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Question 9 (v)
In parallelogram ABCD, two points P and Q are taken on diagonal BD such that DP=BQ (see the given figure) Show that:

APCQ is a parallelogram


Solution

In ΔAPD and ΔCQB,
ADP=CBQ (Alternate interior angles)
AD =CB (Opposite sides of parallelogram ABCD)
DP=BQ (Given)
ΔAPDΔCQB ( using SAS congruence rule)
As we have proved that triangle APD is congruent to triangle  CQB,
So , AP= CQ  (CPCT) and 
AQ=CP
Since opposite sides In quadrilateral APCQ are equal to each other, APCQ is a parallelogram.

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