Why do vertebrate embryos show similarities between organisms that do not appear in the adults?

It is believed that this happens because some signals that regulate development are the same between different species and because they share ancient genes. These ancient genes are expressed during a middle "phylotypic stage" of embryonic development for all species.

For example,  human and animal embryos go through very similar stages of early development and share similar features such as tails and gill-like structures. The major difference appears to be how long it takes to reach each of these same stages.

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