1. **Stating the problem:** We have a circle with two chords intersecting inside it at point O, forming angles $x$, $y$, and $z$. There is an angle of $50^\circ$ between $x$ and $z$ at the intersection point O. We need to find the values of $x$, $y$, and $z$. 2. **Relevant formula:** When two chords intersect inside a circle, the measure of the angle formed is half the sum of the measures of the arcs intercepted by the angle and its vertical angle. In other words, for angles formed by intersecting chords: $$\text{Angle} = \frac{1}{2} (\text{arc}_1 + \text{arc}_2)$$ 3. **Important rule:** The vertical angles formed by the intersecting chords are equal. 4. **Given:** The angle between $x$ and $z$ at O is $50^\circ$. Since $x$, $y$, and $z$ are angles around point O formed by the intersecting chords, and the $50^\circ$ angle is between $x$ and $z$, it implies that $y = 50^\circ$ because vertical angles are equal. 5. **Sum of angles around point O:** The angles $x$, $y$, and $z$ around point O must sum to $180^\circ$ because they form a straight line or linear pair (since the chords intersect inside the circle). 6. **Set up the equation:** $$x + y + z = 180^\circ$$ Substitute $y = 50^\circ$: $$x + 50 + z = 180$$ 7. **Given the symmetry and the problem options, assume $x = z$ (since the $50^\circ$ angle is between $x$ and $z$ and the figure is symmetric):** $$x + 50 + x = 180$$ $$2x + 50 = 180$$ 8. **Solve for $x$:** $$2x = 180 - 50$$ $$2x = 130$$ $$x = \frac{130}{2} = 65^\circ$$ 9. **Find $z$:** Since $x = z$, then $z = 65^\circ$. 10. **Final values:** $$x = 65^\circ, \quad y = 50^\circ, \quad z = 65^\circ$$