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
Skinny triangle based on the small-angle approximations; and, when formula_5 is in radians. The proof of the skinny triangle solution follows from the small-angle approximation by applying the law of sines. Again referring to figure 1; The term formula_7 represents the base angle of the triangle and is this value because the sum of the internal angles of any triangle (in this case the two base angles plus "θ") are equal to "π". Applying the small angle approximations to the law of sines above results in; the desired result. This result is equivalent to assuming that the length of the base of