  We show that when parameterized by triples of angles, the set of similarity
classes of labeled, oriented, possibly degenerate triangles has the natural
structure of the Clifford torus $\sf{T}$, a compact abelian Lie group. On this
torus the main triangle types form distinguished algebraic structures:
subgroups and cosets. The construction relies on a natural definition of
similarity for degenerate triangles.
  We analyze the set of (unrestricted) similarity classes using a uniform
probability measure on $\sf{T}$ and compute the relative measures of the
different triangle types.
  Our computations are compatible with the spherically symmetric probability
distribution analyzed in [Por94] and [ES15], which are based on vertices/side
lengths instead of angles.
