parallelogram. Given
Draw the diagonal Diagonal to create triangles.
Parallel lines
Alternative Angle Theorem
ASA Congruence Theorem
Parallel lines in a parallelogram are congruent.