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We trace the development of arguments for the consistency of non-Euclideangeometries and for the independence of the parallel postulate, showing how thearguments become more rigorous as a formal conception of geometry isintroduced. We use these philosophical views to explain why the certainty ofEuclidean geometry was threatened by the development of what we regard asalternatives to it. Finally, we provide a model that creates the basis forproof and demonstration using natural language in order to study the acuteangles hypothesis on a two right-angled isosceles quadrilateral, with arectilinear summit side.

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