91Ó°ÊÓ

The theorem 5-5 states that if one pair of opposite sides of a quadrilateral are both congruent and parallel, then the quadrilateral is a parallelogram.

The labelled diagram is:

The given statements are: $ABâ‰\dots CD$and $ABâˆ\yen CD$.

To prove statements are: ABCDis a parallelogram.

It is being given that in the triangles, $â–³ABC$and $â–³CDA$,$ABâ‰\dots CD$and $ABâˆ\yen CD$.

In the triangles, $â–³ABC$and $â–³CDA$, it can be noticed that the side AC is common.

Therefore, $ACâ‰\dots AC$.

As $ABâˆ\yen CD$, therefore the angles $∠CAB$and $∠ACD$are alternate interior angles.

Therefore, $∠CABâ‰\dots ∠ACD$.

Therefore, in the triangles $â–³ABC$ and $â–³CDA$, it can be noticed that $ABâ‰\dots CD$, $∠CABâ‰\dots ∠ACD$and $ACâ‰\dots AC$.

Therefore, the triangles $â–³ABC$and $â–³CDA$are congruent triangles by using the SAS postulate.

Therefore, by using CPCT, it can be said that $BCâ‰\dots DA$.

Therefore, it can be noticed that $ABâ‰\dots CD$and $BCâ‰\dots DA$, therefore the pairs of opposite sides of the quadrilateral ABCD are congruent.

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

As $ABâ‰\dots CD$and $BCâ‰\dots DA$, therefore ABCD is a parallelogram.

The two-column proof of the theorem is: