Angle geometry in asymptotic Teichmüller spaces

The angular geometry of asymptotic Teichmüller spaces is studied. Although it has been proved that the angles between any two geodesic rays from the base point of AT(X)AT(X) always exist, it is shown in this paper that there are infinitely many pairs of intersecting geodesic segments such that the angles between them do not exist. The sums of inner angles of geodesic triangles are also studied. It is proved that for any number from 0 to 3π   there exists a geodesic triangle in AT(X)AT(X) such that the sum of its inner angles is equal to such a number.