ABCD is a parallelogram and AP and CQ are perpendiculars from vertices A and C on diagonal BD (see Fig. 8.21). Show that

(i) Δ APB ≅Δ CQD

(ii) AP = CQ

(i) In ΔAPB and ΔCQD,

∠APB = ∠CQD (Each 90°)

AB = CD (Opposite sides of parallelogram ABCD)

∠ABP= ∠CDQ (Alternate interior angles for AB || CD)

ΔAPB ΔCQD (By AAS congruency)

(ii) By using the above result

ΔAPB ΔCQD, we obtain

AP = CQ (By CPCT)