1. **Problem 4:** Given angles around a circle: 137°, $x^\circ$, 223°, and 111.5°. We need to find $x$. 2. The sum of angles around a point on a circle is 360°. 3. Write the equation: $$137 + x + 223 + 111.5 = 360$$ 4. Combine known angles: $$137 + 223 + 111.5 = 471.5$$ 5. Substitute back: $$471.5 + x = 360$$ 6. Solve for $x$: $$x = 360 - 471.5$$ $$x = -111.5$$ 7. Since an angle cannot be negative here, check if the problem involves arcs or angles inside/outside the circle. If these are arcs, the angle $x$ might be the difference between arcs. 8. If $x$ is an angle formed by two arcs, use the formula for angle between arcs: $$x = \frac{|\text{arc}_1 - \text{arc}_2|}{2}$$ 9. Using arcs 137° and 223°: $$x = \frac{|223 - 137|}{2} = \frac{86}{2} = 43$$ 10. So, $x = 43^\circ$. --- 1. **Problem 8:** Given angles 129°, $x^\circ$, 231°, and 115.5° around a circle. 2. Sum of angles around a point is 360°. 3. Write the equation: $$129 + x + 231 + 115.5 = 360$$ 4. Combine known angles: $$129 + 231 + 115.5 = 475.5$$ 5. Substitute back: $$475.5 + x = 360$$ 6. Solve for $x$: $$x = 360 - 475.5 = -115.5$$ 7. Again, negative angle suggests $x$ is angle between arcs. 8. Use formula: $$x = \frac{|231 - 129|}{2} = \frac{102}{2} = 51$$ 9. So, $x = 51^\circ$. **Final answers:** - Problem 4: $x = 43^\circ$ - Problem 8: $x = 51^\circ$