In triangle ABC, AB = AC and the angle at A measures 40°. A point D is taken on side AC so that BD = BC. What is the measure of angle DBC?

Since AB = AC, angle ABC = angle ACB = (180° − 40°)/2 = 70°. As D lies on AC, angle BCD is the same as angle BCA = 70°. In triangle BDC, BD = BC makes it isosceles with angle BDC = angle BCD = 70°, so angle DBC = 180° − 70° − 70° = 40°. The value 30° is angle ABD, the leftover part of angle ABC.