Question on angles in translation surfaces Consider a translation surface $(X,\omega)$ and let $p_1\in X$ be a point of conical singularity and $p,p_2\in X$ non singular points. 
Let $s_1$ and $s_2$ be two geodesics for the singular flat metric (often called saddle connections) respectively from $p$ two $p_1$ and from $p$ to $p_2$ such that $l(s_1)=d(p,p_1)$ and $l(s_2)=d(p,p_2)$ where the length and the distance are the ones induced by $\omega\overline{\omega}$.
Consider the triangle $T\subset \mathbb{R}^2$ with sides of length $d(p,p_1),d(p,p_2),d(p_1,p_2)$. Call $\alpha$ the angle at the vertex between sides of lengths $d(p,p_1)$ and $d(p,p_2)$.
Is $\alpha$ equal to the angle at $p$ in $(X,\omega)$ between $s_1$ and $s_2$?

The angles are different (in general). Let $q$ be a singular point, $p_1$ another singular point. Then you can find regular points $p, p_2$ distinct from $q$ such that the angles between the three segments $qp_1, qp_2, qp$ are equal to $\pi, \pi$ and $2\pi$ respectively. Then the angle at $p$ between $pp_1, pp_2$ is $0$ (both segments pass through $q$), but the corresponding comparison angle $\alpha$ is $\ne 0$. 
Note furthermore that if $T$ is complete and simply-connected, it is a CAT(0) space, hence by the hinge comparison theorem angles in $T$ are $\le$ the comparison angles in $E^2$. As a reference, take a look at Ballmann's book "Lectures on spaces of nonpositive curvature".
