1. **State the problem:** We are given a circle with points E, F, and G on the circumference and two inscribed angles: $\angle E = (12x + 40)^\circ$ and $\angle F = (8x + 10)^\circ$. We need to find the value of $x$. 2. **Recall the property of inscribed angles subtending the same arc:** Inscribed angles that subtend the same arc are equal. Since $\angle E$ and $\angle F$ are inscribed angles in the same circle, and the problem implies a relationship, we set them equal: $$12x + 40 = 8x + 10$$ 3. **Solve the equation:** $$12x + 40 = 8x + 10$$ Subtract $8x$ from both sides: $$12x - \cancel{8x} + 40 = \cancel{8x} + 10 \Rightarrow 4x + 40 = 10$$ Subtract 40 from both sides: $$4x + \cancel{40} - \cancel{40} = 10 - 40 \Rightarrow 4x = -30$$ Divide both sides by 4: $$\frac{\cancel{4}x}{\cancel{4}} = \frac{-30}{4} \Rightarrow x = -\frac{30}{4} = -7.5$$ 4. **Interpretation:** The value $x = -7.5$ is the solution to the equation based on the given angle expressions. **Final answer:** $x = -7.5$