triumph vs cheer

triumph and cheer both are nouns.

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

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

Other hypernyms of triumph include cheer up, chirk up.

triumph (noun)	cheer (noun)
a successful ending of a struggle or contest	the quality of being cheerful and dispelling gloom
the exultation of victory	a cry or shout of approval

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