Halogen light bulb prong broke off in fixture and can not be removed. I inserted another bulb with prong in the socket and left other prong out. The light is now working; however, I need to know if this is safe.
It is NOT safe since as indicated in the question only one prong of the bulb is in its socket contact. Obviously the other one that you left out is merely touching the other broken off prong.
Touching is not good enough as the bulb could move some and turn a touching contact into an intermittent one. This results in arcing and extreme heat generation that can lead to burning the socket, attached wiring and even lead to a fire. The fact that halogen bulbs already generate high heat leads to greater possibility of the bulb to move around as it heats up or cools off.