Let \(H\) be a semisimple weak Hopf algebra and let \(A/B\) be a right weak \(H\)-Galois extension. The author proves that \(A/B\) is a separable extension. This result is used to show that the global dimension of \(A\) is at most the global dimension of \(B\), and the weak dimension of \(A\) is at most the weak dimension of \(B\). A Maschke theorem for weak Hopf-Galois extensions is obtained as a consequence. Also, the global dimension and the weak dimension are computed for weak smash products in the case where the weak Hopf algebra and its dual are semisimple. In particular, a Maschke theorem for weak smash products is given.