  We discuss the existence of the angle between two curves in Teichm\"uller
spaces and show that, in any infinite dimensional Teichm\"uller space, there
exist infinitely many geodesic triangles each of which has the same three
vertices and satisfies the property that its three sides have the same and
arbitrarily given length while its three angles are equal to any given three
possibly different numbers from 0 to $\pi$. This implies that the sum of three
angles of a geodesic triangle may be equal to any given number from 0 to $3\pi$
in an infinite dimensional Teichm\"uller space.
