Let O be the center of the circle, and let OC be the radius of the circle. Since AB is a chord of the circle and the tangents to the circle at A and B intersect at C, it follows that angle AOC is equal to angle BOC (by the Tangent-Chord Angle Theorem).
Since the length of chord AB is equal to the radius OC, it follows that triangle AOB is an isosceles triangle. Therefore, angle AOB is equal to (180 - angle ACB)/2, where angle ACB is the angle we are trying to find.
Combining these two observations, we get:
angle ACB = 2*(angle AOB - angle BOC) = 2*((180 - angle ACB)/2 - angle BOC) = 180 - 2*angle BOC - angle ACB
Simplifying this expression, we get:
angle ACB = 180 - 2*angle BOC
To find angle BOC, we can use the fact that it is half of the central angle subtended by arc AB. Since AB is a chord of the circle and the tangents to the circle at A and B intersect at C, it follows that arc AB is bisected by line OC. Therefore, angle BOC is equal to half of the measure of arc AB.
Since chord AB has the same length as the radius of the circle, it follows that arc AB subtends an angle of 120 degrees (using the Inscribed Angle Theorem). Therefore, angle BOC is equal to 60 degrees.
Substituting this value into the expression we derived earlier, we get:
angle ACB = 180 - 2*60 = 60 degrees.
Therefore, the measure of angle ACB is 60 degrees.