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When talking to a difficult person, stay calm and rational, avoid the temptation to try to win the argument, and don't be drawn into a battle you don't want to wage. You are much more likely to reach a satisfactory outcome if you can stay calm and rational. Think before reacting. Even if someone is extremely angry or rude to you, formulating a calm response is the best approach. It sets boundaries, and sends a message to them that they need to calm down. As previously mentioned, many people behave in a difficult way because they don't feel heard or understood. Just showing that you are listening to what they have to say can sometimes improve the situation.  It's a good idea to let the person know you recognize their feelings. Offer your perceptions of their feelings, and ask for feedback, saying something like "It sounds like you're very angry right now, and I'm sorry you feel that way." This shows your willingness to understand their perspective.  Ask what they are upset about. You can further show your willingness to empathize if you ask them to tell you about their feelings.  Acknowledge valid criticisms. If the person is being extremely critical of you, try to find a kernel of truth in what they are saying, and acknowledge the validity of the point, even if their criticism isn't entirely fair or accurate. This may reduce the person's sense that they are being challenged, even if you then go on to point out where they aren't being fair or accurate. When dealing with a difficult person, it is important to communicate in a clear and open way. Many conflicts arise from misunderstandings.  If you can, try to talk to the person face-to-face rather than via email or other technology. This reduces the risk of miscommunication and may foster more empathy.  If you must engage in a dispute with someone, bring written evidence for your perspective to the table, and try to steer the discussion toward fact-based argument rather than statements of opinion or emotion. Focus your conversation on the issue or problem you have to solve, rather than the person you are dealing with. This helps prevent the conversation from devolving into personal attacks and may lead to more rational thinking on the part of the difficult individual. This approach has the added benefit of projecting yourself as a problem-solver who really cares about the issue at hand and wants to change the situation for the better. Communicate in a way that allows you to express your views and ideas about the situation at hand in a clear way, but without silencing the other person or making them feel that you aren't listening or being rude.  Where possible, ask questions rather than making statements. Difficult people often have strong opinions. If you can lead them to see potential flaws in their reasoning without telling them they are wrong, you can often avoid unnecessary conflict.  For example, politely asking them "have you considered this problem?" might be more productive than saying "your way of thinking about this doesn't account for this issue." Use "I" statements. When you do make statements, make them about yourself rather than the person. This can diffuse the feeling that you are challenging or blaming them.  For example, saying, "I never received that email" is less provocative than "you never sent that email." Similarly, "I felt disrespected by that comment" could be less inflammatory than "you were very rude."
Keep your cool. Acknowledge their feelings. Communicate clearly. Focus on the issue, not the person. Be assertive, but not aggressive.