1.

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

Answer»

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.


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