  We trace the development of arguments for the consistency of non-Euclidean
geometries and for the independence of the parallel postulate, showing how the
arguments become more rigorous as a formal conception of geometry is
introduced. We use these philosophical views to explain why the certainty of
Euclidean geometry was threatened by the development of what we regard as
alternatives to it. Finally, we provide a model that creates the basis for
proof and demonstration using natural language in order to study the acute
angles hypothesis on a two right-angled isosceles quadrilateral, with a
rectilinear summit side.
