1. **State the problem:** We have triangle RST with sides RS = 5, TS = 9, and angle $\angle S = 87^\circ$. We need to find $m\angle R$, side $s = RT$, and $m\angle T$. 2. **Use the Law of Cosines to find side $s$:** The Law of Cosines states: $$s^2 = RS^2 + TS^2 - 2 \cdot RS \cdot TS \cdot \cos(\angle S)$$ Substitute values: $$s^2 = 5^2 + 9^2 - 2 \cdot 5 \cdot 9 \cdot \cos(87^\circ)$$ $$s^2 = 25 + 81 - 90 \cdot \cos(87^\circ)$$ Calculate $\cos(87^\circ) \approx 0.05234$: $$s^2 = 106 - 90 \cdot 0.05234 = 106 - 4.7106 = 101.2894$$ Take the square root: $$s = \sqrt{101.2894} \approx 10.1$$ 3. **Use the Law of Sines to find $m\angle R$:** Law of Sines: $$\frac{\sin(\angle R)}{TS} = \frac{\sin(\angle S)}{s}$$ Rearranged: $$\sin(\angle R) = \frac{TS \cdot \sin(\angle S)}{s}$$ Calculate $\sin(87^\circ) \approx 0.9986$: $$\sin(\angle R) = \frac{9 \cdot 0.9986}{10.1} = \frac{8.9874}{10.1} \approx 0.8893$$ Find $\angle R$: $$\angle R = \arcsin(0.8893) \approx 62.7^\circ$$ 4. **Find $m\angle T$ using the triangle angle sum:** $$\angle T = 180^\circ - \angle S - \angle R = 180^\circ - 87^\circ - 62.7^\circ = 30.3^\circ$$ **Final answers:** $$m\angle R = 62.7^\circ$$ $$s = 10.1$$ $$m\angle T = 30.3^\circ$$