= Solution
A regular hyperbolic hexagon has angle tending to $2\pi/3$ as its radius tends to zero and to zero as its vertices approach the ideal boundary. Continuity therefore gives a radius at which every angle is $\pi/2$.
Two congruent right-angled hexagons glued along three alternating sides form a hyperbolic pair of pants with geodesic boundary. Glue $2g-2$ such pairs of pants along boundary curves in the usual pants decomposition. The right angles match smoothly and produce a closed hyperbolic surface of genus $g$ for every $g\geq2$.
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