An axiom system for absolute plane geometry in terms of collinearity and congruence

We present an axiom system in terms of collinearity and congruence for plane geometry independent of assumptions about parallels. The line reflections and motions of corresponding geometries are studied, and it is shown that the axiom system characterizes the non-elliptic metric planes in the sense of Bachmann with midpoints. If the hyperbolic axiom is added, we show that the familiar algebraic models of hyperbolic planes result.