After a rediscussion of the vanishing theorems for helicity amplitudes in unbroken gauge theories, we study the case of spontaneously broken gauge theories at high energy. The vanishing theorems generalize to a definite pattern of m/E suppression of the amplitudes that vanish in the massless case, where E is the energy scale of the process and m is the mass of the gauge vectors. We use only elementary arguments, and as an application we show how these methods can be employed to understand some aspects of the effective W approximation in the polarized case. DOI: 10.1103/PhysRevD.87.074026