The monotone variational inequalities Ⅵ(Ω, F) have vast applications, including optimal controls and convex programming. In this paper we focus on the Ⅵ problems that have a particular splitting structure and in which the mapping F does not have an explicit form, therefore only its function values can be employed in the numerical methods for solving such problems. We study a set of numerical methods that are easily implementable.Each iteration of the proposed methods consists of two procedures. The first (prediction) procedure utilizes alteating projections to produce a predictor. The second (correction) procedure generates the new iterate via some minor computations. Convergence of the proposed methods is proved under mild conditions. Preliminary numerical experiments for some traffic equilibrium problems illustrate the effectiveness of the proposed methods.