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New exact analytical results are described for surface tension on the triangular Ising lattice. The triangular Ising lattice forms naturally 60° and 120° corners, and a thermodynamic argument is used, which incorporates the exact results, to obtain conditions for the filling transition for both angles. Monte Carlo simulations with Wang–Landau sampling are performed to independently verify the applicability of this argument. Results are in agreement with previous conjectures for the phase boundary in these systems.