From "A.D Alexandrov spaces with curvature bounded below", Burago, Y. and Gromov, M. and Perel'man, G., Russian Mathematical Surveys, 47, 1992, p.5:
A locally complete space $Μ$ with intrinsic metric is called a space with curvature $\ge k$ if in some neighbourhood $U_x$ of each point $x \in M$ the following condition is satisfied: (D) for any four (distinct) points $(a, b, c, d)$ in $U_x$ we have the inequality $\tilde \angle bac + \tilde \angle bad + \tilde \angle cad \le 2\pi$. If the space $Μ$ is a one-dimensional manifold and $k > 0$, then we require in addition that diam $M$ does not exceed $\pi/\sqrt{k}$.
(Here $\tilde \angle pqr$ is the angle at the vertex $\tilde q$ of the triangle $\tilde \triangle pqr$ on the two-dimensional "$k$-plane" of curvature $k$, which has side lengths $|pq|$, $|qr|$, $|rq|$.)
They say in the Introduction,
We are talking, roughly speaking, about spaces with an intrinsic metric, for which the conclusion of Toponogov's angle comparison theorem is true (although only in the small).