The points A, C, B and D lie on a circle in that order, and the chords AB and CD cross at a point P inside the circle. The arc AC that does not contain B or D measures 84°, and the arc BD that does not contain A or C measures 52°. What is the measure of ∠APC?

When two chords cross INSIDE a circle, the angle between them is half the SUM of the two arcs they cut off — the arc between the two near ends and the arc between the two far ends. ∠APC stands on arc AC and on the vertically opposite arc BD, so ∠APC = ½(84° + 52°) = ½ × 136° = 68°. Forgetting to halve gives 136°; using half the DIFFERENCE, ½(84° − 52°) = 16°, applies the rule for a point OUTSIDE the circle; and 42° halves just one of the two arcs.