Saccheri–Legendre theorem implies the characteristic postulate of hyperbolic geometry. Max Dehn gave an example of a non-Legendrian geometry where the angle sum of a triangle is greater than 180 degrees, and a semi-Euclidean geometry where there is a triangle with an angle sum of 180 degrees but Euclid's parallel postulate fails. In Dehn's geometries the Archimedean axiom does not hold. Saccheri–Legendre theorem In absolute geometry, the Saccheri–Legendre theorem states that the sum of the angles in a triangle is at most 180°. Absolute geometry is the geometry obtained from assuming all the axioms that lead to Euclidean geometry with the exception of
Right triangle right triangle, see the hyperbolic triangle of a hyperbolic sector. The values of the trigonometric functions can be evaluated exactly for certain angles using right triangles with special angles. These include the "30-60-90 triangle" which can be used to evaluate the trigonometric functions for any multiple of π/6, and the "45-45-90 triangle" which can be used to evaluate the trigonometric functions for any multiple of π/4. Let "H", "G", and "A" be the harmonic mean, the geometric mean, and the arithmetic mean of two positive numbers "a" and "b" with "a" > "b". If a right triangle has legs "H"