Question

ABCD is a parallelogram AP and CQ are perpendiculars drawn from vertices A and C on diagonal BD (see figure). Show that 1) ΔAPB ≅ ΔCQD ii) AP = CQ

Answer 1

Given that □ABCD is a parallelogram. 

BD is a diagonal. 

AP ⊥ BD and CQ ⊥ BD

i) In ΔAPB and ΔCQD 

AB = CD ( ∵ Opp. sides of //gm ABCD) 

∠APB = ∠CQD (each 90°) 

∠PBA = ∠QDC (alt. int. angles for the lines AB and DC) 

∴ ΔAPB ≅ ΔCQD (AAS congruence) 

ii) From (1) ΔAPB ≅ ΔCQD 

⇒ AP = CQ (CPCT)