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These battles were known as the first battles of the Revolutionary War
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On Christmas Eve, Washington and his men sailed across the Delaware river and surprised Hessians in Trenton. This proved to the British that the American general was better than they thought
(http://cdn.history.com/sites/2/2013/11/Battle-of-Trenton-Princeton-AB.jpeg) -
The Battles of Saratoga are known as the turning point in war. This caused the other European countries to think that the Americans might have a chance.
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King Louis XVI of France signed a treaty with the Americans and agreed to help them with their Independence
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This battle was one of the first losses for the British in the South. It gave the Patriots hope and faith that they would win the Revolution (https://upload.wikimedia.org/wikipedia/commons/thumb/8/85/KingsMountain_DeathOfFerguson_Chappel.jpg/300px-KingsMountain_DeathOfFerguson_Chappel.jpg)
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When Cornwallis set up his base in Yorktown, Washington saw his chance. The French naval fleet came and blocked the British from escaping or receiving supplies. Washington came up from the other side, and together, they trapped the British on the peninsula. They bombarded Yorktown and the British surrendered.
(https://www.landofthebrave.info/images/battle-yorktown-surrender-lord-cornwallis.jpg)