I get along with my father-in-law but my mother-in-law keep feeding him things about me, and then he also starts behaving rude with me. How to handle this?

Let your husband know and see if he can deal with it. I wouldn't confront the MIL if I were you because it will just prove to your FIL that whatever his wife says is the truth. Continue to be kind and respectful to your FIL even if he changes his attitude towards you. Be the bigger person and I hope your FIL will see the truth :) Hang in there and take care!