How did the greeks contributed to the modern theater?

Answer 1

In many ways--
They gave us the modern actor when Thesis stepped forward from the chorus.
They gave us terms like upstage and downstage because their stages were raked.
They gave us excellent dramas and comedies that have survived for thousands of years (Oedipus, Lysistrata, etc.)
They gave us an understanding of the structure of story (read Plato).
And that's just scratching the surface.