1. **State the problem:** We are given three right triangle problems involving tangent ratios: - Find $x$ in $\tan 55^\circ = \frac{x}{10}$. - Find $x$ in $\tan 76^\circ = \frac{x}{354}$. - Find angle $x$ in $\tan x = \frac{125}{72}$. 2. **Formula used:** The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side: $$\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$$ 3. **Solve the first problem:** $$\tan 55^\circ = \frac{x}{10} \implies x = 10 \times \tan 55^\circ$$ Calculate $\tan 55^\circ$: $$\tan 55^\circ \approx 1.4281$$ So, $$x = 10 \times 1.4281 = 14.281$$ Rounded to one decimal place: $$x \approx 14.3$$ 4. **Solve the second problem:** $$\tan 76^\circ = \frac{x}{354} \implies x = 354 \times \tan 76^\circ$$ Calculate $\tan 76^\circ$: $$\tan 76^\circ \approx 4.0108$$ So, $$x = 354 \times 4.0108 = 1419.8$$ Rounded to one decimal place: $$x \approx 1419.8$$ 5. **Solve the third problem:** Given: $$\tan x = \frac{125}{72}$$ Calculate the ratio: $$\frac{125}{72} \approx 1.7361$$ Find angle $x$ by taking arctangent: $$x = \tan^{-1}(1.7361)$$ Calculate: $$x \approx 60.1^\circ$$ **Final answers:** - $x \approx 14.3$ - $x \approx 1419.8$ - $x \approx 60.1^\circ$