1. **State the problem:** We have a triangle with sides 37 mm, 35 mm, and 38 mm, and one angle of 76°. We need to find the size of angle $\theta$ to 2 decimal places. 2. **Identify known elements:** - Side opposite 76° is 37 mm (assumed from description). - Adjacent sides to $\theta$ are 35 mm and 38 mm. 3. **Use the Law of Cosines to find the side opposite $\theta$:** The Law of Cosines states: $$c^2 = a^2 + b^2 - 2ab\cos(C)$$ where $C$ is the angle opposite side $c$. 4. **Calculate the side opposite $\theta$ (call it $c$):** Here, $a=35$, $b=38$, and $C=76^\circ$. $$c^2 = 35^2 + 38^2 - 2 \times 35 \times 38 \times \cos(76^\circ)$$ Calculate each term: $$35^2 = 1225$$ $$38^2 = 1444$$ $$2 \times 35 \times 38 = 2660$$ Calculate $\cos(76^\circ)$: $$\cos(76^\circ) \approx 0.2419$$ So: $$c^2 = 1225 + 1444 - 2660 \times 0.2419$$ $$c^2 = 2669 - 643.654$$ $$c^2 = 2025.346$$ 5. **Find $c$ by taking the square root:** $$c = \sqrt{2025.346} \approx 45.01$$ 6. **Use the Law of Cosines again to find angle $\theta$ opposite side 37 mm:** $$37^2 = 35^2 + 38^2 - 2 \times 35 \times 38 \times \cos(\theta)$$ Substitute values: $$1369 = 1225 + 1444 - 2660 \cos(\theta)$$ $$1369 = 2669 - 2660 \cos(\theta)$$ Rearranged: $$2660 \cos(\theta) = 2669 - 1369$$ $$2660 \cos(\theta) = 1300$$ Divide both sides by 2660: $$\cos(\theta) = \frac{\cancel{1300}}{\cancel{2660}} = \frac{1300}{2660}$$ Simplify fraction: $$\cos(\theta) = \frac{1300}{2660} \approx 0.4887$$ 7. **Find $\theta$ by taking the inverse cosine:** $$\theta = \cos^{-1}(0.4887) \approx 60.75^\circ$$ **Final answer:** $$\boxed{60.75^\circ}$$