An angry customer is simply incapable of rationalizing the situation. They are so caught up in their emotion that it (the anger) filters all their words. Anger is an emotion, and like every emotion, it is felt in the right hemisphere of the brain. Rationalization, problem solving, listening and negotiation are activities of the left hemisphere of the brain, but your belligerent customer is trapped in their right hemisphere.
At this point, don’t expect them to be able to rationalize the situation with you. Solving their issue would be pointless; instead, try to maintain control of the situation, to avoid worsening it even more. And don’t go antagonising a mad customer!