Assume the nucleus is initially stationary, and has zero momentum. If the nucleus broke into only two particles, then by conservation of momentum, the momenta of the two decay products would have to cancel, to keep the final momentum equal to zero. For these 2 momentum vectors to add to zero, they must lie along a line.
Work Step by Step
Observing that the recoil nucleus and the electron don’t fly out in exactly opposite directions implies that at least one other particle must carry some momentum.