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 .