In this paper, a discontinuous finite element method for the positive and symmetric, first-order hyperbolic systems (steady and nonsteady state) is constructed and analyzed by using linear triangle elements, and the (h2)-order optimal error estimates are derived under the assumption of strongly regular triangulation and the H3-regularity for the exact solutions. The convergence analysis is based on some superclose estimates of the inter-polation approximation. Finally, we discuss the Maxwell equations in a two-dimensional domain, and numerical experiments are given to validate the theoretical results.