Saccheri–Legendre theorem

Remarks

The proof starts from a triangle $ABC$ and builds a new triangle with the same angle sum, one of whose angles is at most half of the angle at $A$. Repeating this $n$ times gives a triangle with the same angle sum and an angle at most $2^{-n}$ times the angle at $A$. If the angle sum exceeded two right angles by some angle $\varepsilon$, the axiom of Archimedes would give a step where this small angle is less than $\varepsilon$. Then the other two angles would sum to more than two right angles, which contradicts the exterior angle theorem. The axiom of Archimedes is needed: Max Dehn built non-Archimedean geometries in which the other axioms hold and some triangles have angle sum greater than two right angles.

Going deeper

For a student

The theorem is one of the main results of absolute geometry. With it, the parallel postulate is equivalent to the existence of a single triangle with angle sum $180^\circ$. The defect of a triangle is two right angles minus its angle sum. It is additive when a triangle is cut into two, and in the hyperbolic plane it is proportional to the area.