# ABCD is a parallelogram and AP and CQ are perpendiculars from vertices A and C on diagonal BD (see Fig.).

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ABCD is a parallelogram and AP and CQ are perpendiculars from vertices A and C on diagonal BD (see Fig.). Show that

(i) ∆ APB ≅ ∆CQD

(ii) AP = CQ

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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.