A-5: If a straight line crosses two straight lines the produced inner angle on the similar region is less than two right angles. Thus, the two indefinite straight lines then produced will met on the side, where the sum of the angle will be less than two right angles ("Euclids Axioms and Postulates")The peculiar and unique quality of right angle is its measurement; every right angle will always be of 90 degree. Similarly, their length and direction of arms are insignificant aspects unless they measure complete 90 degree. Through these axioms Euclid aimed to prove few common concepts and developed their foundation; so they could not be argued.
Like for instance, Parallel axioms entail the concept of two parallel lines, which do not intersect at any point even if they are extended. Yet, parallel axiom has been argued amongst the mathematicians due to its very nature. They deemed that there was no need to develop such a notion and it could be used to prove the above four axioms. However, in the eighteen century Euclid’ s axiom were considered the proven fact or obvious truths, but later on they were further researched and challenged. Thus, in nineteen century, Russian Ivanovitch Lobachevski, German Karl Gauss and from Hungary Já nos Bolyai researched on negation of parallel axiom.
These mathematicians proved that negation of parallel axiom resulted in inconsistency and could not be taken away from the first four axioms. Instead these researchers developed a new form of geometry through this research, which is known as hyperbolic geometry. Hyperbolic geometry is used by Albert Einstein in his theory of general
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