How is prejudice and anger overcome in "Twelve Angry Men?" In other words, what evidence proves that prejudice/anger is overcome?

Right from the start, it is clear that juror #10 is a racist bigot. He believes the accused is guilty, and his main argument of support is that the kid isn't white.

You're not going to tell us that we're supposed to believe him, knowing what he is. I've lived among 'em all my life. You can't believe a word they say. You know that.

Juror #9 immediately steps in and calls juror #10 out on his racist and skewed logic, but the other jurors aren't exactly racing to support #9. A few of them are likely in full support of #10's feelings; however, as more and more evidence is brought into question, jurors start changing their vote to "not guilty." #10 simply won't be convinced. He believes that the kid is a liar and absolutely guilty, and #10 repeatedly makes comments that connect his feelings to the defendant's race.

I don't understand you people. How can you believe this kid is innocent? Look, you know how those people lie. l don't have to tell you. They don't know what the truth is. And lemme tell you, they— (NO.5 gets up from table, turns his back to it, and goes to window.)—don't need any real big reason to kill someone either. You know, they get drunk, and bang, someone's lying in the gutter. Nobody's blaming them. That's how they are. You know what I mean? Violent!

As more and more jurors change their vote, #10 gets angrier and angrier.

I’ll tell you something. The crime is being committed right in this room.

Eventually, the logical arguments turn #10's opinion. The evidence that shows that racial prejudice is overcome is the fact that #10 changes his opinion, and his racial prejudice is overcome by the patience, determination, and logic of juror #8.