2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/103805Let M be the interior of a compact 3-manifold with non-empty boundary, and T be an ideal (topological) triangulation of M. This paper describes necessary and sufficient conditions for the existence of angle structures, semi-angle structures and generalised angle structures on (M; T) respectively in terms of a generalised Euler characteristic function on the solution space of normal surface theory of (M; T). This extends previous work of Kang and Rubinstein, and is itself generalised to a more general setting for 3-dimensional pseudo-manifolds.17 pages, 3 figures, to appear in Trans. Amer. Math. SocGeometric Topology57M25; 57N10text