Paul teaches that women must cover their heads in church while men must wear hats, although he does not specify what type of hat.
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Paul teaches that men must not cover their heads in church, although hat-wearing in general is not considered sinful. Women are to pray with their heads covered. Since this injunction is in the same letter to the Corinthians as Paul's prohibition of homosexuality, the sins of hatless praying and sodomy must be considered equivalent.