Define  as the 2-dimensional metric space of constant curvature . So, for example,  is the Euclidean plane,  is the surface of the unit sphere, and  is the hyperbolic plane.

Let  be a metric space. Let  be a triangle in , with vertices ,  and . A comparison triangle  in  for  is a triangle in  with vertices ,  and  such that ,  and .

Such a triangle is unique up to isometry. 

The interior angle of  at  is called the comparison angle between  and  at . This is well-defined provided  and  are both distinct from .

References
 M Bridson & A Haefliger - Metric Spaces Of Non-Positive Curvature, 

Metric geometry