Characters (and people) get angry at all sorts of things that might not make sense to the outside observer: Marty McFly and the word "chicken," words that are terrible slurs to one population but totally normal to others, overreactions due to mental illnesses, etc.
I feel like I have a relatively good grasp of how to portray anger in third person, rational or irrational. Readers are used to other people reacting differently than they would, as long as the character is consistent. However, if the writing is in first-person it seems that I need to take the reader.on the same journey of becoming angry.
How do I show the internal monologue of a character becoming angry? How completely do I need to evoke the same anger in the reader that the character feels?
Edit: @Rand Al'Thor, @Galastel, and @wordsworth all have excellent answers, but I chose @Stilez 's answer because the examples made everything more clear to me. Even though the trigger is irrational, the narrator's justification sounds like something a character would say for rational anger as well.