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In this work we study a family of discontinuous Galerkin methods for the displacement obstacle problem of Kirchhoff plates on convex polyhedral domains, which are characterized as fourth order elliptic variational inequalities of the first kind. We develop a unified approach for DG methods where the weak complementarity form of the variational inequality is used. We derive the optimal error estimate in energy norm for the quadratic method, where the convergence rate is determined by the geometry of the domain. Under additional regularity assumptions on the solution and contact set, we derive an improved error estimate for the cubic method. Numerical experiments demonstrate the performance of the methods and confirm the theoretical results.