When a person is very angry, asking or telling them to be reasonable is doomed to fail — here’s why and what to do instead.

When a person is very angry, the part of their brain associated with being reasonable and articulating reasonable thoughts more or less shuts down. Closed for business. Sign on the door — go away, can’t do this right now.
Yet in the midst of an argument, we still think to ourselves (or even say out loud), “Just be reasonable, can’t you?” Or maybe they say it to us. In a mediation, we may still want to say to a client, “Why can’t you be reasonable?” Or, more judgmentally, as one frustrated mediator put it to me recently, “Why can’t they just put their big boy pants on?”
When someone is choking on a piece of food, we don’t say to them, “Why won’t you finish your sentence?” or “Why can’t you stop that, it’s annoying me.” It would be absurd to expect they could comply with our wishes at that moment.
In the same way, we would be smart to stop expecting someone hijacked by anger to comply with our wishes in that very moment. When the Broca’s area of their brain shuts down during deep upset, they cannot access their reasonableness immediately just because you want them to — or even because they want to (and believe me, they do).
When someone is angry, we can’t just reason them back into reasonableness. The path back to reason isn’t sheer will nor will it have instantaneous results.
The path back usually involves a bit of time and specific actions to calm down. We can help this happen or we can hinder it.
If we are part of the disagreement, and we keep pushing or judging or talking talking talking, we are hindering. We must stop up, take a break (no, don’t blame it on them, just say, “I need a bathroom break”), and give them time and opportunity to calm down. Depending on the person, situation, and what they do in that time, this will usually take anywhere from 30 minutes to an hour or more. Don’t rush it.
If we are the angry person whose Broca’s region is temporarily closed for business, we must shut ourselves up, find a way to get a break, and calm ourselves (some ideas: here and here and here and here).
If we are the mediator, we must help them do the right things to calm down and find ways to stop the things that are feeding the anger.