Question

In parallelogram ABCD, two points P and Q are taken on diagonal BD such that DP = BQ (see Fig. 8.12). Show that: 

(i) ∆APD ≅ ∆CQB   

(ii) AP = CQ 

(iii) ∆AQB ≅∆CPD 

(iv) AQ = CP 

(v) APCQ is a parallelogram

 

Answer 1

(i) In ∆APD and ∆CQB,

∠ADP = ∠CBQ (Alternate interior angles for BC || AD) 

AD = CB (Opposite sides of parallelogram ABCD) 

DP = BQ (Given) 

∠∆APD ∠∆CQB (Using SAS congruence rule) 

(ii) As we had observed that ∆APD ∆CQB,  

∠AP = CQ (CPCT) 

(iii) In ∆AQB and ∆CPD, 

∠ABQ = ∠CDP (Alternate interior angles for AB || CD) 

AB = CD (Opposite sides of parallelogram ABCD) 

BQ = DP (Given) 

∠∆AQB ∠∆CPD (Using SAS congruence rule) 

(iv) As we had observed that ∆AQB ∆CPD, 

∠AQ = CP (CPCT) 

(v) From the result obtained in (ii) and (iv), 

AQ = CP and AP = CQ

Since 

opposite sides in quadrilateral APCQ are equal to each other, 

APCQ is a parallelogram.