  In this paper angular curvature measures are investigated. Our first result
is a complete classification of translation-invariant angular smooth curvature
measures on $\mathbb{R}^n$. Subsequently, we use this result to show that the
class of angular curvature measures on a Riemannian manifold is preserved by
both the pullback by isometric immersions and the action of the
Lipschitz-Killing algebra. The latter confirms the angularity conjecture
formulated by A. Bernig, J.H.G. Fu, and G. Solanes.
