Question

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


Identify the incorrect statement from the following.

• A.
  $\Delta APD\cong \Delta CQB$
• B.
  AP = CQ
• C.
  $\Delta AQB\cong \Delta CPD$
• D.
  AD = CQ

Answer 1

The correct option is D

AD = CQ
In $\Delta APDand\Delta CQB$,
$\angle ADP=\angle CBQ$ (Alternate interior angles)
AD = CB (Opposite sides of parallelogram ABCD)
DP = BQ (Given)
$\therefore \Delta APD\cong \Delta CQB$ ( using SAS congruence rule)
As we have proved that triangle APD is congruent to triangle CQB,
Hence, AP = CQ and AD = CB (CPCT)

In $\Delta AQBand\Delta CPD$,
$\angle ABQ=\angle CDP$ (Alternate interior angles for AB||CD)
AB = CD (Opposite sides of parallelogram ABCD)
BQ = DP( Given)
$\Delta AQB\cong \Delta CPD$ (Using SAS congruence rule)