  We characterize the largest point sets in the plane which define at most 1,
2, and 3 angles. For $P(k)$ the largest size of a point set admitting at most
$k$ angles, we prove $P(2)=5$ and $P(3)=5$. We also provide the general bounds
of $k+2 \leq P(k) \leq 6k$, although the upper bound may be improved pending
progress toward the Weak Dirac Conjecture. Notably, it is surprising that
$P(k)=\Theta(k)$ since, in the distance setting, the best known upper bound on
the analogous quantity is quadratic and no lower bound is well-understood.
