A storyline or feud. The scripted narrative that gives matches meaning. 'The angle is that he stole his girlfriend and now they're fighting about it.' Good angles make you care about who wins. Bad angles make you change the channel. Some angles are subtle character studies. Others involve kidnapping, fake pregnancies, and caskets. Wrestling contains multitudes.
"They've been running this angle for three months. It's time to blow it off at the next premium live event."