Given : ABCD is a parallelogram and AP and CQ are perpendiculars from vertices A and C on BD.
To Prove :
(i) ∆APB ≅ ∆CQD
(ii) AP = CQ
Proof : (i) In ∆APB and ∆CQD, we have
∠ABP = ∠CDQ [Alternate angles]
AB = CD [Opposite sides of a parallelogram]
∠APB = ∠CQD [Each = 90°]
∴ ∆APB ≅ ∆CQD [ASA congruence]
(ii) So, AP = CQ [CPCT] Proved.