  A geometric graph is angle-monotone if every pair of vertices has a path
between them that---after some rotation---is $x$- and $y$-monotone.
Angle-monotone graphs are $\sqrt 2$-spanners and they are increasing-chord
graphs. Dehkordi, Frati, and Gudmundsson introduced angle-monotone graphs in
2014 and proved that Gabriel triangulations are angle-monotone graphs. We give
a polynomial time algorithm to recognize angle-monotone geometric graphs. We
prove that every point set has a plane geometric graph that is generalized
angle-monotone---specifically, we prove that the half-$\theta_6$-graph is
generalized angle-monotone. We give a local routing algorithm for Gabriel
triangulations that finds a path from any vertex $s$ to any vertex $t$ whose
length is within $1 + \sqrt 2$ times the Euclidean distance from $s$ to $t$.
Finally, we prove some lower bounds and limits on local routing algorithms on
Gabriel triangulations.
