1. **State the problem:** We are given two separate circle geometry problems involving angles and need to find the value of $x$ in each. 2. **First problem (Top-left circle):** Given angles at points $M$ and $L$ on the circle: $30^\circ$ and $(2x - 30)^\circ$ respectively. 3. **Key rule:** In a circle, angles subtended by the same chord or arc are equal or supplementary depending on the configuration. Here, since $M$ and $L$ are points on the circumference subtending the same arc, their angles sum to $180^\circ$ (angles in a cyclic quadrilateral sum to $180^\circ$). 4. **Set up the equation:** $$30 + (2x - 30) = 180$$ 5. **Simplify:** $$30 + 2x - 30 = 180$$ $$2x = 180$$ 6. **Solve for $x$:** $$x = \frac{180}{2}$$ $$x = 90$$ 7. **Second problem (Bottom-left circle):** Given an angle inside the circle $125^\circ$ and an external angle $(6x - 11)^\circ$ formed by tangent and chord. 8. **Key rule:** The angle between a tangent and a chord is equal to the angle in the alternate segment of the circle. 9. **Set up the equation:** $$(6x - 11) = 125$$ 10. **Solve for $x$:** $$6x - 11 = 125$$ $$6x = 125 + 11$$ $$6x = 136$$ $$x = \frac{136}{6}$$ $$x = \frac{68}{3} \approx 22.67$$ **Final answers:** - For the first problem, $x = 90$ - For the second problem, $x = \frac{68}{3}$ or approximately $22.67$