The angles of a triangle add up to a straight line

Show that the three interior angles of any triangle in the Euclidean plane add up to $180^{\circ }$.

The usual proof draws a line through one vertex, parallel to the opposite side. There are others: walk once round the triangle and count how far you have turned, for instance.

Whichever you choose, say where the parallel postulate is hiding. On a sphere the angles of a triangle add up to more than $180^{\circ }$, so it has to be hiding somewhere.