Final answer:
The constitutional amendment in the Bill of Rights that is common to both Gideon v. Wainwright and Betts v. Brady is the Sixth Amendment.
Step-by-step explanation:
The constitutional amendment in the Bill of Rights that is common to both Gideon v. Wainwright (1963) and Betts v. Brady (1942) is the Sixth Amendment. The Sixth Amendment ensures fairness in criminal trials, including the right to counsel or legal representation. In Gideon v. Wainwright, the Supreme Court declared that the denial of legal counsel violated Gideon's constitutional right. This decision overturned Betts v. Brady, which previously allowed states to deny counsel to indigent defendants.