  We prove that if $N$ points lie in convex position in the plane then they
determine $N^{1+ 3/23-o(1)}$ distinct angles, provided the points do not lie on
a common circle. This is the first super-linear bound on the distinct angles
problem that has received recent attention. 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, recently used by Solymosi. Convex position also allows us
to divide our set into two large ordered pieces. We use the squeezing lemma and
the level sets of repeated angles to estimate the number of angles formed
between these pieces. We obtain the main theorem using Stevens and Warren's
convexity versus sum-set bound.
