「居直る」は
to become aggressive(攻撃的になる)
to get one's back up(怒って防御的になる)
などと言います。
例:
When I told him about the mistakes he made in his report, he suddenly became aggressive about it.
「彼の報告書の間違いを指摘したら、突然攻撃的になった(=居直った)。」
Even though he was wrong cheating on his wife, he got his back up when she confronted him about it.
「彼が浮気をしているにもかかわらず、妻が浮気について問いただしたら、彼は怒って言い訳をし出した(=居直った)。」
ご参考まで!