Operators for quantized directions

Inspired by the spin geometry theorem, two operators are defined which meas ure angles in the quantum theory of geometry. One operator assigns a discre te angle to every pair of surfaces passing through a single vertex of a spi n network. This operator, which is effectively the cosine of an angle, is d efined via a scalar product density operator and the area operator. The sec ond operator assigns an angle to two 'bundles' of edges incident to a singl e vertex. While somewhat more complicated than the earlier geometric operat ors, there are a number of properties that are investigated including the f ull spectrum of several operators and, using results of the spin geometry t heorem, conditions to ensure that semiclassical geometry states replicate c lassical angles.