SymmetryDuality properties, referred to as dualityDualityproperties, between the kinematic and static problems Dualitykinematic vs static problems are discussed. Referring to general systems, it is proved that, by suitably reducing to the same poles displacements and forces, the congruence matrix and the equilibrium matrices are the transpose of each otherDualitycongruence vs equilibrium matrices. As a consequence, additional duality properties hold between compatibility matrices of over-determined systems and self-solution matrices of under-determined systems. Thus, the kinematic compatibility matrix is the transpose of the self-reactive matrix and the static compatibility matrix is the transpose of the modal matrix. Successively, the Virtual Work PrincipleVirtual Workprinciple is proved as a Theorem, which descends from duality. The Theorem is proved both in vector and in matrix forms. This latter expression permits to identify the Virtual Work Principle with the bilinear identity, which is known from Algebra. In addition to the Theorem, two Corollaries are proved, said of the Virtual DisplacementsVirtual Workvirtual displacements and of the Virtual ForcesVirtual Workvirtual forces. The Corollaries have important practical implications, since they permit to transform a static into a kinematic problem and vice versa, allowing the analyst to chose the simplest one. Strategic applications of the Corollaries to evaluate reactions, solicitations and displacements are illustrated, always solving the dual problem.