1. **Problem statement:** We are given a triangle with sides 102 cm, 91.8 cm, and 100.2 cm, and we need to find the measure of the angle $\theta$ opposite the side of length 100.2 cm to the nearest degree. 2. **Formula used:** We use the Law of Cosines, which relates the sides of a triangle to the cosine of one of its angles: $$\cos(\theta) = \frac{a^2 + b^2 - c^2}{2ab}$$ where $a$ and $b$ are the sides adjacent to angle $\theta$, and $c$ is the side opposite $\theta$. 3. **Assign values:** Let $a = 102$, $b = 91.8$, and $c = 100.2$. 4. **Calculate numerator:** $$a^2 + b^2 - c^2 = 102^2 + 91.8^2 - 100.2^2 = 10404 + 8427.24 - 10040.04 = 8801.2$$ 5. **Calculate denominator:** $$2ab = 2 \times 102 \times 91.8 = 18703.2$$ 6. **Calculate cosine:** $$\cos(\theta) = \frac{8801.2}{18703.2} \approx 0.4705$$ 7. **Find angle $\theta$:** $$\theta = \cos^{-1}(0.4705) \approx 61.9^\circ$$ 8. **Round to nearest degree:** $$\theta \approx 62^\circ$$ **Final answer:** The measure of angle $\theta$ is approximately $62^\circ$.