Question

in a triangle ABC,the bisector of angle A meets opposite side BC at point D. PROVE that AB>BD

Answer 1

I cannot manage to draw figures, so you will have to draw triangle ABC and the bisector of angle BAC (point D is on side BC).
Let angle BAC have a measure of 2x degrees and let angle ACB have a measure of y degrees. Angles BAD and DAC have measures of x degrees. Angle ADC with be 180 - (x+y) degrees. Angle ADB will be x+y degrees.
In triangle ABD, side AB is opposite an angle of x+y degrees, and side BD is opposite a smaller angle of x degrees. By the Law of Sines, AB>BD.