The source of authority cited by the supreme court for applying the bill of rights to the states is

the Fourteenth Amendment to the Constitution

Explanation:

the Fourteenth Amendment to the Constitution specifically created to  Granted citizenship ,  civil Treatments and legal rights to all people in united states regardless of their Race.

All of these rights are covered within the bill of rights, which added ten additional amendments to the constitution. Covering all rights such as the freedom of speech, the freedom to pursue happiness, the Rights to obtain informed consent, The rights to be valued and honored, etc.

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