engagement vs dogfight

engagement and dogfight both are nouns.

Word	Noun	Adjective	Verb	Adverb
engagement	Yes	No	No	No
dogfight	Yes	No	Yes	No

As nouns, dogfight is a hyponym of engagement; that is, dogfight is a word with a more specific, narrower meaning than engagement:

engagement: a hostile meeting of opposing military forces in the course of a war
dogfight: an aerial engagement between fighter planes

Other hyponyms of engagement include Armageddon, pitched battle, naval battle, armed combat, combat, assault.

engagement (noun)	dogfight (noun)
the act of sharing in the activities of a group	a violent fight between dogs (sometimes organized illegally for entertainment and gambling)
a hostile meeting of opposing military forces in the course of a war	an aerial engagement between fighter planes
the act of giving someone a job	disorderly fighting
employment for performers or performing groups that lasts for a limited period of time	a fiercely disputed contest
a mutual promise to marry
contact by fitting together
a meeting arranged in advance

engagement (verb)	dogfight (verb)
	engage in an aerial battle with another fighter plane
	arrange for an illegal dogfight