The Cold War was the result of the build up of tension between the US and USSR following WWII & the expansion of each nation's ideals and beliefs.
The US & USSR were odd allies, who, without Axis aggression, would not have worked together in the first place. Both sides had different economic systems that were against each other, as well as different political beliefs on how government and the rights of people should be laid out. The Cold War was the result of increased tension that led to both sides trying to become more superior to the other (which in the end led to arm races, complex alliances, and proxy wars).
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