1. The problem involves finding the value of the angle $x$ in a circle where two chords intersect, creating angles of $68^\circ$ and $x^\circ$. 2. The key rule for angles formed by intersecting chords inside a circle is: The measure of an angle formed by two chords intersecting inside a circle is half the sum of the measures of the arcs intercepted by the angle and its vertical angle. 3. Let the intercepted arcs be $\alpha$ and $\beta$. Then the angle $x$ satisfies: $$x = \frac{\alpha + \beta}{2}$$ 4. Given one angle is $68^\circ$, its vertical angle is also $68^\circ$, so the intercepted arcs sum to: $$2 \times 68 = 136^\circ$$ 5. Since $x$ is the other angle formed by the intersecting chords, it is: $$x = \frac{\text{arc}_1 + \text{arc}_2}{2} = \frac{136}{2} = 68^\circ$$ 6. Therefore, the value of $x$ is $68^\circ$. Final answer: $$x = 68^\circ$$