1. **State the problem:** We are given a circle with two secant lines intersecting outside the circle and chords intersecting inside the circle, with angles 34°, 18°, x°, and y° labeled. We need to find the missing value of $y$. 2. **Recall the relevant theorems:** - The measure of an angle formed by two secants intersecting outside a circle is half the difference of the intercepted arcs. - 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. **Apply the exterior angle theorem for secants:** If the two secants intersect outside the circle at point $P$, and the arcs intercepted are $34^\circ$ and $18^\circ$, then the angle $x$ formed at $P$ is: $$x = \frac{1}{2} |34^\circ - 18^\circ| = \frac{1}{2} \times 16^\circ = 8^\circ$$ 4. **Apply the interior angle theorem for chords:** For the chords intersecting inside the circle, the angle $y$ is half the sum of the intercepted arcs. The arcs intercepted are $52^\circ$ and $68^\circ$. $$y = \frac{1}{2} (52^\circ + 68^\circ) = \frac{1}{2} \times 120^\circ = 60^\circ$$ 5. **Final answer:** $$\boxed{y = 60^\circ}$$