Question

ABCD is a parallelogram. P and Q are points on diagonal DB such that DP = QB. Prove that ΔAPB ≅ ΔCQD.

Answer 1

Given:

ABCD is a parallelogram.

DP = QB   ...(As given)

AB || CD and AD || BC   ...(Since ABCD is a parallelogram)

To Prove:

△APB ≅ △CQD.

Proof:

△APB and △CQD are congruent using the criteria of congruence.

Step 1: Show that AB = CD

Since ABCD is a parallelogram, opposite sides are equal.

Hence, AB = CD.

Step 2: Show that ∠APB = ∠CQD

In parallelogram ABCD, the angles at the opposite vertices are equal.

Therefore, ∠APB = ∠CQD.

Step 3: Show that DP = QB

This is given in the problem DP = QB.

Step 4: Conclusion (By SAS criterion)

We have:

AB = CD   ...(Step 1)

∠APB = ∠CQD   ...(Step 2)

DP = QB   ...(Step 3)

Thus, by the Side-Angle-Side (SAS) criterion of congruence, we can conclude that △APB ≅ △CQD.

Hence, △APB ≅ △CQD is proved.