  We investigate the Tits boundary of locally compact CAT(0) 2-complexes. In
particular we show that away from the endpoints, a geodesic segment in the Tits
boundary is the ideal boundary of an isometrically embedded Euclidean sector.
As applications, we provide sufficient conditions for two points in the Tits
boundary to be the endpoints of a geodesic in the 2-complex and for a group
generated by two hyperbolic isometries to contain a free group. We also show
that if two CAT(0) 2-complexes are quasi-isometric then the cores of their Tits
boundaries are bi-Lipschitz.
