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So I have 4 collections of lines drawn between points each making a path. The angles are measured. The problem I am attempting to solve is to determine whether or not each of the collections of points are a valid path between some subset of points on a regular 13-gon. So for instance, for the second shape are there 3 vertices in a regular 13 gon such that drawing a line between them gives a 60 degree angle (within 5 degrees of error, see context). I'm having a hard time figuring this out myself. I know the geometry involved but I'm just having a hard time applying it to this problem.

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Some Context

The context of this problem is actually a clue given in the hunt for the secret zoo level of the video game Accounting Plus VR and this was my attempt to solve what these diagrams mean. So because I measured the angles with an actual protractor they have 5 degrees of precision. So if there are valid points with approximate angles that is also a valid answer. I hope this doesn't make the question unanswerable but this is what I have to work with geometry wise. This is effectively an actual "real" problem with imperfect diagrams, rather than an ideal theoretical problem coming out of a book.

And yes the images are the original images I obtained for reference blown up in size. So if one wants to remeasure the angles in case my protractor skill is flawed, they are welcome to.

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  • $\begingroup$ The vertices of a regular $n$-gon break the circle into equal arcs of measure $360^\circ/n$, so any angle joining two vertices to the center must be a multiple of that measure. By the Inscribed Angle Theorem, any angle formed by joining two vertices to another vertex must be a multiple of half that angle, aka $180^\circ/n$. Therefore, for a $13$-gon, you'd expect the angles in the path to be multiples of $13.845\ldots^\circ$. $\endgroup$
    – Blue
    Jan 13 '19 at 7:28
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(Converting a comment to an answer, as requested.)

The vertices of a regular $n$-gon break the circle into equal arcs of measure $360^\circ/n$, so any angle formed by joining two vertices to the center must be a multiple of that measure. By the Inscribed Angle Theorem, any angle formed by joining two vertices to another vertex must be a multiple of half that angle, aka $180^\circ/n$.

Therefore, for a $13$-gon, you'd expect the angles in your paths to be multiples of $13.845\ldots^\circ$.

Taking your measurements of $25^\circ$, $30^\circ$, $40^\circ$, $45^\circ$, $50^\circ$, $60^\circ$ as accurate, you'd seem to need an $n$-gon that allows for inscribed angles that are multiples of $5^\circ$; in that case, $n=36$. This is close to $39$, which is a multiple of $13$. Note that $26$ is also a multiple of $13$ (as well as being the number of letters in the English alphabet), and it allows for inscribed angles that are a multiple of $6.932\ldots^\circ$.

In any event, you can use the $180^\circ/n$ formula to check the viability of your guesses for $n$ and/or the accuracy of your angle measurements.

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