Summarize:

Argumentative people will often argue out of a desire to see an emotional response from their opponent. Don't show if the interaction is making you upset or angry. Smile. Show your antagonist you aren't going to let an argument bring you down. Sometimes that's all an angry, argumentative person is looking to do. Instead of trying to convince an argumentative person to see things your way, ask them questions to explain their viewpoint. Often this can help you get to root of the argument. For example you can ask "What is your reasoning behind that idea?" Aside from not having to justify your own position to your opponent, you are allowing them an opportunity to vent and feel their voice is heard. Many times this will be enough to calm an argumentative person, even if it doesn't resolve the disagreement. If you feel a situation is not going to be resolved in a positive manner, just step away from it. You can always pick up the topic again at a later time when you feel it can be discussed in a reasonable manner.
Stay calm. Ask questions. Know when to walk away.