They are homologous in the sense that all tetrapods share that forelimb structure. They are analogous because both bats and birds adapted flight to their local environmental conditions. One being a mammal, that has the finger extension type wing and one being aves with the full forelimb extending the wing. They have the flight in common, but not the structures, so are analogous. They have an ancestral condition leading to the similarity of forelimb structure and so are homologous.
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A bat's wing can be considered a homologous structure because it shares a common evolutionary origin with the arms and hands of humans and other mammals. It can also be considered an analogous structure because it serves a similar function, flight, to the wings of birds, which have a different evolutionary origin.
The bird's wing has a fairly rigid bone structure, and the main flying muscles move the bones at the point where the wing connects to the body. A bat has a much more flexible wing structure. It is very much like a human arm and hand, except it has a thin membrane of skin (called the patagium) extending between the "hand" and the body, and between each finger bone. Bats can move the wing like a hand, essentially "swimming" through the air. The "thumb" extends out of the wing as a small claw, which bats use to climb up trees and other structures. This helps them reach a high "launching point" for flight takeoff. Appropriately, the order of bats is called Chiroptera, Greek for "hand-wing."
they are both homologous structures
In the diploid cell nucleus there are homologous chromosomes. During meiosis homologous chromosomes pair each other and bring about crossing over to inter change genetic material. This is how new combination of genes are created in the male and female gametes.
An analogous trait is a trait that is similar in function or purpose between different organisms but has evolved independently. For example, wings in birds and wings in bats are considered analogous traits because they serve the same purpose of flight but evolved separately.
Analogous structures are body parts or organs in different species that have a similar function but not necessarily a common evolutionary origin. For example, the wings of birds and bats are analogous structures because they both serve the purpose of flight, even though they evolved independently.