in accordance with truth or fact; correctly: to see rightly; to understand rightly.
in accordance with morality or equity; uprightly.
properly, fitly, or suitably: to be rightly dressed.
Informal. with certainty; positively: I can't rightly say.

before 900; Middle English; Old English rihtlīce. See right, -ly

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Based on the Random House Dictionary, © Random House, Inc. 2014.
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World English Dictionary
rightly (ˈraɪtlɪ)
1.  in accordance with the facts; correctly
2.  in accordance with principles of justice or morality
3.  with good reason; justifiably: he was rightly annoyed with her
4.  properly or suitably; appropriately: rightly dressed for a wedding
5.  informal (used with a negative) with certainty; positively or precisely (usually in the phrases I don't rightly know, I can't rightly say)

Collins English Dictionary - Complete & Unabridged 10th Edition
2009 © William Collins Sons & Co. Ltd. 1979, 1986 © HarperCollins
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Example sentences
Students and employers are rightly wary of the quality of online courses.
They see themselves, rightly so, as leading the emerging global civilization.
Everyone knows that candidates cherry-pick their submitted evaluations, and
  rightly so.
Their publisher would accept nothing less, and rightly so.
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