Internal angle of a vertex of degree $d$ in $\mathbb{E}^2$ and $\mathbb{S}^2$

I am currently working on determining the maximum number of times the minimum spherical distance can occur among $n$ points in $\mathbb{S}^2$, and I have the following question.

In $\mathbb{E}^2$, I cite from Pach and Agarwal's Combinatorial Geometry: "The internal angle of a simple closed polygon $C$ which bounds a graph $G$ at a vertex of degree $d$ is at least $(d-1)\frac{\pi}{3}$." How does this statement generalize to $\mathbb{S}^2$?

In particular, I am not sure how the statement is arrived at in $\mathbb{E}^2$, and for this reason I am unable to generalize it to $\mathbb{S}^2$.

EDIT: To make the question clear, I am asking for a proof or explanation of the statement quoted in $\mathbb{E}^2$, and maybe an idea for how to generalize it to $\mathbb{S}^2$. An explanation of the statement in $\mathbb{E}^2$ is sufficient for an accepted answer though. When I try constructing a graph the $(d-1)\frac{\pi}{3}$ makes sense and gives a lower bound for the angle at a vertex of degree $d$ but I don't know where it is coming from.

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