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How do triangle angle sums differ in Euclidean, spherical, and hyperbolic geometry?
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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 geometry, the sum of a triangle's interior angles depends on the curvature of the space in which the triangle exists.1
- Euclidean Geometry: In this flat, zero-curvature space, the sum of the interior angles of a triangle is always exactly 180 degrees (or radians).21
- Spherical Geometry: On a surface with constant positive curvature, such as a sphere, the sum of the interior angles of a triangle is always greater than 180 degrees. The amount by which the sum exceeds 180 degrees is known as the spherical excess.31
- Hyperbolic Geometry: On a surface with constant negative curvature, the sum of the interior angles of a triangle is always less than 180 degrees.1
These differences arise because the parallel postulateâwhich holds true in Euclidean geometryâdoes not apply to non-Euclidean spaces, where the number of lines parallel to a given line through a point varies.1