sum of angles of triangle in Euclidean geometry
Proof. Let be an arbitrary triangle with the interior angles , , . In the plane of the triangle we set the lines and such that and . Then the lines do not intersect the line . In fact, if e.g. would intersect in a point , then there would exist a triangle where an exterior angle (http://planetmath.org/ExteriorAnglesOfTriangle) of an angle would equal to an interior angle of another angle which is impossible. Thus and are both parallel to . By the parallel postulate, these lines have to coincide. This means that the addition of the triangle angles , , gives a straight angle.
See also http://www.cs.bham.ac.uk/research/projects/cogaff/misc/triangle-sum.html#pardoe-proofthis intuitive proof!
- 1 Karl Ariva: Lobatsevski geomeetria. Kirjastus “Valgus”, Tallinn (1992).
|Title||sum of angles of triangle in Euclidean geometry|
|Date of creation||2013-09-26 10:00:16|
|Last modified on||2013-09-26 10:00:16|
|Last modified by||pahio (2872)|