Two angles that have a common vertex and a common side are called supplementary angles. true or false

It can be true. Just not always.
Short False
Draw what looks to be a 60 degree angle. Draw a line from the vertex of the 60 degree to somewhere between the two arms of the 60 degree angle. You have drawn what you question describes, but your angle is 60 degrees not 180. If you draw a line from the vertex of a 180 degree angle you will get two angles that are supplementary. 

I think your safest answer is false.

Bis the correct answer