1. **State the problem:** We have a circle with center O and two chords intersecting at O forming a 50° angle. We want to find the values of $x$, $y$, and $z$ which are angles related to the chords and triangle inside the circle. 2. **Recall important rules:** - The angle between 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 angles in a triangle is 180°. - Angles subtended by the same chord in the circle are equal. 3. **Analyze the given angle:** The angle at O between the two chords is 50°. 4. **Find $x$ and $z$:** Since $x$ and $z$ are points on the circle connected to O, and the angle between chords at O is 50°, the arcs intercepted by these chords relate to this angle. The angle formed by two chords intersecting inside the circle is half the sum of the intercepted arcs. 5. **Express the intercepted arcs:** Let the arcs opposite to $x$ and $z$ be $a$ and $b$ respectively. Then, $$50 = \frac{a + b}{2} \implies a + b = 100$$ 6. **Find $y$:** The triangle near $z$ has an angle $y$ opposite the chord segment ending at $z$. Since $y$ is an inscribed angle subtending arc $a$, then $$y = \frac{a}{2}$$ 7. **Use the sum of arcs in the circle:** The total circle is 360°, so the remaining arc is $360 - (a + b) = 360 - 100 = 260$. 8. **Assuming symmetry or equal arcs for simplicity:** If $a = b = 50$, then $$y = \frac{a}{2} = \frac{50}{2} = 25$$ 9. **Therefore, the values are:** - $x = 50°$ (angle at point $x$ on the circle) - $y = 25°$ - $z = 50°$ This solution assumes equal arcs intercepted by the chords for simplicity based on the given 50° angle at O. **Final answers:** $$x = 50°, \quad y = 25°, \quad z = 50°$$