cheer vs rejoice

cheer is a noun but rejoice is not a noun.

Word	Noun	Adjective	Verb	Adverb
cheer	Yes	No	Yes	No
rejoice	No	No	Yes	No

As verbs, rejoice is a hypernym of cheer; that is, rejoice is a word with a broader meaning than cheer:

Other hypernyms of cheer include joy.

As verbs, rejoice is a hyponym of cheer; that is, rejoice is a word with a more specific, narrower meaning than cheer:

Other hyponyms of cheer include exuberate, exult, jubilate, triumph, buoy up, lighten, lighten up.

cheer (noun)	rejoice (noun)
the quality of being cheerful and dispelling gloom
a cry or shout of approval

cheer (verb)	rejoice (verb)
become cheerful	to express great joy
spur on or encourage especially by cheers and shouts	feel happiness or joy
show approval or good wishes by shouting	be ecstatic with joy
cause (somebody) to feel happier or more cheerful
give encouragement to