Etymology
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Words related to wag

halter (n.)
Old English hælfter "rope for leading a horse," from Proto-Germanic *halftra- "that by which something is held" (source also of Old Saxon haliftra "halter," Old High German halftra, Middle Dutch halfter), from suffixed form of PIE *kelp- "to hold, grasp" (see helve). Also "hangman's noose" (mid-15c.). In women's clothing sense, originally "strap attached to the top of a backless bodice and looped around the neck," 1935, later extended to the tops themselves.
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*wegh- 

Proto-Indo-European root meaning "to go, move, transport in a vehicle."

The root wegh-, "to convey, especially by wheeled vehicle," is found in virtually every branch of Indo-European, including now Anatolian. The root, as well as other widely represented roots such as aks- and nobh-, attests to the presence of the wheel — and vehicles using it — at the time Proto-Indo-European was spoken. [Watkins, p. 96]

It forms all or part of: always; away; convection; convey; convex; convoy; deviate; devious; envoy; evection; earwig; foy; graywacke; impervious; invective; inveigh; invoice; Norway; obviate; obvious; ochlocracy; ogee; pervious; previous; provection; quadrivium; thalweg; trivia; trivial; trivium; vector; vehemence; vehement; vehicle; vex; via; viaduct; viatic; viaticum; vogue; voyage; wacke; wag; waggish; wagon; wain; wall-eyed; wave (n.); way; wee; weigh; weight; wey; wiggle.

It is the hypothetical source of/evidence for its existence is provided by: Sanskrit vahati "carries, conveys," vahitram, vahanam "vessel, ship;" Avestan vazaiti "he leads, draws;" Greek okhos "carriage, chariot;" Latin vehere "to carry, convey," vehiculum "carriage, chariot;" Old Church Slavonic vesti "to carry, convey," vozŭ "carriage, chariot;" Russian povozka "small sled;" Lithuanian vežu, vežti "to carry, convey," važis "a small sled;" Old Irish fecht "campaign, journey," fen "carriage, cart;" Welsh gwain "carriage, cart;" Old English wegan "to carry;" Old Norse vegr, Old High German weg "way;" Middle Dutch wagen "wagon."

waggish (adj.)
"willing to make a fool of oneself, and fond of doing so to others," 1580s, from wag (n.) + -ish. Related: Waggishly; waggishness.
waggle (v.)
late 15c. (implied in waggling), frequentative of wag (v.). Compare Dutch waggelen "to waggle," Old High German wagon "to move, shake," German wackeln "to totter." Transitive sense from 1590s. Related: Waggled.
wagtail (n.)

c. 1500, kind of small bird that has its tail in continuous motion (late 12c. as a surname), earlier wagstart (mid-15c.), from wag (v.) + tail (n.). From 18c. as "a harlot," but this sense seems to be implied much earlier:

If therefore thou make not thy mistress a goldfinch, thou mayst chance to find her a wagtaile. [Lyly, "Midas," 1592]