Let me tell you a few secrets

First, anger uses up social energy. They get it back from your response to them... If you don't let them read any emotion from you, they'll tire themselves out very quickly.

You just have to control your body language, keep your tone calm, and let them talk. Make it clear you're paying attention to them, but otherwise give them nothing

You don't have to listen to what they say, they're just making angry human noises. Just listen to their tone, it'll rise and fall in energy cyclically until they run out of energy

When they stop talking, just give them a few moments of silence so they can feel embarrassed, then disregard their little temper tantrum and progress the conversation like it never happened, focusing on solutions

And that's the second secret - you can prompt-break a human. In every interaction, humans take on roles. Customer-employee, public official-citizen, manager-worker... Humans naturally fall into roles

You can pull a human out of their role by not playing your part, and in that moment of confusion you can recontectualize the interaction

In this case, you change the conversation roles from "you being mad at me" to "I'm the expert helping you fix your problem"

Obviously, if you just say that, people will generally just get more upset. But if you pull them off balance and start acting a new role, they'll take on the counterpart role

It's all third path conflict resolution, it's honestly harder to explain then it is to do it