In this thesis the axiomatic basis will be a general absolute plane in the sense of H. Karzel. The word “general” means that no claim is made on any kind of continuity assumptions. We show that an absolute geometry has either a singular, a hyperbolic or an elliptic congruence. By this notions we get a complete characterization of different possibilities which can occur in a general absolute plane studying the value of the fourth angle of any Lambert-Saccheri quadrangle or, equivalently, the sum of the angles of any triangle. This yields, in particular, a Archimedse-free proof a statement generalizing the In the following way, we get some characterizations to an absolute geometry with triangles and Lambert-Saccheri quadrangles by using the distance function defined in absolute plane which are equivalent to the validty of the parallel postulate.