  Interior and exterior angle vectors of polytopes capture curvature
information at faces of all dimensions and can be seen as metric variants of
$f$-vectors. In this context, Gram's relation takes the place of the
Euler-Poincar\'e relation as the unique linear relation among interior angles.
We show the existence and uniqueness of Euler-Poincar\'e-type relations for
generalized angle vectors by building a bridge to the algebraic combinatorics
of geometric lattices, generalizing work of Klivans-Swartz.
  We introduce flag-angles of polytopes as a geometric counterpart to
flag-$f$-vectors. Flag-angles generalize the angle deficiencies of
Descartes-Shephard, Grassmann angles, and spherical intrinsic volumes. Using
the machinery of incidence algebras, we relate flag-angles of zonotopes to
flag-$f$-vectors of graded posets. This allows us to determine the linear
relations satisfied by interior/exterior flag-angle vectors.
