Page 6 - ln1

Example:

In a parallelogram ABCD, two point P and Q are taken on its diagonal BD such that DP =
BQ. Prove that APCQ is a parallelogram.

Given: P and Q are two points on diagonal BD such that DP = BQ.

To prove: APCQ is a parallelogram.
In triangles APD and CQB, we have,
DP = BQ (given)
AD = BC (opposite sides of parallelogram ABCD)
and ADP = CBQ (alternate angles when BD intersects parallel lines AD and BC)
Therefore, APD CQB (by SAS criterion of congruency)
Therefore, AP = CQ (by cpct)
Similarly, taking triangles CPD and AQB, we can prove that
CP = AQ
Therefore, APCQ is a parallelogram. (by the property 5 i.e. if the opposite sides are equal,
then it is a parallelogram)

6