A square and an equilateral triangle are both inscribed in the same circle, and they share the vertex A. Going round the circle in one chosen direction, starting from A, the first vertex of the square reached is P and the first vertex of the triangle reached is Q. What is the measure of ∠PAQ?

The square cuts the circle into four arcs of 360/4 = 90° each, so P is 90° round from A. The triangle cuts it into three arcs of 360/3 = 120°, so Q is 120° round from A in the same direction. The arc PQ is therefore 120° − 90° = 30°, and ∠PAQ is an inscribed angle standing on that arc, so it is half of it: 15°. The trap 30° quotes the arc without halving, 45° halves the square's 90° arc instead of the difference, and 60° halves the triangle's 120° arc.