If you can get away from them try to, sometimes the situation requires a temporary ceasefire as you will, adding fuel to the fire doesn't help if both sides are freaking out on each other(that isn't a conversation).
A lot of the time the main problem is they are not respecting you, instead they are being rude/emotional/freaking/condescending to you, YOU do not deserve that. You are worth more than that. Do not settle for bad behavior that a child would try to get away with. People who argue like this only do so because other people let them get away with it, NO! Do not let it continue, stop it now and work towards getting it to be better/nicer/less yelling etc. Remember, when a child throws a tantrum we don't give in and agree with the child to get them to shut up. THIS IS BAD BEHAVIOR THAT NEEDS TO BE CORRECTED
If you have to engage with them or want to try to calm things down(because you think it is appropriate) when they are being emotional and ranting about something, you need to tell them something like "I am sorry you feel that way". Really emphasize they are spouting off feelings(if they are). Call it for what it is and it will help get things moving in the correct direction. An emotional person may think they are making rational logical arguments, but they aren't and you are not doing yourself a favor by letting this bad behavior continue!!! This bad behavior needs to be dealt with consistently and thoroughly, think of it as dealing with a child, the child doesn't know any better so you have to be patient/loving/kind/respectful and explain everything as best you can without getting emotional. If they are not treating you with respect then you need to let it be known that you do not like what they are doing and that is is unacceptable. Put your foot down. If you are talking politics with someone who is going to turn into a child, let them know it insults your intelligence when they make emotional appeals. Tell them how you are offended that they do that to you(turns the tables because it is now your emotion vs theirs and they are being the asshole). Sometimes you need to speak their language more and if/when needed you may want to speak in emotional terms to get them to understand it better. There are of course many ways and variations of what you could say, I am only giving some examples I have used. If you really need some kind of therapist please note that I am not one and am merely talking of my multiple successful experiences with others.
TL:DR Logical arguments can't usually beat emotions(muh feels). But a consistent approach to change the other persons behavior can lead to better conversations in the future and in the long term help rectify the communication gap. :)
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And they all pride themselves on being tuff.