In the Treaty of Paris, the British Crown formally recognized American independence and ceded most of its territory east of the Mississippi River to the United States, doubling the size of the new nation and paving the way for westward expansion.
Each was a natural trading partner.
Taking a side in that conflict would have had bad consequences for the U.S., the proverbial no-win situation.
The American civil War was a war fought from 1861 to 1865. It was one of the bloodiest wars.
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.
Hope this helps!