  In this paper we study properties of lattice trigonometric functions of
lattice angles in lattice geometry. We introduce the definition of sums of
lattice angles and establish a necessary and sufficient condition for three
angles to be the angles of some lattice triangle in terms of lattice tangents.
This condition is a version of the Euclidean condition: three angles are the
angles of some triangle iff their sum equals \pi. Further we find the necessary
and sufficient condition for an ordered n-tuple of angles to be the angles of
some convex lattice polygon. In conclusion we show applications to theory of
complex projective toric varieties, and a list of unsolved problems and
questions.
