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1. In baseball, to tap a runner with the ball in order to remove them from the field and end the play. A noun or pronoun can be used between "tag" and "out." He began rounding third base but was tagged out just before getting to home. The ball bounced right back to the pitcher who immediately tagged the batter out.
2. In tag-team professional wrestling, to tag one's teammate (usually on the hand) and then exit the ring in order to allow the teammate to begin wrestling the opponent. Everyone thought he was done for after being smashed by that suplex, but he managed to tag out at the last minute. The crowd all booed as she tagged out.
3. slang To allow someone else to take over one's current activity, especially to give oneself a reprieve. Sorry George, but I've been trying to get the baby to sleep for over an hour—I need to tag out. Just let me know if you want to tag out at any point.
4. slang To take over someone else's current activity, especially to give them a reprieve. In this usage, a noun or pronoun is used between "tag" and "out." Tom's been looking after the kids all afternoon, so I'm going to go tag him out. Tom arrives at 8 AM to tag out the late-night security guard.
tag someone out
[in baseball] to touch with the ball, and thereby put someone out. The shortstop tagged the runner out and retired the side. He tagged out the runner.
To touch some base runner with the ball in order to put that player out: The shortstop tagged out the runner at second. I tagged the player out and ended the inning.