91Ó°ÊÓ

The theorem 5-4 states that if both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

The labelled diagram is:

The given statements are: $TSâ‰\dots QR$and $TQâ‰\dots SR$.

To prove statements are: TSRQ is a parallelogram.

It is being given that in the triangles, $â–³TSQ$and $â–³RQS$, it can be noticed that $TSâ‰\dots QR$and $TQâ‰\dots SR$.

In the triangles, $â–³TSQ$and $â–³RQS$, it can be noticed that the side QS is common.

Therefore, $QSâ‰\dots QS$.

Therefore, the triangles $â–³TSQ$and $â–³RQS$are congruent triangles by using SSS postulate.

Therefore, by using CPCT, it can be said that $∠TSQâ‰\dots ∠RQS$and $∠TQSâ‰\dots ∠RSQ$.

From the diagram it can be noticed that the angles $∠TSQ$and $∠RQS$are the alternate interior angles and the angles $∠TQS$and $∠RSQ$are the alternate interior angles.

Therefore, $TSâˆ\yen QR$and $TQâˆ\yen SR$.

If both pairs of opposite sides of the quadrilateral, then the quadrilateral is a parallelogram.

As $TSâˆ\yen QR$and $TQâˆ\yen SR$, therefore TSRQ is a parallelogram.