1. **Problem Statement:** Find the measure of angle $\angle I$ in triangle $\triangle IJK$ where sides $i=96$ cm, $j=22$ cm, and $k=85$ cm. 2. **Formula Used:** We use the Law of Cosines to find an angle when all three sides are known: $$\cos(\angle I) = \frac{j^2 + k^2 - i^2}{2jk}$$ 3. **Substitute the known values:** $$\cos(\angle I) = \frac{22^2 + 85^2 - 96^2}{2 \times 22 \times 85}$$ 4. **Calculate the squares:** $$22^2 = 484, \quad 85^2 = 7225, \quad 96^2 = 9216$$ 5. **Plug in the values:** $$\cos(\angle I) = \frac{484 + 7225 - 9216}{2 \times 22 \times 85} = \frac{7709 - 9216}{3740} = \frac{-1507}{3740}$$ 6. **Simplify the fraction:** $$\cos(\angle I) = -\frac{1507}{3740}$$ 7. **Calculate the cosine value:** $$\cos(\angle I) \approx -0.4029$$ 8. **Find the angle using inverse cosine:** $$\angle I = \cos^{-1}(-0.4029)$$ 9. **Calculate the angle in degrees:** $$\angle I \approx 114.7^\circ$$ **Final answer:** The measure of $\angle I$ is approximately **114.7 degrees** to the nearest tenth.