In parallelogram ABCD, two points P and Q are taken on diagonal BD such that DP = BQ see Figure. Show that: (i) ∆ APD ≅ ∆ CQB (ii) AP = CQ (iii) ∆ AQB ≅∆ CPD (iv) AQ = CP (v) APCQ is a parallelogram

(i) In ΔAPD and ΔCQB,
DP = BQ [∵ Given]
∠ADP = ∠CBQ [∵ Alternate angle]
AD = BC [∵ Opposite sides of a parallelogram]
Hence, ΔAPD ≅ ΔCQB [∵ SAS Congruency rule]

(ii) ΔAPD ≅ CQB [∵ Prove above]
AP = CQ …(1) [∵ CPCT]

(iii) In ΔAQB and ΔCPD,
QB = DB [∵ Given]
∠ABQ = ∠CDP [∵ Alternate angle]
AB = CD [∵ Opposite sides of a parallelogram]
Hence, ΔAQB ≅ ΔCPD [ SAS Congruency rule]

(iv) ΔAQB ≅ ΔCPD [∵ Prove above]
AQ = CP …(2) [∵ CPCT]

(v) In APCQ,
AP = CQ [∵ From (1)]
AQ = CP [∵ From (2)]
The opposite sides of quadrilateral APCQ are equal.
Hence, APCQ is a parallelogram.