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How do Euclidean, spherical, and hyperbolic geometry differ in their parallel lines?
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Euclidean geometry
This diagram depicts three basic theorems that triangles are congruentâthat is, they are of equal shape and sizeâif: (1) two sides and the included angle are equal, designated SAS; (2) two angles and the included side are equal, designated ASA; or (3) all three sides are equal, designated SSS.
In Euclidean geometry, the behavior of parallel lines is defined by the parallel postulate, which states that through any given point not on a line, there passes exactly one line parallel to the given line in the same plane.12
When this postulate is altered, it results in non-Euclidean geometries, which differ in how they handle parallel lines:12
- Spherical geometry: In this system, there are no lines parallel to a given line through a point not on that line. Because the "straight lines" are great circles, any two lines will eventually intersect.2
- Hyperbolic geometry: In this system, there are infinitely many lines passing through a given point that do not intersect the given line.2
These differences arise because each geometry assumes a different curvature for the space in which it operates: Euclidean geometry assumes a flat plane, spherical geometry involves constant positive curvature, and hyperbolic geometry involves constant negative curvature.23