  Let $C(L)$ be the right-angled Coxeter group defined by an abstract
triangulation $L$ of $\mathbb{S}^2$. We show that $C(L)$ is isomorphic to a
hyperbolic right-angled reflection group if and only if $L$ can be realized as
an acute triangulation. The proof relies on the theory of CAT(-1) spaces. A
corollary is that an abstract triangulation of $\mathbb{S}^2$ can be realized
as an acute triangulation exactly when it satisfies a combinatorial condition
called "flag no-square". We also study generalizations of this result to other
angle bounds, other planar surfaces and other dimensions.
