Non-Euclidean geometry

Non-Euclidean geometry replaces the fifth postulate of the Euclidean geometry by one of two alternative postulates. The first postulate allows two parallels through any external point and is known as the hyperbolic geometry which is attributed to Lobachevski and Bolyai independently. The second postulate allows no parallels through any external point and is known as the elliptic geometry attributed to Riemann. The results of these two types of non-Euclidean geometry contradict those of Euclidean geometry only when dealing with parallel lines, either explicitly or implicitly, which alters the theorem of the sum of the angles in a triangle.

Reference: (2009). Non-Euclidean geometry. Columbia Electronic Encyclopedia, 6th Edition, 1. http://search.ebscohost.com/
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