Birds lived alongside Cretaceous dinosaurs. Birds as a seprate group diverged in the very late Jurrassic. The way evolutionary descent works means that we are Reptiliomorpha, because our ancestors decended from permian therapsids like the ugly Gorgonapsid. So yes, they are an extension of the dinosaurs.If all life perishes, there would be nothing on the planet. In the case of the dinosaurs, it is likely that a few very lucky dinos survived the meteorite, and over time evolved into birds. The meteorite did not kill all living beings, otherwise the earth would most likely be a barren wasteland. Therefore, the birds of today all trace lineage to those very lucky dinos millions of years ago, making them an extension of the dinosaur family tree.