Question 32946
If the measure of the arc is x degrees, the measure of your angle is x/2 degrees.
Draw the picture you described; Draw a central angle that subtends the 
same arc.  The triangle formed is equilateral because two of the sides 
are radii.  The supplementary angle at the center is 180-x degrees.
Your angle is 1/2(180-(180-x))= (1/2)x degrees.
Cheers,
Stan H>