How is a direct democracy and a republic the same?

The terms are not always the same. What is the same is this; in a democracy and in a republic, citizens have to the right to vote. In government, democracies do not exist. It would come down to the "mob rule" Greek definition. In a republic, on the other hand, elected officials carry out the policies thy believe are in the best interests of a nation. The parts of a republic are designed to protect citizens with a minority viewpoint by having various types of legislative, judicial and executive "parts" along with a written constitution. There are many forms of a republic. The example given is one of many types.