91Ó°ÊÓ

The given diagram is:

From the given diagram it can be noticed that:

$∠DCA=∠BAC$.

It can be noticed that the angles $∠DCA$and $∠BAC$are the alternate interior angles and the transversal is AC.

As, $∠DCA=∠BAC$, therefore, it can be said that the alternate interior angles $∠DCA$and $∠BAC$are equal.

Therefore, the lines AB and CD are parallel.

Therefore, $ABâˆ\yen CD$.

From the given diagram it can be noticed that:

$∠DAC=∠BCA$.

It can be noticed that the angles $∠DAC$and $∠BCA$are the alternate interior angles and the transversal is AC.

As, $∠DAC=∠BCA$, therefore, it can be said that the alternate interior angles $∠DAC$and $∠BCA$are equal.

Therefore, the lines AD and BC are parallel.

Therefore, $ADâˆ\yen BC$.

Therefore, $ABâˆ\yen CD$and $ADâˆ\yen BC$.

That implies both pairs of opposite sides are parallel.

Therefore, the given quadrilateral ABCD is a parallelogram.

Yes, the given figure ABCD is a parallelogram.

The theorem that applies is that if both pairs of opposite sides are parallel then the quadrilateral is a parallelogram.