  This is the second in a series of papers in which we investigate ideal
triangulations of the interiors of compact 3-manifolds with tori or Klein
bottle boundaries. Such triangulations have been used with great effect,
following the pioneering work of Thurston. Ideal triangulations are the basis
of the computer program SNAPPEA of Weeks and the program SNAP of Coulson,
Goodman, Hodgson and Neumann. Casson has also written a program to find
hyperbolic structures on such 3-manifolds, by solving Thurston's hyperbolic
gluing equations for ideal triangulations.
  In this second paper, we study the question of when a taut ideal
triangulation of an irreducible atoroidal 3-manifold admits a family of angle
structures.
  We find a combinatorial obstruction, which gives a necessary and sufficient
condition for the existence of angle structures for taut triangulations. The
hope is that this result can be further developed to give a proof of the
existence of ideal triangulations admitting (complete) hyperbolic metrics.
  Our main result answers a question of Lackenby. We give simple examples of
taut ideal triangulations which do not admit an angle structure. Also we show
that for `layered' ideal triangulations of once-punctured torus bundles over
the circle, that if the manodromy is pseudo Anosov, then the triangulation
admits angle structures if and only if there are no edges of degree 2. Layered
triangulations are generalisations of Thurston's famous triangulation of the
Figure 8 knot space. Note that existence of an angle structure easily implies
that the 3-manifold has a CAT(0) or relatively word hyperbolic fundamental
group.
