  We consider a generalized angle in complex normed vector spaces. Its
definition corresponds to the definition of the well known Euclidean angle in
real inner product spaces. Not surprisingly it yields complex values as
`angles'. This `angle' has some simple properties, which are known from the
usual angle in real inner product spaces. But to do ordinary Euclidean geometry
real angles are necessary. We show that even in a complex normed space there
are many pure real valued `angles'. The situation improves yet in inner product
spaces. There we can use the known theory of orthogonal systems to find many
pairs of vectors with real angles, and to do geometry which is based on the
Greeks 2000 years ago.
