Lines 120-125: How did the nurse refer to Dexter? How did the nurse's comment and Judy's reaction to Dexter's response make him feel? What character trait would cause him to feel that way?

The nurse refers to Dexter in a subservient manner, like he is a servant.

The nurse's comment and Judy's reaction to Dexter's response makes him feel that he must take action so that he will never be in such a position again.

His determination is what causes him to feel this way.

Explanation:

In Winter Dreams by F. Scott Fitzgerald Dexter encounters both Judy and her nurse for the first time, and the nurse treated Dexter like a servant: "Miss Jones is to have a little caddy, and this one says he can't go."

When Dexter explains his reason for not being able to go due to his responsibility, Judy simply dismisses it for her own selfish pleasure: "Well he's here now.", then she dropped her bag and set off in haughty manner towards the tee.

This treatment makes Dexter convinced to just quit his job at the club, he believes he he must never put himself in a position to be treated that way again.

The trait which made him feel that way is his determination, and he uses that determination later to "win" Judy and to achieve a sense of status and position that will enable him to have the best experience.