Comparison triangle

Define $$M_{k}^{2}$$ as the 2-dimensional metric space of constant curvature $$k$$. So, for example, $$M_{0}^{2}$$ is the Euclidean plane, $$M_{1}^{2}$$ is the surface of the unit sphere, and $$M_{-1}^{2}$$ is the hyperbolic plane.

Let $$X$$ be a metric space. Let $$T$$ be a triangle in $$X$$, with vertices $$p$$, $$q$$ and $$r$$. A comparison triangle $$T*$$ in $$M_{k}^{2}$$ for $$T$$ is a triangle in $$M_{k}^{2}$$ with vertices $$p'$$, $$q'$$ and $$r'$$ such that $$d(p,q) = d(p',q')$$, $$d(p,r) = d(p',r')$$ and $$d(r,q) = d(r',q')$$.

Such a triangle is unique up to isometry.

The interior angle of $$T*$$ at $$p'$$ is called the comparison angle between $$q$$ and $$r$$ at $$p$$. This is well-defined provided $$q$$ and $$r$$ are both distinct from $$p$$.