Question

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

(i) ∆APB ≅ ∆CQD 

(ii) AP = CQ.

Answer 1

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.