1. **State the problem:** We have a triangle with sides $TS=4$ cm, $TR=12.9$ cm, and angle $\angle TSR=77^\circ$. We need to find the other two angles of the triangle. 2. **Known information:** - Side $TS=4$ cm - Side $TR=12.9$ cm - Angle $\angle TSR=77^\circ$ 3. **Use the Law of Cosines to find side $SR$:** $$SR^2 = TS^2 + TR^2 - 2 \times TS \times TR \times \cos(77^\circ)$$ 4. **Calculate $SR^2$:** $$SR^2 = 4^2 + 12.9^2 - 2 \times 4 \times 12.9 \times \cos(77^\circ)$$ $$= 16 + 166.41 - 103.2 \times \cos(77^\circ)$$ 5. **Evaluate $\cos(77^\circ)$:** $$\cos(77^\circ) \approx 0.224951$$ 6. **Substitute and simplify:** $$SR^2 = 16 + 166.41 - 103.2 \times 0.224951$$ $$= 182.41 - 23.21 = 159.20$$ 7. **Find $SR$ by taking the square root:** $$SR = \sqrt{159.20} \approx 12.61 \text{ cm}$$ 8. **Use the Law of Sines to find another angle, say $\angle STR$ opposite side $SR$:** $$\frac{\sin(77^\circ)}{SR} = \frac{\sin(\angle STR)}{TR}$$ 9. **Rearranged to solve for $\sin(\angle STR)$:** $$\sin(\angle STR) = \frac{TR}{SR} \times \sin(77^\circ) = \frac{12.9}{12.61} \times \sin(77^\circ)$$ 10. **Calculate $\sin(77^\circ)$:** $$\sin(77^\circ) \approx 0.974370$$ 11. **Substitute values:** $$\sin(\angle STR) = 1.023 \times 0.974370 = 0.996$$ 12. **Find $\angle STR$ by taking inverse sine:** $$\angle STR = \sin^{-1}(0.996) \approx 85.7^\circ$$ 13. **Find the last angle $\angle TRS$ using the triangle angle sum:** $$\angle TRS = 180^\circ - 77^\circ - 85.7^\circ = 17.3^\circ$$ **Final answer:** $$\angle TSR = 77^\circ, \quad \angle STR \approx 85.7^\circ, \quad \angle TRS \approx 17.3^\circ$$