We show that the singularity of the free energy of Ising models in the absence of a magnetic field on the triangular, square, and honeycomb lattices is related to zeros of the pseudopartition function on an elementary cycle. Using the Griffiths' smoothness postulate, we extend these results to the case in a magnetic field and derive a formula of the critical line of an Ising antiferromagnet, which is in good agreement with the numerical results.