  By using non-positively curved cubings of prime alternating link exteriors,
we prove that certain ideal triangulations of their complements, derived from
reduced alternating diagrams, are non-degenerate, in the sense that none of the
edges is homotopic relative its endpoints to a peripheral arc. This guarantees
that the hyperbolicity equations for those triangulations for hyperbolic
alternating links have solutions corresponding to the complete hyperbolic
structures. Since the ideal triangulations considered in this paper are often
used in the study of the volume conjecture, this result has a potential
application to the volume conjecture.
