We extend the classical Gauss-Bonnet theorem for the Euclidean, elliptic, hyperbolic, and Lorentzian planes to the other three Cayley-Klein geometries of dimension two, all three of which are absolute-time spacetimes, providing one proof for all nine geometries. Suppose that M is a polygon in any one of the nine geometries. Let Γ, the boundary of M, have length element ds, discontinuities θ_i, and signed geodesic curvature κ_g, where M and Γ are oriented according to Stokes' theorem. Let K denote the constant Gaussian curvature of the geometry with area form dA. Then ∫_Γ κ_g ds + ∑_i θ_i + ∫∫_M K dA = 2\pi for the nonspacetimes and ∫_Γ κ_g ds + ∑_i θ_i + ∫∫_M K dA = 0 for the spacetimes, where we assume that Γ is timelike.