In this paper, we establish the convergence of a nonconforming triangular Morley element for the plate bending problem on degenerate meshes. An explicit bound for the interpolation error is derived for arbitrary triangular meshes without any assumptions. The optimal convergence rates of the moment error and rotation error are derived for triangular meshes satisfying the maximal angle condition. Our results can also be extended to the three dimensional Morley element presented recently in [41]. Finally, some numerical results are reported that confirm our theoretical results.