When Oedipus and Creon are arguing, Oedipus primarily relies on ethos, or an appeal to his authority as king:
OEDIPUS: Quite sane—in my behalf.
CREON: Not just as much in mine?
OEDIPUS: You—my mortal enemy?
CREON: What if you’re wholly wrong?
OEDIPUS: No matter—I must rule.
CREON: Not if you rule unjustly.
OEDIPUS: Hear him, Thebes, my city!
Oedipus supports his argument—namely, that Creon should be sentenced to death for a treachery he did not commit—by referencing his own authority as a king who "must rule." When Creon retorts that Oedipus should not "rule unjustly," Oedipus calls out to Thebes, which he refers to as "my city," once again referencing his authority as ruler of Thebes. This use of ethos juxtaposes Oedipus with Creon, who primarily uses logos, or logic, in their discourse.
The use of ethos reveals Oedipus's hubris—his excessive pride—which ultimately leads to his downfall. While Oedipus was doomed by fate, Oedipus also has numerous opportunities throughout the play to figure out the truth of the prophecy: for example, when Tiresias shares his own riddle which accurately describes the conclusion of the play. Moreover, figuring out the riddles of Apollo and his prophets would reenact the very process by which Oedipus became king: by solving the riddle of the Sphinx. Oedipus then relies on his own authority as king despite being unable to replicate what gave him that authority in the first place. In other words, it is Oedipus's intelligence that made him king, and it is his ignorance that causes his tragic downfall.