We ask the question of how angular momentum is conserved in electroweak interaction processes. To introduce the problem with a minimum of mathematics, we first raise the same issue in elastic scattering of a circularly polarized photon by an atom, where the scattered photon has a different spin direction than the original photon, and note its presence in scattering of a fully relativistic spin-1/2 particle by a central potential. We then consider inverse beta decay in which an electron is emitted following the capture of a neutrino on a nucleus. While both the incident neutrino and final electron spins are antiparallel to their momenta, the final spin is in a different direction than that of the neutrino-an apparent change of angular momentum. However, prior to measurement of the final particle, in all these cases angular momentum is indeed conserved. The apparent nonconservation of angular momentum arises in the quantum measurement process in which the measuring apparatus does not have an initially well-defined angular momentum, but is localized in the outside world. We generalize the discussion to massive neutrinos and electrons, and examine nuclear beta decay and electron-positron annihilation processes through the same lens, enabling physically transparent derivations of angular and helicity distributions in these reactions.