is bat and bird are homologous ? if yes then why?

Bat wings consist of flaps of skin stretched between the bones of the fingers and arm. Bird wings consist of feathers extending all along the arm. These structural dissimilarities suggest that bird wings and bat wings were not inherited from a common ancestor with wings.

Bird and bat wings are analogous—that is, they have separate evolutionary origins, but are superficially similar because they evolved to serve the same function. Analogies are the result of convergent evolution.

Interestingly, though bird and bat wings are analogous as wings, as forelimbs they are homologous. Birds and bats did not inherit wings from a common ancestor with wings, but they did inherit forelimbs from a common ancestor with forelimbs.