The shear shallow water model is a higher order model for shallow flows which includes some shear effects that are neglected in the classical shallow models. The model is a non-conservative hyperbolic system which can admit shocks, rarefactions, shear and contact waves. The notion of weak solution is based on a path but the choice of the correct path is not known for this problem. In this paper, we construct exact solution for the Riemann problem assuming a linear path in the space of conserved variables, which is also used in approximate Riemann solvers. We compare the exact solutions with those obtained from a path conservative finite volume scheme on some representative test cases.