Focus on resolution, your peace of mind, your day, and your goals. Your peace is above people's mismanaged ego and pending frustration.
Don't consider the other person superior or inferior; both bring your ego into play. Treat them like an immature kid; you need to handle them and not fight. Remember, aging doesn't bring adulthood; most people in the market are overgrown kids, pity them, manage them, and don't get angry.
Answer people’s doubts and questions, communicate your thoughts well, and your communication stands complete; beyond this nothing productive would happen. React to people’s thoughts and intentions and not emotions.
The moment an argument takes a heated turn, stop speaking and make it clear that you aren't up for a fight. If you both can handle it maturely, the discussion can continue, or you would walk away. Be clear that you don't want a fight.
Realize that not having control on one's thoughts and behavior is a mental weakness and lack of self-control. It's not something to be proud of; they failed the test, and you don't need to stoop to their grades.
Empathize with the other person, realize their loss, the reason for their emotions, that otherwise, they aren't a devil but an ordinary human like you. A true winner is a person who calms the other person down and does not ignite a fight. Also, maintain clear boundaries; once your moral boundaries are breached, do no further discussions, walk away, or take action, and action doesn't mean falling at their level, something that keeps them away from you.