beautiful, handsome, pretty
姣 centers on physical attractiveness: a person, especially a woman, who is beautiful, handsome, or pretty.
姣 combines 女 (woman) with 交, which likely contributes the sound and may suggest a sense of attractiveness or fine appearance.
A woman 女 with a 交 shape on the right: imagine a woman whose features are so well-proportioned that they seem to intersect perfectly, making her beautiful.
For コウ, picture a beautiful woman named Kou standing at an intersection: Kou -> コウ, and her beauty is as striking as the crossing paths.