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In parallelogram ABCD, two points P and Q are taken on diagonal BD such that DP = BQ (see Fig.). Show that : 

(i) ∆ APD ≅ ∆CQB 

(ii) AP = CQ 

(iii) ∆ AQB ≅ ∆CPD 

(iv) AQ = CP 

(v) APCQ is a parallelogram

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Given : ABCD is a parallelogram and P and Q are points on diagonal BD such that DP = BQ. 

To Prove : 

(i) ∆APD ≅ ∆CQB 

(ii) AP = CQ 

(iii) ∆AQB ≅ ∆CPD 

(iv) AQ = CP 

(v) APCQ is a parallelogram 

Proof : 

(i) In ∆APD and ∆CQB, we have 

AD = BC [Opposite sides of a ||gm] 

DP = BQ [Given] 

∠ADP = ∠CBQ [Alternate angles] 

∴ ∆APD ≅ ∆CQB [SAS congruence] 

(ii) ∴ AP = CQ [CPCT] 

(iii) In ∆AQB and ∆CPD, we have 

AB = CD [Opposite sides of a ||gm] 

DP = BQ [Given] 

∠ABQ = ∠CDP [Alternate angles] 

∴ ∆AQB ≅ ∆CPD [SAS congruence] 

(iv) ∴ AQ = CP [CPCT] 

(v) Since in APCQ, opposite sides are equal, therefore it is a parallelogram. Proved.

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