Theory:

Theorem VIII
A quadrilateral is a parallelogram if a pair of opposite sides is equal and parallel.

Given: $$ABCD$$ is a quadrilateral, where $$AB = CD$$ and $$AB \ || \ CD$$.

To prove: $$ABCD$$ is a parallelogram.

Proof: In $$\Delta ADC$$ and $$\Delta CBA$$:

$$AB = CD$$ [Given]

$$\angle BAC = \angle ACD$$ [Alternate interior angles]

$$CA = CA$$ [Common side]

Therefore, $$\Delta ADC \cong \Delta CBA$$ [by $$ASA$$ congruence criterion].

$$\Rightarrow AD = BC$$ [by CPCT] - - - - - (I)

We have, $$\angle BAC = \angle ACD$$.

These are alternate angles.

The alternate interior angles theorem states that, "the alternate interior angles are congruent when the transversal intersects two parallel lines".

Here, $$AC$$ is the transversal line for the lines $$AD$$ and $$BC$$.

So, $$AD \ || \ BC$$ - - - - - (II)

From (I) and (II), we get:

$$AD \ || \ BC$$ and $$AD = BC$$

Thus, both pair of opposite sides are equal and parallel.

Therefore, $$ABCD$$ is a parallelogram.