Triangle of these conditions hold: (a) each of the angles is positive, and (b) the angles sum to 180°. If degenerate triangles are permitted, angles of 0° are permitted. Three positive angles "α", "β", and "γ", each of them less than 180°, are the angles of a triangle if and only if any one of the following conditions holds: the last equality applying only if none of the angles is 90° (so the tangent function's value is always finite). There are thousands of different constructions that find a special point associated with (and often inside) a triangle, satisfying some unique property:
Sum of angles of a triangle sum of all three exterior angles, that equals to 360° in the Euclidean case (as for any convex polygon), is less than 360° in the spherical case, and is greater than 360° in the hyperbolic case. In the differential geometry of surfaces, the question of a triangle's angular defect is understood as a special case of the Gauss-Bonnet theorem where the curvature of a closed curve is not a function, but a measure with the support in exactly three points – vertices of a triangle. Sum of angles of a triangle In several geometries, a triangle has three "vertices" and