adagio vs dancing

adagio and dancing both are nouns.

Word	Noun	Adjective	Verb	Adverb
adagio	Yes	Yes	No	Yes
dancing	Yes	No	No	No

As nouns, dancing is a hypernym of adagio; that is, dancing is a word with a broader meaning than adagio:

adagio: a slow section of a pas de deux requiring great skill and strength by the dancers
dancing: taking a series of rhythmical steps (and movements) in time to music

Other hypernyms of adagio include dance, saltation, terpsichore.

adagio (noun)	dancing (noun)
a slow section of a pas de deux requiring great skill and strength by the dancers	taking a series of rhythmical steps (and movements) in time to music
(music) a composition played in adagio tempo (slowly and gracefully)

adagio (adjective)	dancing (adjective)
(of tempo) leisurely

adagio (adverb)	dancing (adverb)
slowly