In the paper published in 1987 by Mitchell, Mount and Paradimitrou we find the following statement:
LEMMA 3.5. The general form of a geodesic path is a path which goes through an alternating sequence of vertices and (possibly empty) edge sequences such that the unfolded image of the path along any edge sequence is a straight line segment and the angle of the path passing through a vertex is greater than or equal to $\pi$. The general form of an optimal path is the same as that of a geodesic path, except that no edge can appear in more than one edge sequence and each edge sequence must be simple.
I keep reading the paper over and over again and I don't understand what "the angle of the path passing through a vertex is greater than or equal to $\pi$" means. The angle with respect to what? A line passing through a point doesn't make an angle.
Let vertex $v$ be given. Then we have the set of all faces $\{f_i\}$ touching $v$, which we may order $f_1,\dots,f_k$ such that all pairs $(f_i,f_{i+1})$ and also $(f_k,f_1)$ are edge-adjacent. This could be a clockwise or counterclockwise ordering; we will say increasing $i$ is the clockwise direction. The ordering of faces induces an ordering on the adjacent edges of $v$ by $e_1,\dots,e_k$, each $e_i$ connecting $(f_i,f_{i+1})$ and $e_k$ connecting $(f_k,f_1)$. Then the total angle of all faces at $v$ is: $$TA(v) = \sum_{i=1}^{k-1} \angle(e_i,e_{i+1}) + \angle(e_k,e_1)$$
Where $\angle$ means the unsigned angle between the two edges. Now consider the situation in which a path $p$ passes through a vertex $v$. The ordering of faces $f_i$ is cyclic, so wlog let's say $p$ passes through faces $f_1$, then $v$, then through face $f_j$. Analogous to the paper's notation, let $\alpha_1$ be the part of the path on $f_1$ and $\alpha_j$ be the part of the path on $f_j$. We define the clockwise angle as the angle: $$cw(p) = \angle(\alpha_1,e_1) + \sum_{i=1}^{j-1}\angle(e_i,e_{i+1}) + \angle(e_j,\alpha_j)$$
And the counterclockwise angle by: $$ccw(p) = \angle(\alpha_1,e_k) + \sum_{i=1}^{k-j+1} \angle(e_{k-i+1},e_{k-i}) + \angle(e_{j+1},\alpha_j)$$
Note that $cw(p) + ccw(p) = TA(v)$. The paper then defines the "angle made by $p$ at $v$" by $\angle p = \min\{cw(p),ccw(p)\}$. If $v$ is a vertex like figure 2 in the paper, this angle can be much greater than $\pi$, which is why lemma 3.5 is stated the way it is (figure 4 shows you why).