Did the supercontinent Pangea really exist?

Many scientists today believe in the concept that all the continents were connected at some point in the past and that they separated over time.  This supercontinent is called Pangea.  They claim that the continents look like puzzle pieces that should fit together and that plate tectonics separated them.  While it does look like South America can fit right into the side of Africa, in actuality, the idea simply does not work.  For instance, one is not told that in order to get Pangea most of the land masses were distorted and Central America is completely missing, along with many islands.  Also, if one took all the water off the earth, you would find that all the land is connected.  While plate tectonics (breaking of the earth’s crust that causes earthquakes and volcanoes) does exist, it does not have the power to move these land masses as described.