Abstract. For every two points z 0 , z 1 in the upper-half plane H, consider all elements γ in the principal congruence group Γ(N ), acting on H by fractional linear transformations, such that the hyperbolic distance between z 1 and γz 0 is at most R > 0. We study the distribution of angles between the geodesic rays [z 1 , γz 0 ] as R → ∞, proving that the limiting distribution exists independently of N and explicitly computing it. When z 1 = z 0 this is found to be the uniform distribution on the intervalˆ− π 2 , π 2˜.