Data: ABCD is a parallelogram and AP and CQ are perpendiculars from vertices A and C on diagonal BD.
To Prove: (i) ∆APB ≅ ∆CQD
(ii) AP = CQ.
Proof: ABCD is a parallelogram. BD is diagonal.
AP ⊥ BD and CQ ⊥ BD.
(i) In ∆APB and ∆CQD, ∠APB = ∠CQD = 90°
AB = CD (opposite sides)
∠ABP = ∠CQD (alternate angles)
∴ ∆APB ≅ ∆CQD (AAS postulate)
(ii) ∴ AP = CQ.