Etymology
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hinge (n.)

late 14c., "movable joint of a gate or door," not found in Old English, cognate with Middle Dutch henghe "hook, handle," Middle Low German henge "hinge," from Proto-Germanic *hanhan (transitive), *hangen (intransitive), from PIE *konk- "to hang" (see hang (v.)). The notion is the thing from which a door hangs. Figurative sense of "that on which events, etc., turn" is from c.1600. Stamp-collecting sense is from 1883.

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revolving (adj.)

"making revolutions, rotating," 1690s, present-participle adjective from revolve (v.). Revolving door is attested from 1856 in industrial processes, 1896 in buildings.

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knocker (n.)
late 14c., agent noun from knock (v.). Sense of "door banger" is by 1590s. Knockers "a woman's breasts" is slang attested from 1941.
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valve (n.)
late 14c., "one of the halves of a folding door," from Latin valva (plural valvae) "section of a folding or revolving door," literally "that which turns," related to volvere "to roll," from PIE root *wel- (3) "to turn, revolve." Sense extended 1610s to "membranous fold regulating flow of bodily fluids;" 1650s to "mechanical device that works like an anatomical valve;" and 1660s in zoology to "halves of a hinged shell." Related: Valved.
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slammer (n.)
"jail, prison," 1952, perhaps from earlier U.S. slang sense of "door" (by 1943), agent noun from slam (v.). As "one who slams," from 1892.
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entrance (n.)

1520s, "act of entering," from French entrance, from entrer (see enter). Sense of "door, gate" first recorded in English 1530s. Meaning "a coming of an actor upon the stage" is from c. 1600.

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sash (n.2)

framed part of a window, into which the panes are fitted, 1680s, sashes, a mangled Englishing of French châssis "frame" of a window or door (see chassis).

The word was mistaken as a plural and further mangled by loss of the -s by 1704. Sash-door, one having panes of glass to admit light, is by 1726; sash-weight, attached by cords to either side of a sash to balance it and make it easier to raise and lower, is attested by 1737.

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Katie 
fem. proper name, diminutive form of Kate. Noun Katie-bar-the-door "a brouhaha, a turbulent and combative situation" is by 1888; the notion is "get ready for trouble."
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staircase (n.)
also stair-case, 1620s, originally the enclosure of the stairs, from stair + case (n.2) in its sense "frame;" compare former window-case, door-case.
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prudential (adj.)

"involving or characterized by prudence," mid-15c., prudencial, from Medieval Latin prudentialis, from Latin prudentia "a foreseeing, foresight" (see prudence). Related: Prudentially.

Prudential, the U.S. insurance company, dates to the 1870s; its logo featuring the Rock of Gibraltar dates from c. 1900 and was widely known 20c. The Prudential of Great Britain is a different company, founded 1848 to provide loans to professional and working people, noted for its door-to-door agents ("the Man from the Pru").

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