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Parallelogram Opposite Angles Theorem

Proof

Parallelogram Opposite Angles Theorem

In a parallelogram, the opposite angles are congruent.

For the parallelogram the following statement holds true.

Proof

This theorem can be proved by using congruent triangles. Consider the parallelogram and its diagonal

Opposite sides of a parallelogram are parallel. Therefore, by the Alternate Interior Angles Theorem it can be stated that and Furthermore, by the Reflexive Property of Congruence, is congruent to itself.

Two angles of and their included side are congruent to two angles of and their included side. By the Angle-Side-Angle Congruence Theorem, and are congruent triangles. Since corresponding parts of congruent figures are congruent, and are congruent angles.

By drawing the diagonal and using a similar procedure, it can be shown that and are also congruent angles.

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 angles and

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. As the triangles are congruent, all their corresponding parts are congruent. Therefore, the angles  and are congruent as well.

Since the angles  and are the sum of the same two congruent angles, they are congruent.

The proof can be summarized in a flowchart proof.