What is the relationship between nou and kyogen?
They are like twins.
A form of comic imitations, originating from sangaku which is also the origin of nou, developed over the years into a farcical drama based on comic dialogue which then became kyogen.
On the other hand, it was the more serious song-and-dance dramas known as utai and mai which became the nou theater.
Today, stagings of nou usually present plays in the order of nou, kyogen and nou.