Hi Olga You've just asked a very interesting question. Some teachers would say there is no difference and you can use them interchangeably. To a certain extent I would agree. Because it's such a common mistake using both has become acceptable in conversation. However, grammatically I feel it is correct to use with when talking about anger towards a person. It implies a general state of anger. Angry at would imply you directing your anger at an object or situation. I hope this helps.

You can use either one, but "with" is better because you and yourself are so closely associated. Using the preposition "at" is similar to pointing at something. You can point at yourself, but it's not natural. Normally you point at something else.

I think either is okay here. I've heard native speakers using both. Generally: he looked angrily at her/shouted angrily at her. Here the anger is like an arrow and she is the target. The anger is moving in her direction, hence the word at. We say to look at, or to throw a brick at a car. With is the standard phrase if it is anger at another person. He was angry with his children for leaving their toys on the floor, so he shouted angrily at them. But if you are angry with yourself, you can be internally directing your own anger towards yourself, so here it can be at or with. You can say, "she felt angry at herself for making such a bad decision. Although grammatically it is better to say, "she was angry at herself ... - she was directing her feelings of anger towards herself/at herself. Hope this helps. If you want to delve into prepositions the best resource I know of is the Cambridge Online Dictionary. Look up words like with and at, then look for the advanced uses which are listed in the CEFR levels of B1, B2 and C1. Particularly, for academic IELTS, I always recommend my students do this for the words at and by because it can help them in task 1 writing. Kind regards, IELTS Coach Kate