1. **State the problem:** Calculate the area of triangle STR where side ST = 4 cm, side TR = 12.9 cm, and the included angle $\angle STR = 77^\circ$. 2. **Formula used:** The area of a triangle given two sides and the included angle is $$\text{Area} = \frac{1}{2}ab\sin(C)$$ where $a$ and $b$ are the sides and $C$ is the included angle. 3. **Apply the formula:** Here, $a = 4$ cm, $b = 12.9$ cm, and $C = 77^\circ$. 4. **Calculate the area:** $$\text{Area} = \frac{1}{2} \times 4 \times 12.9 \times \sin(77^\circ)$$ 5. **Simplify step-by-step:** $$= 2 \times 12.9 \times \sin(77^\circ)$$ 6. **Calculate $\sin(77^\circ)$:** $$\sin(77^\circ) \approx 0.97437$$ 7. **Multiply:** $$\text{Area} \approx 2 \times 12.9 \times 0.97437 = 25.13$$ 8. **Final answer:** $$\boxed{25.1 \text{ cm}^2}$$ This confirms the correct setup and calculation of the area using the sine formula for triangles.