Distribution of angles between geodesic rays associated with hyperbolic lattice points

For every two points z₀, Z₁ 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 z₁ and γz₀ is at most R > 0. We study the distribution of angles between the geodesic rays [z ₁, γz₀] as R → ∞, proving that the limiting distribution exists independently of N and explicitly computing it. When z₁ = z₀, this is found to be the uniform distribution on the interval [-π/2, π/2].