How does the u. s. supreme court affect the rights of citizens?

They interpret the Constitution and any laws they may be deciding on, and determine if those laws violate anything that is in the Constitution. Many of those cases deal with the rights the Constitution grant to the citizens of the U.S. For example, a recent decision that dealt with the 2nd Amendment (the right to bear arms) said that the law in Washington DC that banned most people from owning handguns violated the 2nd Amendment right to bear arms.

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