The fifth postulate of Euclid states that if a transversal falls on two straight lines such that the interior angles on one side sum to less than two right angles, the two lines meet at a finite point on that side. In this paper we prove the converse: if two straight lines meet at a finite point, then a transversal cutting the two lines forms interior angles on the side of the meeting point that sum to less than two right angles. The proof is elementary and operates strictly within the framework of Euclid’s Book I. This paper is a companion to our earlier preprint on Euclid’s Fifth Postulate.
Radhakrishnamurty Padyala (Fri,) studied this question.