Special right triangle are in the ratio 1 : 2 : 3 and respectively measure 30° (), 60° (), and 90° (). The sides are in the ratio 1 : : 2. The proof of this fact is clear using trigonometry. The geometric proof is: The 30°–60°–90° triangle is the only right triangle whose angles are in an arithmetic progression. The proof of this fact is simple and follows on from the fact that if "α", , are the angles in the progression then the sum of the angles = 180°. After dividing by 3, the angle must be 60°. The right angle
Angular defect to less than 360°, on a spherical triangle the interior angles always add up to more than 180° (the exterior angles add up to "less" than 360°), and the angles in a hyperbolic triangle always add up to less than 180° (the exterior angles add up to "more" than 360°). In modern terms, the defect at a vertex or over a triangle (with a minus) is precisely the curvature at that point or the total (integrated) over the triangle, as established by the Gauss–Bonnet theorem. For a polyhedron, the defect at a vertex equals 2π minus the sum of all