If each pair of opposite sides of a quadrilateral is equal, then it is a parallelogram. 

Given: ABCD is a quadrilateral, each pair of opposite sides of quadrilateral ABCD are AB and CD and also sides AD and BC are equal.

AB = CD

BC = AD

To prove: ABCD is a parallelogram.

Construction: Join A to C that is AC, is a diagonal, the diagonal AC divides parallelogram ABCD into two triangles △ ABC and △ CDA.

Proof:  In △ ABC and △ CDA

AB = CD (Given)

BC = DA (Given) and 

AC = CA (Common)

These triangles △ ABD and △CDA are congruent,

△ ABC ≅ △CDA

So, △ ABC ≅ △CDA (By SSS congruent)

So, ∠BAC = ∠DCA (CPCT)….(1)

and ∠BCA = ∠DAC (CPCT)….(2)

From lins AB and DC and AC is a transversal.

∠BAC = ∠DCA (Alternate angles and are equal)

So, lines AB and DC are parallel.

i.e. AB ∥ DC

From lins AD and BC and AC is a transversal.

∠BCA = ∠DAC (Alternate angles and are equal)

So, lines AD and BC are parallel.

i.e. AD ∥ BC

Thus, in quadrilateral ABCD both pairs of opposite sides are equal and parallel.

∴ ABCD is a parallelogram.

 Hence proved.