sum of angles of triangle in Euclidean geometry

The parallel postulateMathworldPlanetmath (in the form given in the parent entry ( allows to prove the important fact about the trianglesMathworldPlanetmath in the Euclidean geometryMathworldPlanetmath:

Theorem.  The sum of the interior anglesMathworldPlanetmath of any triangle equals the straight angleMathworldPlanetmath.

Proof.  Let ABC be an arbitrary triangle with the interior angles α, β, γ.  In the plane of the triangle we set the lines AD and AE such that  BAD=β  and  CAE=γ.  Then the lines do not intersect the line BC.  In fact, if e.g. AD would intersect BC in a point P, then there would exist a triangle ABP where an exterior angle ( of an angle would equal to an interior angle of another angle which is impossible.  Thus AD and AE are both parallelMathworldPlanetmathPlanetmath to BC.  By the parallel postulate, these lines have to coincide.  This means that the addition of the triangle angles α, β, γ gives a straight angle.

See also intuitive proof!


  • 1 Karl Ariva: Lobatsevski geomeetria.  Kirjastus “Valgus”, Tallinn (1992).
Title sum of angles of triangle in Euclidean geometry
Canonical name SumOfAnglesOfTriangleInEuclideanGeometry
Date of creation 2013-09-26 10:00:16
Last modified on 2013-09-26 10:00:16
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 4
Author pahio (2872)
Entry type Theorem
Classification msc 51M05