Two projections P and Q on a Hilbert space are called acute if . We utilize the von Neumann alternating projection theorem to prove that if P and Q are acute, then . Conversely, if and PQ is a compact operator, then P and Q are acute. An example is presented to show that the assumption of compactness is necessary. Let be a von Neumann algebra, be the lattice of all projections in , and . A pair (P, Q) is called modular in if for every with . We present several characterizations of modular pairs of projections in a von Neumann algebra. In particular, for a factor of type I or III, we investigate certain modularity conditions.

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