  Inspired by the spin geometry theorem, two operators are defined which
measure angles in the quantum theory of geometry. One operator assigns a
discrete angle to every pair of surfaces passing through a single vertex of a
spin network. This operator, which is effectively the cosine of an angle, is
defined via a scalar product density operator and the area operator. The second
operator assigns an angle to two ``bundles'' of edges incident to a single
vertex. While somewhat more complicated than the earlier geometric operators,
there are a number of properties that are investigated including the full
spectrum of several operators and, using results of the spin geometry theorem,
conditions to ensure that semiclassical geometry states replicate classical
angles.
