To an absolute plane (E,L, there will be associated an ordered commutative group (W , + , ) such that (W , + ) is a subgroup of the corresponding K-loop (E, + ) of the absolute plane and a cyclic ordered commutative group ( , · , ? ) where ( , · ) is isomorphicto a rotation group fixing a point. (W , + , ) , resp. ( , · , ? ) , will serve to introduce a distance ? describing the congruence and satisfying the triangular inequality or resp. a measure ? for angles describing the congruence and conjugacy of angles. Keywords: Absolute plane; Distance function; Measure of angles; Group; Loop