I was thinking about the name "Triangle", when I realized that although we usually think of polygons in terms of the number of their sides. However, when I searched the origin of the word "polygon", Oxford Languages states the following: "late 16th century: via late Latin from Greek polugōnon, neuter (used as a noun) of polugōnos ‘many-angled’."
Therefore, I was wondering whether polygons would always have the same number of edges and vertices. In formulating a proof, I considered a simply connected graph in $\mathbb{R}^2$, and using Euler's characteristic formula $V+F-E=2$, I concluded that this statement should always hold since there is 1 face formed on the inside of the simply connected graph and 1 on the outside, resulting in 2 faces in total and giving the desired identity.
Please correct me if this is not rigorous, as I have not studied graph theory formally and am only basing this on my limited knowledge of it.
Finally, I'd like to ask whether there are other known useful relationships between vertices in faces in $\mathbb{R}^2$. Also, is there an angle for each vertex in $\mathbb{R}^2$? What about in $\mathbb{R}^n$ (is there an angle for each vertex)?