Following Schmidt, Thurnheer and Bugeaud–Kristensen, we study how Dirichlet’s theorem on linear forms needs to be modified when one requires that the vectors of coefficients of the linear forms make a bounded acute angle with respect to a fixed proper non-zero subspace V of ℝⁿ. Assuming that the point of ℝⁿ that we are approximating has linearly independent coordinates over ℚ, we obtain best possible exponents of approximation which surprisingly depend only on the dimension of V. Our estimates are derived by reduction to a result of Thurnheer, while their optimality follows from a new general construction in parametric geometry of numbers involving angular constraints.