Abstract: Let H be a Hilbert space and P, Q orthogonal projections on B(H).It is proved that M_θ(P,Q) is non-empty if and only if the dimension of the intersection of PH and (I-Q) H is the same as the dimension of the intersection of QH and (I-P) H, where M_θ(P,Q) is the set of projections on Hwhich are within distance of sinθ from P and cosθ from Q.The operator matrix decomposition form of Halmos' two projection theory is applied to prove the sufficiency.On the other hand, a linear bijection between the intersection of PH and (I-Q) H and the intersection of QH and (I-P) H is constructed to give the necessity.