From my perspective, you can use "angry with" + somebody, which means you feel angry because of the person. So, "-I'm angry with my sister" and "She's angry with me " are correct. When you use "angry at" + something, you mean you feel angry because of something. For example, I feel angry at being abandoned/ he feels angry at what he saw.

You can use "angry with" and "angry at" when directed towards a person. For example: "I'm angry at you for not telling me sooner." "I'm angry with you for not telling me sooner." "Angry at" can also be used when directing anger towards a non-living thing. For example: "I got angry at my computer for not working."

If you said any of these, people would understand what you are trying to say. I think I've heard all of those at least once.

"Angry with" is correct; it just sounds very formal. More often in casual conversation you will hear "mad at" rather than "angry with". This may not be the case in other English speaking parts of the world, but it has been my experience living in America. I cannot think of a situation where you would hear "angry at".

Angry with someone. She is angry with me . I am angry with you .