The independence of the parallel postulate and the acute angles hypothesis on a two right-angled isosceles quadrilateral

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.