In parallelogram ABCD, two points P and Q are taken on diagonal BD such that DP = BQ (see Fig. 8.20). Show that:

(i) Δ APD ≅Δ CQB

(ii) AP = CQ

(iii) Δ AQB ≅Δ CPD

(iv) AQ = CP

(v) APCQ is a parallelogram

Question

Philoid · Accepted Answer

(i) In ΔAPD and ΔCQB,

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

AD = CB (Opposite sides of parallelogram ABCD)

DP = BQ (Given)

ΔAPD ΔCQB (Using SAS congruence rule)

(ii) As we had observed that,

ΔAPD ΔCQB

AP = CQ (CPCT)

(iii) In ΔAQB and ΔCPD,

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

AB = CD (Opposite sides of parallelogram ABCD)

BQ = DP (Given)

ΔAQB ΔCPD (Using SAS congruence rule)

(iv) As we had observed that,

ΔAQB ΔCPD,

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

In parallelogram ABCD, two points P and Q are taken on diagonal BD such that DP = BQ (see Fig. 8.20). Show that:(i) Δ APD ≅Δ CQB(ii) AP = CQ(iii) Δ AQB ≅Δ CPD(iv) AQ = CP(v) APCQ is a parallelogram