Saccheri–Legendre theorem Saccheri–Legendre theorem In absolute geometry, the Saccheri–Legendre theorem states that the sum of the angles in a triangle is at most 180°. Absolute geometry is the geometry obtained from assuming all the axioms that lead to Euclidean geometry with the exception of the axiom that is equivalent to the parallel postulate of Euclid. The theorem is named after Giovanni Girolamo Saccheri and Adrien-Marie Legendre. The existence of at least one triangle with angle sum of 180 degrees in absolute geometry implies Euclid's parallel postulate. Similarly, the existence of at least one triangle with angle sum of less than 180 degrees
Internal and external angles simple polygon and a line extended from an adjacent side. The interior angle concept can be extended in a consistent way to crossed polygons such as star polygons by using the concept of directed angles. In general, the interior angle sum in degrees of any closed polygon, including crossed (self-intersecting) ones, is then given by 180("n"–2"k")° where "n" is the number of vertices and the non-negative number "k" is the number of total revolutions of 360° one undergoes walking around the perimeter of the polygon. In other words, 360"k"° represents the sum of all the exterior angles. For example, for