Etymology
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hang in (v.)

"persist through adversity," 1969, usually with there; see hang (v.) + in (adv.).

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hang on (v.)

1860, "to remain clinging," 1860, especially "cling fondly to" (1871); see hang (v.) + on (adv.). As a command to be patient, wait a minute, from 1936, originally in telephone conversations.

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hang up (v.)

c. 1300, "suspend (something) so that it is supported only from above;" see hang (v.) + up (adv.); telephone sense by 1911. The noun hang-up "psychological fixation" is first attested 1959, from notion of being suspended in one place.

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hang out (v.)

c. 1400, intransitive (as of the tongue, from the mouth); transitive use by 1560s; see hang (v.) + out (adv.). Colloquial meaning "to be found" is recorded from 1811, "in allusion to the custom of hanging out a sign or 'shingle' to indicate one's shop and business" [Century Dictionary]. As a noun (often hangout) "residence, lodging" attested from 1893; earlier "a feast" (1852, American English).

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