We prove that if N points lie in convex position in the plane then they determine \(\Omega (N^{5/4})\) distinct angles, provided that the points do not lie on a common circle. This is derived from a more general claim that if N points in the convex position in the real plane determine KN distinct angles, then \(K=\Omega (N^{1/4})\) or \(\Omega (N/K)\) points are co-circular. The proof makes use of the implicit order one can give to points in convex position and relies on a slightly more general order assumption. The assumption enables one to reduce the issue to counting incidences between points and a multiset of cubic curves, with special attention being paid to the case when the curves are reducible.
