This is the question I am given, and I have a model answer for it as well... but I am having difficulty understanding it.
What I can see is that the points are on a unit circle. Of course I can also see that $360°$$/$$72$ $=$ $5°$.
Other than that I don't know how to approach this question.
As the points are on the unit circle, all you have to show is the polar angles of the $72$ vertices are attained.
In particular, you should have $k\cdot 35\equiv 5\mod 360\,$ for some $k\in\mathbf Z$. This is equivalent to: $$7k\equiv 1\mod72$$ So you have to find the inverse of $7$ modulo $72$. This is done through the Extended Euclidean Algorithm. You'll get $$k\equiv 31\mod 72$$ From ther, it is not difficult to obtain the polar angles of the other vertices.