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Euclidean and Non-Euclidean GeometriesMany people do not know that a triangle can be made with three right angles. It may seem impossible to most people as it is in Euclidean geometry. However. a triangle with three right angles can be made in non-Euclidean geometry such as hyperbolic and elliptic geometry. This fact proves that Euclidean and non-Euclidean geometries are radically different. In order to contrast and compare the three geometries, one must look at how they are created, the theorems that apply to them, and their applications in real life starting with Euclidean geometry.Euclidean geometry is the most widely known geometry out of the others. It is the study based on the axioms and theorems made by Greek mathematician Euclid. The mathematician created The Elements written c.300 BC which consisted of 13 books which are collections of definitions, axioms, propositions, and proofs of the propositions. There are some theorems postulates that are unique to the geometry such as the shortest distance between two points is one unique straight line and that the sum of the angles in any triangle equals 180 degrees. However, they may not apply to other geometries. Euclidean geometry has a limited amount of applications in real life because it is the study of flat space. While useful for architecture and to survey land, Euclidean geometry does not hold up as well when moved to the third dimension which contributes to the creation of non-Euclidean geometry. Euclids most famous postulate, which is his fifth, is that if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, if produced indefinitely, meet on that side on which are the angles less than the two right angles. Today, most know this postulate as given a line and a point not on the line, it is possible to draw exactly one line through the given point parallel to the line. His fifth postulate contributed...

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