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Best way... build a cage with a Dino gate and lure it in and you can tranq and tame it in peace

An argentavis can't fit through a ceiling with one tile missing, but a pteranadon or lymantria can. Build a trap with one corner ceiling missing, aggro the argentavis, fly in one side and out the missing ceiling tile, and land. Dismount, run around, and close the entry door. Tame safely at your leisure.

Don't forget to build a taming hut / room and lure the bird in. They fly off and don't come back if you start tranqing them in the wild.