The common ancestor thing is hard to wrap my brain around. Dawkins gives a cool thought experiment where he says imagine a card catalog with photos of you and your ancestors in chronological order. If you look back 10 generations, you'll find a human. If you go back 100 generations, you'll find a human. If you go back 5,000 generations, you'll still see a human! However, they probably won't look exactly like a moden human. If you go back 15,000 generations, you'll find something human-like, but not really a modern homo sapiens. All of those cards along the way have miniscule,imperceptible differences. If you go back far enough, you'll find something like a rodent. But the number of cards you need to flip through to find that rodent is extremely large. Something like 200 million generations. Keep in mind the more ancient animals had shorter life spans.

So t-rex and chickens may have come from the same branch, but there are millions of "cards" between them.

i think a better way to look at it is; the chicken and the t-rex share a common ancestor

an animal existed that branched into 2 different paths, one towards birds and the other big ol lizard things (conjecture, am stupid)

Over the last 15 years, scientists have updated the theorized appearance of thousands of species of dinosaurs to have feathers. Most of these species are in the theropod family. You’ll be seeing updates in natural history museums as time goes on.

https://www.amnh.org/exhibitions/dinosaurs-among-us/feathers

There’s also the chicken’s talons to consider. Birds with similar claws are called raptors due to their similarity in appearance to the dinosaur. This does not mean they are direct descendants, however, but that they have a genealogically shared ancestor.

That's a mental shortcut, the birds came from dinosaurs, the birds cladistically ARE dinosaurs; birds, dinosaurs and crocodilians are archosaurs, but birds are closest dinosaurs relatives

https://terriblelizards.libsyn.com/tls02e07_bird_origins

Terrible Lizards Podcast episode on bird origins. (Dinosaur means Terrible Lizard)

Not exactly ancestors, as others have said.

DNA doesn't last nearly long enough. Scientists have made great strides in analyzing ancient DNA (aDNA), They have decoded the genomes of Neanderthals and other extinct human species. But that aDNA is only tens of thousands of years old. IIRC the theoretical maximum is something like 1 million years. No chance on dinosaur DNA.

As to how what evidence there is, I think that's already sufficiently answered, and better than I could.

https://www.sciencedirect.com/science/article/pii/S0960982215009458#:~:text=Birds%20evolved%20from%20theropod%20dinosaurs,in%20one%20burst%20of%20innovation.

This seems to be a decent intro. Basically therapods, of which T-Rex was one, are ancestors of the birds.

Half science half click bait, completely awesome

The best evidence for their common ancestry is from 2008, but it doesn’t look like there have been any new developments since then.

Molecular analysis of a shred of 68-million-year-old Tyrannosaurus rex protein — along with that of 21 modern species — confirms that dinosaurs share common ancestry with chickens, ostriches, and to a lesser extent, alligators.

Source Harvard Gazette

They put a plunger-looking appendage on a chicken's ass and discovered that after this was done it walked more like a T-Rex does...seriously.