Every school teaches that the three angles of a triangle add to a hundred and eighty degrees, and every school is right. What is usually left out is the condition attached to it: the triangle has to be drawn on a flat surface.
Here is a triangle that is not. Start at the North Pole and walk south until you reach the equator. Turn left through a right angle and walk along the equator for a quarter of the way round the world. Turn left through a right angle again and walk north, and you will arrive back at the Pole.
You have walked a closed three-sided figure, and you turned through a right angle at the equator twice. The third angle, at the Pole, is also a right angle, because the two southward paths left the Pole a quarter of the world apart. Three right angles: two hundred and seventy degrees.
Nothing is broken. The hundred-and-eighty rule was never a rule about triangles in general; it is a rule about triangles on a plane. On the surface of a sphere the total is always more than a hundred and eighty, and how much more depends on how much of the sphere the triangle covers.
This matters to anyone who flies or sails a long way. It does not matter on an exam paper, where every triangle is flat — but knowing why the rule holds is worth more than knowing that it does.