Di Francesco introduced Aztec triangles as combinatorial objects for which their domino tilings are equinumerous with certain sets of configurations of the twenty-vertex model that are the main focus of his article. We generalize Di Francesco's construction of Aztec triangles. While we do not know whether there is again a correspondence with configurations in the twenty-vertex model, we prove closed-form product formulas for the number of domino tilings of our generalized Aztec triangles. As a special case, we obtain a proof of Di Francesco's conjectured formula for the number of domino tilings of his Aztec triangles, and thus for the number of the corresponding configurations in the twenty-vertex model.