Given: A quadrilateral ABCD in which AB ║ DC and AB = DC.
To Prove: ABCD is a parallelogram
i.e., AB ║ DC and AD ║ BC.
Construction: Join A and C.
Proof : Since AB is parallel to DC and AC is transversal
∠BAC = ∠DCA [Alternate angles]
AB = DC [Given]
And AC = AC [Common side]
⇒ ∆BAC ≅ ∆DCA [By SAS]
⇒ ∠BCA = ∠DAC [By cpctc]
But these are alternate angles and whenever alternate angles are equal, the lines are parallel.
⇒ AD ║ BC
Now, AB ║ DC (given) and AD ║ BC [Proved above]
⇒ ABCD is a parallelogram
