  Consider a graph with a rotation system, namely, for every vertex, a circular
ordering of the incident edges. Given such a graph, an angle cover maps every
vertex to a pair of consecutive edges in the ordering -- an angle -- such that
each edge participates in at least one such pair. We show that any graph of
maximum degree 4 admits an angle cover, give a poly-time algorithm for deciding
if a graph with no degree-3 vertices has an angle-cover, and prove that, given
a graph of maximum degree 5, it is NP-hard to decide whether it admits an angle
cover. We also consider extensions of the angle cover problem where every
vertex selects a fixed number $a>1$ of angles or where an angle consists of
more than two consecutive edges. We show an application of angle covers to the
problem of deciding if the 2-blowup of a planar graph has isomorphic thickness
2.
