In a way that cheers or encourages; in a manner that provides hope or comfort.
From 'cheer' plus the present participle '-ing' plus the adverb suffix '-ly,' creating an adverb from the verb 'to cheer.' This double-suffix construction shows how English can stack grammatical elements.
Words like 'cheeringly' are wonderful examples of English's flexibility—we can take almost any verb, add '-ing' to make it an adjective, then add '-ly' to make an adverb, creating shades of meaning without new base words.
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