Question

how to solve this question

Answer 1

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.

Answer 2

1) triangle APD =~ triangle CQB BECAUSE THEY ARE BISECTED BY THE diagnol bd and halfly moulded by APQC

2)AP =CQ because oppsite side of rectangle are equall or by CPCT (congruent side of congruent triangle BQC and APD)

3) triangle AQB=~ triangle CPD by CPCT

4) AQ=CP because oppsite side or rectangle are equall ( rectangle AQCP)

5) ACPQ is a parallelogram because all the properties of parallelogram in this figure is identified and verified

hence all statements proved
hope it helps you dear
have a great day ^_^