A regular pentagon ABCDE is inscribed in a circle, its vertices named in order round the circle. What is the measure of ∠ACB?

The five equal sides cut the circle into five equal arcs of 360°/5 = 72°, so the arc AB not containing C measures 72°. ∠ACB is an inscribed angle standing on that arc, and an inscribed angle is half the arc it stands on, so ∠ACB = 72°/2 = 36°. The trap 72° reports the arc, or equivalently the CENTRAL angle ∠AOB, instead of the angle at the circumference; 108° is the pentagon's interior angle ∠ABC; and 54° halves 108°.