# In parallelogram ABCD, two points P and Q are taken on diagonal BD such that DP = BQ (see the given figure). Show that: - Mathematics

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

(i) ΔAPD ≅ ΔCQB

(ii) AP = CQ

(iii) ΔAQB ≅ ΔCPD

(iv) AQ = CP

(v) APCQ is a parallelogram

#### Solution

(i) In ΔAPD and ΔCQB,

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.

Concept: Another Condition for a Quadrilateral to Be a Parallelogram
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#### APPEARS IN

NCERT Class 9 Maths