prove that a quadrilateral is a parallelogram if a pair of opposite sides are equal and parallel

Let ABCD is a quadrilateral in which we have  AB = CD  and  AB || CD.

Join AC.

Since AB || CD and AC is a transversal,

∠DCA = ∠BAC  [alternate interior angles]

In ΔDCA and ΔBAC

 DC = AB  [given]

∠DCA = ∠BAC  [proved above]  

 AC = AC  [ common ]

 ΔDCA is congruent to ΔBAC  by  SAS.

⇒ ∠DAC = ∠BCA    [CPCT]

But ∠DAC and ∠BCA are alternate interior angles made by the transversal AC with the lines DA and CB and are also equal.

⇒ DA || CB

In quadrilateral ABCD, we have

 AB || CD  [given]

DA || CB  [proved above]

⇒ quadrilateral ABCD is a parallelogram.