Saccheri–Legendre theorem

In absolute geometry, the Saccheri–Legendre theorem tells us that the sum of the angles in a triangle is at most 180°. This idea comes from geometry that follows all the usual rules except one special rule called the parallel postulate of Euclid.

The theorem is named after two mathematicians, Giovanni Girolamo Saccheri and Adrien-Marie Legendre. Saccheri wrote about it in a book in 1733 called Euclid Freed of Every Flaw, but his work was mostly forgotten. Later, when Legendre rediscovered the idea, it was often called Legendre's theorem.

This theorem helps us understand different types of geometry. If there is a triangle with angles adding up to exactly 180°, it means we are working with Euclid's parallel postulate. If a triangle's angles add up to less than 180°, it shows we are dealing with the rules of hyperbolic geometry. One way to prove this theorem uses the Archimedean axiom. Max Dehn showed examples of special geometries where the angle sum can be more than 180°.