The four corners of a quadrilateral PQRS, taken in that order, all lie on one circle. The angles at P and at Q are in the ratio 2 : 3, and the angle at R measures 100°. What is the size of the angle at Q?

The angles at $P$ and R sit at opposite corners, so they are supplementary: ∠$P=180$° − 100° = 80°. The ratio ∠$P$ : ∠$Q=2$ : 3 then gives ∠$Q=80\times 3/2=120$°. Everything checks: ∠$S=180$° − 120° = 60°, and 80 + 120 + 100 + 60 = 360°. The trap 60° answers ∠$S$, the angle opposite ∠$Q$; 80° answers ∠$P$; and 108° comes from splitting 180° in the ratio 2 : 3, which wrongly treats the ADJACENT angles $P$ and $Q$ as the supplementary pair — it is the opposite corners that add to 180°.