  The orbit closure of the unfolding of every rational right and isosceles
triangle is computed and the asymptotic number of periodic billiard
trajectories in these triangles is deduced. This follows by classifying all
orbit closures of rank at least two in hyperelliptic components of strata of
Abelian and quadratic differentials. Additionally, given a fixed set of angles,
the orbit closure of the unfolding of all unit area rational parallelograms,
isosceles trapezoids, and right trapezoids outside of a discrete set is
determined.
