you have considered the case that the diagonals intersect at the centre of the circle. But if the number of vertices is odd the diagonals don't intersect at the centre of the circle.
If you have a regular n-gon then you have:
from one vertix are starting (n-1) lines (2 sides and (n-3) diagonals)
these lines are forming (n-2) angles which have the same size.
the size of these angles can be calculated by:
In your case:
That means: You have a 18-gon and between the diagonals in question there are 5 steps à 10°