In triangle ABC, AB = AC. A point D is taken on side BC so that AD = BD. If angle BAD = 25°, what is the measure of angle DAC?

In triangle ABD, AD = BD, so the angles opposite those sides are equal: angle ABD = angle BAD = 25°. Hence angle ABC = 25°, and since AB = AC we also get angle ACB = 25°, so angle BAC = 180° − 25° − 25° = 130°. Therefore angle DAC = 130° − 25° = 105°. Check with triangle ADC: angle ADC = 180° − (180° − 50°) = 50°, and 50° + 25° + 105° = 180°.