  This paper is dedicated to a lattice analog to the classical ``sum of
interior angles of a polygon theorem''. In 2008, the first formula expressing
conditions on the geometric continued fractions for lattice angles of triangles
was derived, while the cases of $n$-gons for $n > 3$ remained unresolved. In
this paper, we provide the complete solution for all integer $n$. The main
results are based on recent advances in geometry of continued fractions.
