In what geometry is the sum of the interior angles of a triangle equal to 180 degrees?

Fact

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In plane Euclidean geometry, the sum of the three interior angles of a triangle is 180 degrees. This result arises in particular from the postulate of parallels and appears in the geometric tradition associated with Euclid’s “Elements”. However, it is not universal for all surfaces. On a sphere, a triangle formed by arcs of great circles can have a sum greater than 180 degrees; in hyperbolic geometry, it is lower. These differences are essential in modern geometry and in the study of curved spaces. The word “always” must therefore be limited explicitly to the Euclidean plane.