We consider the Dirichlet problem for a class of fully nonlinear elliptic equations on a bounded domain . We assume that is symmetric about a hyperplane and convex in the direction perpendicular to . By a well-known result of Gidas, Ni and Nirenberg and its generalizations, all positive solutions are reflectionally symmetric about and decreasing away from the hyperplane in the direction orthogonal . For nonnegative solutions, this result is not always true. We show that, nonetheless, the symmetry part of the result remains valid for nonnegative solutions: any nonnegative solution is symmetric about . Moreover, we prove that if , then the nodal set of divides the domain into a finite number of reflectionally symmetric subdomains in which has the usual Gidas-Ni-Nirenberg symmetry and monotonicity properties. Examples of nonnegative solutions with nontrivial nodal structure will also be given.