Expert advice on the right way to argue without anger

Divide and conquer. Take things on one at a time

1. Stop thinking about what you are going to say next. This is not effective, as it distracts you from listening to the points the other person is trying to make. But, it is a good idea to plan ahead so if you must argue you are prepared.
2. Listen to what the other person has to say. Try not to become angry, just listen quietly until it is your turn to speak. Do not try to cut off the other person or interrupt.
3. When it is your turn to speak, use your words to convey your feelings, not your tone. Say things like, “When you say that, I feel _____”, or, “It hurts me when you _____”. This makes the other person feel like they are not at fault, and they don’t have to become defensive. If you said to them, “I hate it when you _______”, it makes them feel like they have to defend themselves and both of you end up getting more angry and farther from a resolution.
4. Respect what the other person has to say. If you do not like it, that doesn’t matter. That is their opinion and you should respect it, whether you agree or not. (Although this presupposes that opinion cannot be wrong, and that there is no objective truth, which is not a universally agreed to philosophy. Indeed, if opinion were truly subjective and never wrong, there would be no reason to debate in the first place. The subjectivist philosophy is a premise in itself which lacks all proof.)
5. Keep the dialogue going, with each person taking turns speaking and saying what they feel and think in the way described above. You will both eventually understand what the other is trying to say, and you will both see a clearer picture of the other’s feelings.