  Within the synthetic-geometric framework of Lorentzian (pre-)length spaces
developed in Kunzinger and S\"amann (Ann. Glob. Anal. Geom. 54(3):399--447,
2018) we introduce a notion of a hyperbolic angle, an angle between timelike
curves and related concepts like timelike tangent cone and exponential map.
This provides valuable technical tools for the further development of the
theory and paves the way for the main result of the article, which is the
characterization of timelike curvature bounds (defined via triangle comparison)
with an angle monotonicity condition. Further, we improve on a geodesic
non-branching result for spaces with timelike curvature bounded below.
