What angered Hector, making him face Achilles in Homer's Iliad?
Achilles became angered by Hector when in Book 17 Hector killed Achilles' dear friend and half brother Patroclus. In Book 17, Achilles sends Patroclus to help defend the Achaian ships from the Trojans. When the Trojans see an army being led by Patroclus and Myrmidon, they begin to retreat, thinking that Achilles has again joined the war. Patroclus successfully kills many Trojans; however, Achilles also warned Patroclus not to press on past the ships, which he does, nearly taking the whole city of Troy. As a result, Apollo who sides with the Trojans makes Patroclus vulnerable, knocking his helmet off and opening him up to be attacked by Hector who killed him with a spear. As a result of Patroclus's death, by Book 18, Achilles has vowed to avenge himself on Hector for Patroclus's death. We see Achilles make this vow to his mother immediately after learning of Patroclus's death, as we see in his lines:
I will not live nor go about among mankind unless Hector fall by my spear, and thus pay me for having slain Patroclus son of Menoetius. (Bk. 18)
Hence, by Book 22, Achilles pursues Hector around the city of Troy until they both face off in a duel. Hector, on the other hand, has not been angered by Achilles and does not face Achilles out of anger. In fact, Hector is actually too scared to face Achilles, which is why Achilles chases him around the city walls. Hector does not have the courage to face Achilles until the goddess Athene appears beside Hector in the form of his brother Deiphobos. Thinking that Deiphobos has come to help him fight, Hector finally has the courage to face Achilles in a duel, which Hector loses.