IN A PARALLELOGRAM ABCD, THE ANGLE BISECTOR OF ANGLE A BISECTS BC. WILL ANGLE BISECTOR OF B ALSO BISECT AD ?

To determine if the angle bisector of angle B in parallelogram ABCD bisects AD as well, we need to consider the properties of a parallelogram.

In a parallelogram, opposite angles are congruent. Therefore, angle A is congruent to angle C, and angle B is congruent to angle D.

Now, since the angle bisector of angle A bisects side BC, it must divide the angle A into two equal halves. Let's call the point where the angle bisector intersects BC as point X.

Now, we can look at triangle ABX. Since the angle bisector AX divides angle A into two equal halves, it means that angle ABX and angle XBC are congruent. Therefore, triangle ABX is an isosceles triangle.

Now, let's consider triangle CDX. We know that angle C is congruent to angle A, and angle XBC is congruent to angle ABX. Since angle ABX and angle XBC are congruent, it means that angle BCX and angle CDX are congruent. Therefore, triangle CDX is an isosceles triangle as well.

From this, we can conclude that sides AX and DX are congruent in length (because they are the respective angles bisectors of sides BC and AD in the parallelogram ABCD). Hence, the angle bisector of angle B also bisects side AD in parallelogram ABCD.