Trekantsligningen

Trekantsligningen, or the triangle angle-sum equation, is a fundamental result in Euclidean geometry stating that the three interior angles of any triangle add up to a straight angle. In common notation, if a triangle has interior angles A, B and C, then A + B + C = 180 degrees, or in radians α + β + γ = π.

A standard proof uses parallel lines. Extend one side of the triangle and draw a line through the opposite vertex parallel to that side; the alternate interior angles show that angle A equals the sum of angles B and C, hence A + B + C = 180°. An equivalent view uses the general polygon interior angle sum formula, which for a triangle (n = 3) gives (3 − 2)π = π, corresponding to 180 degrees.

The statement is specific to Euclidean geometry. In non-Euclidean geometries the sum of the interior angles of a triangle differs from 180 degrees: in spherical geometry it is greater than 180 degrees, while in hyperbolic geometry it is less. This makes the triangle angle-sum a diagnostic for the curvature of the underlying space.

Applications of trekantsligningen include solving triangles, determining unknown angles, and validating measurements in surveying, architecture, and computer graphics. It also underpins many trigonometric methods used to relate angles and side lengths.

In summary, trekantsligningen expresses a key invariant of triangles in flat geometry, while diverging in curved spaces, highlighting the link between geometry and the nature of space.