  We classify rational triangles which unfold to Veech surfaces when the
largest angle is at least $\frac{3\pi}{4}$. When the largest angle is greater
than $\frac{2\pi}{3}$, we show that the unfolding is not Veech except possibly
if it belongs to one of six infinite families. Our methods include a criterion
of Mirzakhani and Wright that built on work of M\"oller and McMullen, and in
most cases show that the orbit closure of the unfolding cannot have rank 1.
