Triangle of his "Elements", around 300 BC. The sum of the measures of the interior angles of a triangle in Euclidean space is always 180 degrees. This fact is equivalent to Euclid's parallel postulate. This allows determination of the measure of the third angle of any triangle given the measure of two angles. An "exterior angle" of a triangle is an angle that is a linear pair (and hence supplementary) to an interior angle. The measure of an exterior angle of a triangle is equal to the sum of the measures of the two interior angles that are not adjacent to
Schwarz triangle function case the region in the sector between the radii "OA" and "OB" outside the edge "AB" is convex as the intersection of three convex regions. By induction the angles at "A" and "B" are greater than or equal to /3. Thus the geodesics to "C" from "A" and "B" start off in the region; by convexity, the triangle "ABC" lies wholly inside the region. The quadrilateral "OACB" has all its angles less than (since "OAB" is a geodesic triangle), so is convex. Hence the radius "OC" lies inside the angle of the triangle "ABC" near "C". Thus the angles between