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If the email does not require a response, not sending one could be the best option. Consider not responding if you receive emails that are off topic and don’t require your attention. For example, coworkers may sometimes make assumptions about your work habits. Don’t respond to an email that says, “It must be nice to get to take extra long lunches,” or “Everyone knows that your meetings don’t last all afternoon.” As long as your manager knows the truth, you're better off staying out of office drama. If a response is needed, the tone should be professional, even if the sender was not professional. Read your email out loud before sending it to make sure that it sounds professional rather than rude or emotional. Avoid being blunt in your response, as that can sound rude. Your response needs to be calm and professional, so don’t write it in a rush. Most emails are not time-sensitive, so you can take all day to respond if you need to.  After you write your response, set it aside for about an hour. If you know that you’re writing out of emotion, switch to a different task until you calm back down. Add the email address last to avoid accidentally sending it early. Don’t try to respond to everything, and don’t provide a long personal defense. If you want to squash the rude behavior, you need to keep your email focused on work topics. Choose one work-related topic to focus on. One way to ensure that your email is professional is to only discuss facts. Facts include specific things like what work has been assigned and completed, due dates, timelines, data, and related topics.  For example, if the rude email asked about the status of a report, say, "Thanks for checking in. I received the new data this morning, so I'm updating my report. The new report will be available this afternoon." This keeps the focus of the response on the work you were assigned and the progress of your project, not on your reaction to your rude coworker. It’s tempting to get defensive or rude in response to an offensive email, but it will only worsen the situation. Instead, take the professional high road.  Don’t accuse them of being rude. Keep your feelings out of the email. Before you hit send, ask yourself if any part of the email is a defense of yourself.
Avoid responding if no response is required. Stay professional. Take your time writing your response. Keep your message short and simple. Stick to the facts and don't elaborate. Leave the emotion out of your response.