How do you find the measure of interior angles?

The sum of all of the interior angles can be found using the formula S = (n - 2)*180

If the diagram is to scale (and you don't have any information given), you could use a protractor and just make sure that the sum of the angles is correct.
(180 degrees is the sum of the angles for a triangle, 360 degrees for a quadrilateral ALWAYS)

If you are with a problem like- Angle 1= 3x.  Angle 2=5x+1. etc, then you need to add the angles up, combine like-terms, solve for x, and plug in the x value to find out.  

If you are given the value of one angle, then you need to solve for it by adding up all the given angles and subtracting it from the sum (180-triangle, 360 quadrilateral, what the shape should equal in interior angles).

Hope I helped in all my rambling. :)