The main reason that France signed a formal aliance with the colonies during the American Revolution was

because they wanted to lower the influence and strength of Britain, which was regarded as the main opponent of France in this period.

Explanation:

After France was defeated in a war with Britain (French and Indian war) they lost all the territories in North America. Additionally, they were practically embarrassed with that treaty and were just waiting for the opportunity to revenge on them. That opportunity finally came in the American Revolution after the battle of Saratoga. At the end peace treaty was once more signed in Paris, this time France was on the side of winners.

for the united states of america, 1945 to 1964 was a time of high economic growth and general prosperity. it was also a time of confrontation as the capitalist united states and its allies politically opposed the soviet union and other communist countries; the cold war had begun.

explanation: