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{{ printedBook.courseTrack.name }} {{ printedBook.name }} The opposite sides in a parallelogram are congruent.

That is, if $PQRS$ is a parallelogram, then $PQ ≅SRandQR ≅PS.$

This can be proven using the ASA Congruence Theorem.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 following angles are congruent: $∠PQS≅∠QSRand∠PSQ≅∠RQS.$

Since the triangles have two congruent angles and share one side, $QS ,$ the ASA Congruence Theorem applies. It states that two triangles are congruent if two angles and their including side are congruent. $△PQS≅△RSQ$ 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 |

$PQRS$ is a parallelogram. | Given |

Draw the diagonal $QS .$ | Diagonal to create triangles. |

$PQ∥SR,$ $QR∥PS$ | Parallel lines |

$∠PQS≅∠QSR,$ $∠PSQ≅∠RQS$ | Alternative Angle Theorem |

$△PQS≅△RSQ$ | ASA Congruence Theorem |

$PQ≅SR,$ $QR≅PS$ | Parallel lines in a parallelogram are congruent. |