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In a parallelogram, the opposite angles are congruent.
For the parallelogram PQRS, the following statement holds true.
∠Q≅∠Sand∠P≅∠R
This theorem can be proved by using congruent triangles. Consider the parallelogram PQRS and its diagonal PR.
Opposite sides of a parallelogram are parallel. Therefore, by the Alternate Interior Angles Theorem it can be stated that ∠QPR≅∠SRP and ∠QRP≅∠SPR. Furthermore, by the Reflexive Property of Congruence, PR is congruent to itself.
By drawing the diagonal QS and using a similar procedure, it can be shown that ∠P and ∠R are also congruent angles.
∠Q≅∠Sand∠P≅∠R
In the parallelogram PQRS, the diagonal QS is drawn.
According to the definition of a parallelogram, the opposite sides are parallel, PQ∥SR and QR∥PS.
Then, by the Alternate Interior Angles Theorem, the angles ∠PQS≅∠QSR and ∠PSQ≅∠RQS.
Since the angles ∠PQR and ∠QRS are the sum of the same two congruent angles, they are congruent.
The proof can be summarized in a flowchart proof.