I believe this question has already been answered by CPhill here: https://web2.0calc.com/questions/geometry-help_68
For convenience sake, I have posted it here:
1) In triangle ABC, AB=AC and D is a point on line AC so that line BD bisects angle ABC. If BD=BC, what is the measure, in degrees, of angle A?
Call the each bisected angle. a
And call angle BCA, b
And since BD = BC...then angle BDC = angle BCD = angle BCA = b
So....since AB = AC....then angle ABC = angle BCA ⇒ 2a = b ⇒ 4a = 2b
So..in triangle BDC....
a + 2b =180
a + 4a = 180
5a = 180
a = 36
And b = 2a = 72
So...in triangle ABC
Angle ABC + Angle BCA + Angle A = 180
2a + b + Angle A = 180
72 + 72 + Angle A = 180
144 + Angle A = 180
Angle A = 180 - 144 = 108°
This is all CPhill's work, so it is all his.