The more annoyed you are the harder it is to listen, and the more needful it is. There are people who, as soon as they open their mouths, you want to shut them up. We often forget that others besides ourselves think their ideas are good, worthy of attention, and right-headed. When they are people with whom we vehemently disagree, the temptation is to just blow them off with some shriveling words of dismissal and pat ourselves on the back for our self-perceived (or should I say self-deceived) intellectual prowess. This blowing-off accomplishes nothing good and often exacerbates any further productive communication.
It is amazing what can be accomplished when you do listen, ask questions, and respectfully counter-argue. Almost never do you change a mind, but you can change an attitude, which may be just as important. Our increasingly polarized culture is creating blow-off attitudes the size of Mt. Everest. Stepping back, clearing our hearts of antagonism, clearing our heads of fuzziness, and showing a sincere willingness to hear the other person out does more for building peace than digging in and taking snarky shots at our opponents.