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.