1. **Problem statement:** We need to find the measure of angle $x$ formed on the circumference by two chords intersecting inside a circle. 2. **Relevant theorem:** The angle formed on the circumference by two chords intersecting inside a circle is half the sum of the measures of the two opposite angles formed inside the circle where the chords intersect. 3. **Given:** The two angles inside the circle where the chords intersect are $80^\circ$ and $60^\circ$. 4. **Formula:** $$x = \frac{1}{2} (\text{angle}_1 + \text{angle}_2)$$ 5. **Substitute the values:** $$x = \frac{1}{2} (80^\circ + 60^\circ)$$ 6. **Calculate the sum inside the parentheses:** $$x = \frac{1}{2} (140^\circ)$$ 7. **Simplify:** $$x = 70^\circ$$ **Final answer:** The measure of angle $x$ is $70^\circ$.