Q16 of 109 Page 300

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.


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