Should birds be considered dinosaurs?

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Yes, they are Avian Dinosaurs. If one wants to describe dinosaurs without including our feathered friends, they should use the term "non-avian dinosaur"

There are two major divisons of dinosaurs, the bird-hipped clade (Ornithischians) and the lizard-hipped clade (Saurischea). in retrospect the names are bad, because Birds are more closely related to the latter group than the supposedly Bird hipped bunch. The Saurischea are further divided into Sauropodomorpha (stuff like Diplodocus and other long necked dinosaurs) and Therapods. This last group splits further into a few classes until we get to the Birds. In fact, one of the birds closest relatives is, of all things, the Tyranosaur group.

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Actually Crocodiles and Alligators, although the closest living relative of dinosaurs (including birds) are not actually dinosaurs. They belong to a clade called Archnosaur, which includes the dinosaurs and the pterosaurs.

Sharks are not dinosaurs but cartiliginous fish. They branch off at a far earlier point.