When you encounter someone with whom you deeply disagree, and the tone of a conversation turns hostile, there are a few possibilities. When emotions run high, recognize what is going on, take a step back, and examine the situation.
People are not always skilled in rational discourse.
Keep your cool and make calm explanations in defense of your position. If someone's reply seems to miss the mark, describe why you believe their response is erroneous or non-responsive. Treat them like adults, though, and try not to be condescending. Schools and the media have failed to prepare most people for rational debate.
People are not always interested in rational discourse.
Some people just want to rant. State your case and then let them vent. Brush it off. They may eventually cool down and consider the conversation after they are past the heat of the moment. If you can keep your cool, onlookers and interlocutors alike will be able to reflect on your words when they are in the right frame of mind.
People may be invested in their own position.
Discuss the ideas, and defend your position while seriously considering theirs. Maybe they have some mistaken preconceptions. There's even the possibility you are wrong yourself, at least in part, so don't discount that. Prejudices and preconceptions can cloud our reasoning even in the best circumstances. We may want to think we are dispassionate, objective observers of reality, but we are usually emotionally or financially invested in our beliefs, too. This is why dissent is often treated as a personal attack. Keep your arguments civil, and you might minimize this response.
It's a troll out to sow discord
Drop the conversation. Don't feed the troll.
