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also hang-nail, 1670s, apparently from hang (v.) + (finger) nail, but probably folk etymology from Old English agnail "a corn on the foot, painful spike (in the flesh)" from Proto-Germanic *ang- "compressed, hard, painful" (from PIE *angh- "tight, painfully constricted, painful;" see anger) + Old English nægl "spike" (see nail (n.)).
hangnail hang·nail (hāng'nāl')
A small piece of dead skin at the side or the base of a fingernail that is partly detached from the rest of the skin.