From "[A.D. Alexandrov spaces with curvature bounded below][1]",
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_, p.1,
> 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).


  [1]: http://iopscience.iop.org/0036-0279/47/2/R01;jsessionid=D6F567EBB64F297A7AAD9AC0AF206739.c1