since no conditions are given for the sides of triangle, i think you can draw a triangle with arbitrary length of sides,,
then, name the corners/angles as A, B, C (or any three letter you like)
then, choose 2 angles,, for example you chose A and B. to bisect the angle, get a compass and measure the angle and get its half (if for instance angle A is 40, then draw an angle bisector which is 40/2 or 20 degrees from one side),, do the same for angle B,, extend the angle bisectors of A and B until they intersect. this is now the point of intersection.
lastly, i think you mean "measure the ANGLE I didn't bisect".. so measure angle C using compass,,
so there,, =)