When are edges adjacent?
Two edges are adjacent if they're incident on the same vertex:

Two edges are adjacent if they have exactly one common endvertex.
—Bollobás, Modern Graph Theory (Springer, 1998; page 2).
Two edges $e\neq f$ are adjacent if they have an end in common.
—Diestel, Graph Theory (2nd Edition, Springer, 2000; page 3).