1. **State the problem:** We have a triangle with sides of lengths 8, 10, and 12 units. We want to find the approximate measure of angle $G$. 2. **Identify the sides relative to angle $G$:** The side opposite angle $G$ is the side of length 8 (left side), the side adjacent to angle $G$ is the side of length 12 (top side), and the hypotenuse is 10 (side $EG$). 3. **Use the Law of Cosines or Trigonometric Ratios:** Since we know all three sides, the Law of Cosines is appropriate: $$\cos(G) = \frac{a^2 + b^2 - c^2}{2ab}$$ where $c$ is the side opposite angle $G$, and $a$ and $b$ are the other two sides. 4. **Plug in the values:** $$a = 12, \quad b = 10, \quad c = 8$$ $$\cos(G) = \frac{12^2 + 10^2 - 8^2}{2 \times 12 \times 10} = \frac{144 + 100 - 64}{240} = \frac{180}{240}$$ 5. **Simplify the fraction:** $$\cos(G) = \frac{\cancel{180}}{\cancel{240}} = \frac{3}{4} = 0.75$$ 6. **Find angle $G$ using inverse cosine:** $$G = \cos^{-1}(0.75) \approx 41.4^\circ$$ 7. **Conclusion:** The approximate measure of angle $G$ is $41.4^\circ$, which corresponds to option C.