Author’s introduction: ”The main part of the classical trigonometry is devoted to revealing the relations between the sides and the angles associated with a triangle in a space form. In a series of papers we are going to generalize such relations to the case in which the ambient space is a Riemannian manifold. For the sake of simplicity, we first discuss geodesic triangles in a two dimensional oriented Riemannian manifold, thus can unify the trigonometries of Euclidean geometry, Riemannian elliptic geometry and Lobachevskij geometry.”