The fifth axiom of Euclidean geometry says that for any given point T, that does not lie on a line p, there exists exactly one line through T that does not intersect p. If we disregard this axiom, we get different non-Euclidean geometries. We will investigate the hyperbolic geometry, where for each line an infinite number of parallel lines can be drawn through a given point. First, we will define the hyperbolic plane, then we will explore isometries of hyperbolic plane and prove that every isometry that preserves orientation can be written in the form of a Möbius transformation. We will show that isometries that preserve orientation form a group of isometries in the hyperbolic plane. In the end, we will prove some of the fundamental theorems of hyperbolic trigonometry.
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