In parallelogram ABCD, two points P and Q are taken on diagonal BD such that DP = BQ. Show that (i) ΔAPD ≅ ΔCQB (ii) AP = CQ (iii) ΔAQB ≅ ΔCPD (iv) AQ = CP (v) APCQ is a parallelogram

Construction: JoinACto meetBDinO.Therefore,OB=ODandOA=OC.. (1)(Diagonals of a parallelogram bisect each other)ButBQ=DP...given∴OB–BQ=OD–DP∴OQ=OP(2)Now, in□APCQ,OA=OCfrom (1)OQ=OPfrom (2)∴□APCQis a parallelogram.In△APDand△CQB,AD=CBopposite sides of a parallelogramAP=CQopposite sides of a parallelogramDP=BQ...given△APD≅△CQB...By SSS test of congruence∴AP=CQ...c.s.c.t.AQ=CP...c.s.c.t. .. (3)In△AQBand△CPD,AB=CDopposite sides of a parallelogramAQ=CP...from (3)BQ=DP...given∴△AQB≅△CPDBy SSS test of congruence