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

**(i)**

**(ii)** AP = CQ

**(iii)** **(iv)** AQ = CP

**(v)** APCQ is a parallelogram

**Solution (i)**

In and , we have

DP=BQ ** (Given)**

** (Alternate Interior angles, )**

AD=BC **(Opposite sides of parallelogram are equal)**

Therefore, **by SAS congruence rule**,

**Solution (ii)**

We have already showed in **solution (i)** that .

**(Corresponding parts of congruent triangles are equal)**

**Solution (iii)**

In and , we have

BQ=DP **(Given)**

** (Alternate Interior angles, )**

AB=CD ** (Opposite sides of parallelogram are equal)**

Therefore, **by SAS congruence rule**,

**Solution (iv)**

We have showed in **solution (iii)** that

**(Corresponding parts of congruent triangles are equal)**

**Solution (v)**

We have showed in** solution (ii)** and **solution (iv)** that AP=CQ and AQ=CP.

It means that APCQ is a paralleogram.

**(If in a quadrilateral, each pair of opposite sides is equal, then it is a parallelogram.)**