Triangle Angle Sum Theorem

The Triangle Angle Sum Theorem states that for any triangle lying in the Euclidean plane that the sum of the angles is equal to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 180^o} or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2 \pi } radians.

This is one of the facts that characterizes Euclidean geometry: it is not, however, true in other geometries. The proof for this theorem is reliant on the parallel postulate, so does not generalize into other geometries. For example, any triangle constructed on the surface of the earth (known as a spherical triangle) the sum of the angles is always greater than Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 180^o } .