The bisectors of ∠A and ∠B of the parallelogram ABCD intersect at P on the side CD. If BC = 3 cm, then AB =

1. 4 cm

2. 5 cm

3. 6 cm

4. 8 cm

In parallelogram ABCD, we know that AB ∥ DC. The line AP is a transversal.

∠PAB = ∠APD (Alternate interior angles are equal) ….(1)

Since AP is the bisector of ∠A:

∠PAB = ∠PAD …….(2)

From equation (1) and (2), we get :

∴ ∠PAD = ∠APD

In triangle APD, since two angles are equal, the triangle is isosceles.

AD = DP

ABCD is a parallelogram, AD = BC = 3 cm.

Thus, DP = 3 cm.

Similarly, for the bisector BP and transversal BP:

∠PBA = ∠BPC (Alternate interior angles are equal) …….(3)

Since BP bisects ∠B:

∠PBA = ∠PBC ……….(4)

From equation (3) and (4), we get :

∴ ∠BPC = ∠PBC

In triangle BCP, since two angles are equal, the triangle is isosceles.

PC = BC = 3 cm.

Length of CD = DP + PC = 3 + 3 = 6 cm.

AB = CD = 6 cm [Opposite sides of parallelogram are equal]

Hence, option 3 is the correct option.