1. **State the problem:** The user wants to solve for the values of $x$, $y$, and $z$ related to angles formed by chords intersecting inside a circle with a given 50° angle at the intersection point. 2. **Recall the key formula:** The 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. **Express the given angle:** Given angle at intersection $O$ is $50^\circ$, so $$50 = \frac{a + b}{2} \implies a + b = 100$$ where $a$ and $b$ are the measures of the intercepted arcs. 4. **Relate $y$ to arc $a$:** The inscribed angle $y$ subtends arc $a$, so $$y = \frac{a}{2}$$ 5. **Relate $x$ and $z$ to arcs:** Since $x$ and $z$ are angles subtended by arcs $b$ and $a$ respectively, $$x = \frac{b}{2}, \quad z = \frac{a}{2}$$ 6. **Use the sum of arcs:** Since $a + b = 100$, then $$x + z = \frac{b}{2} + \frac{a}{2} = \frac{a + b}{2} = 50$$ 7. **Express $y$ in terms of $a$:** From step 4, $y = \frac{a}{2}$. 8. **Solve for $x$, $y$, and $z$ in terms of $a$:** $$x = \frac{100 - a}{2}, \quad y = \frac{a}{2}, \quad z = \frac{a}{2}$$ 9. **Without additional information, $a$ can vary, so $x$, $y$, and $z$ depend on $a$.** **Final answer:** $$x = \frac{100 - a}{2}, \quad y = \frac{a}{2}, \quad z = \frac{a}{2}$$ where $a$ is the measure of one intercepted arc and $0 < a < 100$.