Miranda v. Arizona has a very significant impact today. This case ruled that any arrested person must be read his/her Miranda Rights (You have the right to remain silent. Anything you say can and will be used against you in a court of law. Etc.). This is due to the fact that many people self-incriminated themselves because they were unaware of their rights. This lead to innocent people being accused of crimes.
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Do you want vague details? Westward expansion was when the colonists began pushing West and migrating past the mountains further into Northern America. As this happened relationships with the Native Americans intensified as colonists began forcing more and more Native Americans off of their lands.
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"Manifest destiny was a term that came to describe a widespread belief in the middle of the 19th century that the United States had a special mission to expand westward.