On what single issue did Atticus build his case? Why did Mayella resent Atticus?
I think if you have to sum his case up in one word, you could say injury. Mayella's injuries were the focus of every testimony. Of Bob and Heck, he asked about if anyone ever called a doctor. He wanted an answer to this question because it could have provided another layer of evidence. It also would have shown a seriously concerned father. His lack of regard for her injuries shows a disrespect for his daughter. Her testimony focuses on where her bruises were and what was done to her... in her words. Tom's testimony then demonstrates his inability to inflict these injuries.
Mayella didn't like Atticus for pointing out the truth. There was not any trickery, just cold hard factual evidence. She thought his courtesy was rudeness. That just demonstrates how out of touch she was with society.
The single issue that Atticus builds his case on is the fact that Mayella Ewell seemed to have been beaten by a left handed person. Bob Ewell, her father, was left handed. Tom Robinson, who she said raped her, had a withered left hand that had no strength in it. So the implication is that he could never have beaten her in that way.
She resents Atticus because he is out to make her look like a liar. He has already made her dad look like he was lying and she does not want him to treat her like that.