Data: In parallelogram ABCD, two points P and Q are taken on diagonal BD such that DP = BQ.
To Prove:
(i) ∆AAPD ≅ ∆CQB
(ii) AP = CQ
(iii) ∆AQB ≅ ∆CPD
(iv) AQ = CP
(v) APCQ is a parallelogram.
Proof: ABCD is a parallelogram.
(i) In ∆APD and ∆CQB,
AD = BC (opposite sides)
∠ADP = ∠CBQ (alternate angles)
DP = BQ (Data)
∴ ∆APD ≅ ∆CQB (ASA Postulate)
(ii) AP = CQ
(iii) In ∆AQB and ∆CPD,
AB = CD (opposite sides)
∠ABQ = ∠CDP (alternate angles)
BQ = DP (Data)
∴ ∆AQB ≅ ∆CPD (ASA Postulate)
(iv) ∴ AQ = CP
(v) In ∆AQPand
∆CPQ, AP = CQ (proved)
AQ = PC PQ is common.
∴ ∆AQP ≅ ∆CPQ
∴ ∠APQ = ∠CQP
These are pair of alternate angles.
∴AP || PC
Similarly, AQ || PC
∴ APCQ is a parallelogram.
