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.
|is a parallelogram.||Given|
|Draw the diagonal||Diagonal to create triangles.|
|Alternative Angle Theorem|
|ASA Congruence Theorem|
|Parallel lines in a parallelogram are congruent.|