We report on the theoretical investigation of the topological properties of a periodically quenched one-dimensional dimerized lattice where a piecewise constant Hamiltonian switches from h1 to h2 at a partition time tp within each driving period T. We examine different dimerization patterns for h1 and h2 and the interplay with the driving parameters that lead to the emergence of topological states both at zero energy and at the edge of the Brillouin-Floquet quasienergy zone. We illustrate different phenomena, including the occurrence of both edge states in a semimetal spectrum, the topological transitions, and the generation of zero-energy topological states from trivial snapshots. The role of the different symmetries in our results is also discussed.