  For a locally finite set in $\mathbb{R}^2$, the order-$k$ Brillouin
tessellations form an infinite sequence of convex face-to-face tilings of the
plane. If the set is coarsely dense and generic, then the corresponding
infinite sequences of minimum and maximum angles are both monotonic in $k$. As
an example, a stationary Poisson point process in $\mathbb{R}^2$ is locally
finite, coarsely dense, and generic with probability one. For such a set, the
distribution of angles in the Voronoi tessellations, Delaunay mosaics, and
Brillouin tessellations are independent of the order and can be derived from
the formula for angles in order-$1$ Delaunay mosaics given by Miles in 1970.
