ABCD is a cyclic quadrilateral, its vertices named in order round the circle, in which AB = AD. If ∠BCD = 130°, what is the measure of ∠ABD?

Opposite angles of a cyclic quadrilateral are supplementary, so ∠BAD = 180° − 130° = 50°. Now ignore the circle and look at triangle ABD alone: AB = AD makes it isosceles, so its base angles are equal and ∠ABD = ∠ADB = (180° − 50°)/2 = 65°. Answering 50° stops at ∠BAD, which is the apex angle of that triangle rather than one of its base angles, and 25° halves it once too often.