Four points A, B, C and D lie on a circle in that order, and they cut the circle into four arcs whose measures are in the ratio arc AB : arc BC : arc CD : arc DA = 1 : 3 : 5 : 3. What is the measure of ∠ABC?

The twelve equal parts share out 360°, so one part is 30° and the four arcs measure 30°, 90°, 150° and 90°. ∠ABC sits at B and stands on the arc from A round to C the other way, which is arc AD + arc DC = 90° + 150° = 240°. An inscribed angle is half its arc, so ∠ABC = 120°. Check: the opposite angle ∠ADC stands on arc AB + arc BC = 120°, giving 60°, and 120° + 60° = 180° as a cyclic quadrilateral demands. The trap 240° quotes the arc without halving, 60° answers the OPPOSITE angle ∠ADC, and 150° simply reads off the largest arc.