Usually, eaves. the overhanging lower edge of a roof.
Often, eaves. the overhanging edge of anything, as a hat.

before 1000; Middle English eves, Old English efes; cognate with Old High German obisa, Gothic ubizwa hall; cf. above, over

eaved, adjective
uneaved, adjective
Dictionary.com Unabridged
Based on the Random House Dictionary, © Random House, Inc. 2014.
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Word Origin & History

late O.E., from S.W. Midlands dial. eovese (sing.), from O.E. efes "edge of a roof," from P.Gmc. *ubaswa-/*ubiswa (cf. O.H.G. obasa "porch, hall, roof," Goth. ubizwa "porch," Ger. oben "above"), from the root of over. Treated as plural and new singular form eave emerged c.1580.
Online Etymology Dictionary, © 2010 Douglas Harper
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Example sentences
It elevates the ridge board so that there is air movement from the eave all the
  way up and out through the ridge.
The reason he gave was that both roof vents and eave vents had been
  accidentally installed.
Doubling or tripling the starter course at the eave gave added protection to
  this exposed surface.
Eave overhangs are framed by extending the top chords of the trusses beyond the
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