In 1983 Frank Morgan conjectured an ``angle criterion'': a pair of oriented m-planes in \(R^ n\) is area-minimizing if and only if the characterizing angles between them satisfy the inequality \(\beta_ m\leq \beta_ 1+...+\beta_{m-1}.\) \textit{F. Morgan} [Math. Ann. 261, 101-110 (1982; Zbl 0549.49029)] proved the conjecture for \(m=2\). Further results were obtained for \(m=3\) and \(m=4\) by \textit{J. Dadok} and \textit{R. Harvey} [Duke Math. J. 50, 1231-1243 (1983; Zbl 0535.49030)] and \textit{J. Dadok}, \textit{R. Harvey} and \textit{F. Morgan} [Trans. Am. Math. Soc. 307, No.1, 1-40 (1988)] and for \(m=5\) by M. Messaoudene (unpublished). \textit{D. Nance} [Math. Ann. 279, 161-164 (1987)] proved that for all m, any pair of m-planes satisfying the criterion is area-minimizing. This paper completes the proof of the angle conjecture in all dimensions, by providing surfaces of less area for pairs not satisfying the condition.