The magnon pairing mechanism is derived to explain the high-temperature superconductivity of both the La2-xSrxCu(1)O(4) and Y(1)Ba(2)Cu(3)O(7) systems. Critical features include (i) a one- or two-dimensional lattice of linear Cu-O-Cu bonds that contribute to large antiferromagnetic (superexchange) coupling of the Cu(II)(d(9)) orbitals; (ii) holes in the oxygen ppi bands [rather than Cu(III)(d(8))] leading to high mobility hole conduction; and (iii) strong ferromagnetic coupling between oxygen ppi holes and adjacent Cu(II)(d(9)) electrons. The ferromagnetic coupling of the conduction electrons with copper d spins induces the attractive interaction responsible for the superconductivity, leading to triplet-coupled pairs called "tripgems." The disordered Heisenberg lattice of antiferromagnetically coupled copper d spins serves a role analogous to the phonons in a conventional system. This leads to a maximum transition temperature of about 200 K. For La(1.85)Sr(0.15)Cu(1)O(4), the energy gap is in excellent agreement with experiment. For Y(1)Ba(2)Cu(3)O(7), we find that both the CuO sheets and the CuO chains can contribute to the supercurrent.