The possibility of inducing topological superconductivity with cuprate high-temperature superconductors (HTSC) is studied for various heterostructures. We first consider a ballistic planar junction between a HTSC and a metallic ferromagnet. We assume that inversion symmetry breaking at the tunnel barrier gives rise to Rashba spin-orbit coupling in the barrier and allows equal-spin triplet superconductivity to exist in the ferromagnet. Bogoliubov-de Gennes equations are obtained by explicitly modeling the barrier and taking account of the transport anisotropy in the HTSC. By making use of the self-consistent boundary conditions and solutions for the barrier and HTSC regions, an effective equation of motion for the ferromagnet is obtained where Andreev scattering at the barrier is incorporated as a boundary condition for the ferromagnetic region. For a ferromagnet layer deposited on a (100) facet of the HTSC, triplet p-wave superconductivity is induced. For the layer deposited on a (110) facet, the induced gap does not have the p-wave orbital character, but has an even orbital symmetry and an odd dependence on energy. For the layer on the (001) facet, an exotic f-wave superconductivity is induced. We also consider the induced triplet gap in a one-dimensional half-metallic nanowire deposited on a (001) facet of a HTSC. Due to the breaking of translational symmetry in the direction perpendicular to the wire axis, the expression for the gap receives contributions from different perpendicular momentum eigenstates in the superconductor. We find that for a wire axis along the a axis, these different contributions constructively interfere and give rise to a robust triplet p-wave gap. For a wire oriented 45 degrees away from the a axis, the different contributions destructively interfere and the induced triplet p-wave gap vanishes. For the appropriately oriented wire, the induced p-wave gap may give rise to Majorana fermions at the ends of the half-metallic wire. In light of the recent experimental progress in the quest for realizing Majorana fermions, we also discuss inducing superconductivity in the spin-orbit-coupled nanowire using a HTSC. Based on our result, topological superconductivity in a semiconductor nanowire may be possible given that it is oriented along the a axis of the HTSC. DOI: 10.1103/PhysRevB.87.014504