Foundations of geometry and therefore are valid theorems in absolute geometry. Proposition I.31 proves the existence of parallel lines (by construction). Also, the Saccheri–Legendre theorem, which states that the sum of the angles in a triangle is at most 180°, can be proved. The theorems of absolute geometry hold in hyperbolic geometry as well as in Euclidean geometry. Absolute geometry is inconsistent with elliptic geometry: in elliptic geometry there are no parallel lines at all, but in absolute geometry parallel lines do exist. Also, in elliptic geometry, the sum of the angles in any triangle is greater than 180°. Logically, the axioms do
Schwarz triangle function since the case of two cusps is obtained by doubling a triangle with one cusp and non-zero angles /"a" and /2. So consider the geodesic triangle in the upper half plane with angles 0, /"a", /"b" with "a", "b" integers greater than one. The interior of such a triangle can be realised as the region "X" in the upper half plane lying outside the unit disk |"z"| ≤ 1 and between two lines parallel to the imaginary axis through points "u" and "v" on the unit circle. Let Γ be the triangle group generated by the three reflections in the