Under the anger there is usually something you wanted and did not get, or something you are afraid of losing. Say that, and you have handed the other person something they can actually do.

What the research says

In a laboratory experiment, people who recalled an anger provoking event ended up in different states depending on how they thought about it: reappraising what had happened reduced anger, while going over it kept the anger going. It used a recalled memory in a controlled setting rather than a live argument. Still, putting the event into terms of a need is a form of reappraisal, and reappraisal is the side of that comparison that helped.

Strength of evidence: one line of research. A single study, or a theoretical account.