What is the adjective that describes the way someone speaks as if it was obvious?

Are you looking for "patronizing"? (it does not convey the irritation streak, though)
MW defines patronizing as

"showing that you believe you are more intelligent or better than other people."

e.g.

  • She spoke to us in a patronizing tone.
  • A patronizing comment.

from TFD - "to behave towards (someone) in a way which is kind and friendly but which nevertheless shows that one thinks oneself to be more important, clever etc than that person."