1. **Problem statement:** In triangle $\triangle XYZ$, given $\angle T = 69^\circ$, side $t = 8$ cm, and side $x = 5$ cm, find the requested value (assumed to be side or angle related to these). 2. **Formula and rules:** To solve triangles with two sides and an included angle or two angles and a side, we use the Law of Cosines or Law of Sines. 3. **Using Law of Cosines:** $$c^2 = a^2 + b^2 - 2ab \cos(C)$$ where $c$ is the side opposite angle $C$. 4. **Applying values:** Here, $a = 8$, $b = 5$, and $C = 69^\circ$. Calculate: $$c^2 = 8^2 + 5^2 - 2 \times 8 \times 5 \times \cos(69^\circ)$$ $$c^2 = 64 + 25 - 80 \times \cos(69^\circ)$$ 5. Calculate $\cos(69^\circ) \approx 0.3584$: $$c^2 = 89 - 80 \times 0.3584 = 89 - 28.672 = 60.328$$ 6. Find $c$: $$c = \sqrt{60.328} \approx 7.77 \text{ cm}$$ **Final answer:** The side opposite $69^\circ$ is approximately $7.77$ cm.