The measure of an exterior angle of a regular polygon is given. find the measure of an interior angle. then find the number of sides. given angle: 30.

If the exterior angle is 30, then its supplement, the base angle INSIDE the polygon is 180-30 which is 150.  Now we need to concentrate on the triangles involved in this n-gon.  If we take one triangle out to assess, the base angle will measure half of the 150, or 75.  Now each base angle is 75, so by the triangle angle-sum theorem, 180-75-75 = 30.  The vertex angle of this single triangle is 30 degrees (it's not a coincidence that the vertex angle is the same measure as the exterior angle, btw).  If the vertex angle measures 30, we can divide 360 by the vertex angle which will give us the number of angles in the center of this polygon.  360/30 = 12.  If there are 12 central angles in this polygon, there are also 12 sides.  There you go!