You use 'to be/get angry at what somebody does' when you want to give the reason using the noun clause [what sb does]
You use 'to be/get angry with somebody for doing something' when you want to give the reason for your being/getting (with sb) using the prepositional phrase including the gerund [for doing sth]
And finally, you use 'to be/get angry (with somebody) about something' while giving the reason for such a behaviour of yours using the prepositional phrase excluding the gerund [about sth]:
I got really angry at what you said to me that night.
I got really angry with her for saying that I was stupid./I got really angry for her saying that I was stupid.
I got angry (with him) about his sick attitude towards elderly people.