In this paper, we present a theory on the formation of exceptional points (EPs) of resonance states in two-dimensional periodic structures. Exceptional points are spectral degeneracies that may occur in non-Hermitian systems for some values of the system parameters. At an EP, two or more eigenfunctions coalesce and their eigenvalues coincide. Due to the local behavior of the eigenvalues and eigenfunction at and near the EPs, they have found important applications in sensing, lasing, unidirectional operations, etc. In this work, we employ the Dirichlet-to-Neumann (DtN) maps and mode matching methods to study EPs of resonance states on two different periodic structures; namely, a slab with a periodic array of circular holes and a slab with two-segment piece-wise constant periodic permittivity. The study reveals that a family of EPs exist continuously as the periodic structures tend to a uniform slab. Furthermore, the EP path continues to exist below the light line and eventually ends at a point that can be accurately determined.