Question

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

Answer 1

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.