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how is the us a democracy like athens

1. Athenian democracy was exclusive.  Women, slaves, children, and resident aliens did not have citizenship and could not vote.  They did not have any influence on the government at this time. 
2.  Alienation of political power was absurd to the Greek people. Elections were held through a lottery/chance technique, and they only personally selected their military leaders (like Pericles).  "The quintessentially democratic method was selection by lot, a practice that embodies a criterion of selection in principle opposed to the alienation of citizenship and to the assumption that the demos is politically incompetent" 

Popular sovereignty or the authority of the people referenced in the permeable of the constitution . this principle states that government power comes from the consent of the citizens.