Which is the most appropriate/correct usage?
- Are you angry on me?
- Are you angry with me?
- Are you angry at me?
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I have never heard the phrase "angry on" used before.
Regarding the other two, I would say that it depends upon the context.
If directed at a person "angry with" should always be used.
e.g. I was very angry with her.
If directed at a situation I believe "angry at" would also be acceptable.
e.g. I was very angry at how rainy it was.
The key is who the anger is aimed at. "Angry at" is not really aimed at anyone.
So in the example you give, "Are you angry with me?" is correct.
Not sure how much use Google NGrams is here. It does at least show that "angry on" is used very rarely: