The angular resolution of a straight-line drawing of a graph is the smallest angle formed by any two edges incident to a vertex. The angular resolution of a graph is the supremum of the angular resolutions of all straight-line drawings of the graph. We show that testing whether a graph has angular resolution at least $\pi/(2k)$ is complete for $\exists\mathbb{R}$, the existential theory of the reals, for every fixed $k \geq 2$. This remains true if the graph is planar and a plane embedding of the graph is fixed.