1. **State the problem:** We are given a circle with marked arcs of 53°, 60°, and 80°, and two variables $x$ and $y$ representing angles inside the circle. We need to find the value of $x$. 2. **Recall the relevant circle theorems:** - 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. - The sum of arcs around a circle is 360°. 3. **Identify the arcs related to $x$:** - The angle $x$ is formed by two chords intersecting inside the circle. - The intercepted arcs for angle $x$ are the arcs opposite to it, which are 53° and 60°. 4. **Apply the formula for the angle formed by two chords:** $$x = \frac{1}{2} (\text{arc}_1 + \text{arc}_2)$$ where $\text{arc}_1 = 53^\circ$ and $\text{arc}_2 = 60^\circ$. 5. **Calculate $x$:** $$x = \frac{1}{2} (53 + 60) = \frac{1}{2} (113) = 56.5^\circ$$ 6. **Final answer:** $$\boxed{56.5^\circ}$$