. In parallelogram ABCD, two points P and Q are taken on diagonal BD such that DP = BQ (see Fig. 8.20). Show that: (i) ΔAPD ≅ ΔCQB (ii) AP = CQ (iii) ΔAQB ≅ ΔCPD (iv) AQ = CP (v) APCQ is a parallelogram
Solution: (i) In ΔAPD and ΔCQB, DP = BQ (Given) ∠ADP = ∠CBQ (Alternate interior angles) AD = BC (Opposite sides of a parallelogram) Thus, ΔAPD ≅ ΔCQB [SAS congruency] (ii) AP = CQ by CPCT as ΔAPD ≅ ΔCQB. (iii) In ΔAQB and ΔCPD, BQ = DP (Given) ∠ABQ = ∠CDP (Alternate interior angles) AB = CD (Opposite sides of a parallelogram) Thus, ΔAQB ≅ ΔCPD [SAS congruency] (iv) As ΔAQB ≅ ΔCPD AQ = CP [CPCT] (v) From the questions (ii) and (iv), it is clear that APCQ has equal opposite sides and also has equal and opposite angles. , APCQ is a parallelogram.
