Looking at the diagram BAC is an isosceles triangle.
According to the isosceles triangle theorem. angle BAC and Angle BCA are equal.
BAC = 55
BCA = 55
Since we know two of the angles we will use the angle addition postulate to find angle ABC.
ABC + BAC + BCA = 180
ABC + 55 + 55 = 180
ABC + 110 = 180
ABC = 70
Line BD is angle bisector of angle B.
SO angle DBC is half the measure of angle ABC.
DBC = 70/2
DBC = 35
The answer is D.
Hope this helps :)