In the adjoining figures, is a parallelogram. If and are points on and respectively such that and prove that is a parallelogram.

ABCD is parallelogram

We know that opposite sides and angles of parallelogram are equal.

∴ ∠B = ∠D and AD = BC and AB = DC

Also, AD || BC and AB|| DC

It is given that and

Hence, ​AP = CQ … ∵ AD = BC

In ∆​DPC and ∆​BQA, we have,

AB = CD

∠B = ∠D

DP = QB … as and

Hence, by SAS test for congruency,

∆​DPC ≅ ∆​BQA

∴ PC = QA … by cpct

Hence, from above, in AQCP, we have,

AP = CQ and PC = QA

∴ AQCP is a parallelogram.