D is a point lying inside triangle ABC. The angle BAC measures 50°, angle ABD measures 20° and angle ACD measures 25°. What is the measure of angle BDC?

Produce BD to meet AC at P. In triangle ABP the angle at P is an exterior angle, so angle BPC = angle BAP + angle ABP = 50° + 20° = 70°. Now in triangle DPC the angle BDC is an exterior angle, so angle BDC = angle DPC + angle DCP = 70° + 25° = 95°. In general the angle at an interior point equals the vertex angle plus the two small angles, 50 + 20 + 25.