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Here are the first four axioms of Euclidean Geometry

* Any two points can be joined by a straight line.

* Any straight line segment can be extended indefinitely in a straight line.

* Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as centre.

* All right angles are congruent.

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In Non-Euclidean Geometry we use the first four axioms of

Euclidean geometry and then replace the fifth postulate

(axiom) with an alternative version.

One

point outside it, there is exactly one line through the point

parallel with the given line. We therefore get two alternative

versions:

* No possible parallel lines

** Spherical Geometry

** Angles in triangle add up to more than 180 degrees

* Many possible parallel lines

** Hyperbolic geometry

** Angles in triangle add up to less than 180 degrees

*** See Poincares Disc