1. **Problem statement:** Find the measure of the indicated angle in each right triangle using the given sides. 2. **Formula and rules:** In a right triangle, use trigonometric ratios: sine, cosine, or tangent. - $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$ - $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$ - $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$ Use inverse trig functions to find the angle: $\theta = \sin^{-1}(x)$, $\cos^{-1}(x)$, or $\tan^{-1}(x)$. 3. **Calculations:** **(1) Top-left triangle:** legs 7 (opposite), 4 (adjacent), angle at left base. $$\theta = \tan^{-1}\left(\frac{7}{4}\right) = \tan^{-1}(1.75) \approx 60.3^\circ$$ **(2) Top-right triangle:** hypotenuse 7, base 3.7 (adjacent), angle at left base. $$\theta = \cos^{-1}\left(\frac{3.7}{7}\right) = \cos^{-1}(0.5286) \approx 58.1^\circ$$ **(3) Center-left triangle:** legs 14 and 14, angle at left base. $$\theta = \tan^{-1}\left(\frac{14}{14}\right) = \tan^{-1}(1) = 45^\circ$$ **(4) Center-right triangle:** legs 6 and 6, angle at top-left. $$\theta = \tan^{-1}\left(\frac{6}{6}\right) = 45^\circ$$ **(5) Lower-left triangle:** hypotenuse 10, base 8 (adjacent), angle at bottom-right. $$\theta = \cos^{-1}\left(\frac{8}{10}\right) = \cos^{-1}(0.8) \approx 36.9^\circ$$ **(6) Lower-right triangle:** legs 3 km (opposite), 2 km (adjacent), angle at bottom-right. $$\theta = \tan^{-1}\left(\frac{3}{2}\right) = \tan^{-1}(1.5) \approx 56.3^\circ$$ **(7) Lower-left triangle:** base 15 km (adjacent), height 8 km (opposite), angle at left base. $$\theta = \tan^{-1}\left(\frac{8}{15}\right) = \tan^{-1}(0.5333) \approx 28.1^\circ$$ **(8) Lower-right triangle:** hypotenuse 14 m, leg 8 m (adjacent), angle at bottom-right. $$\theta = \cos^{-1}\left(\frac{8}{14}\right) = \cos^{-1}(0.5714) \approx 54.5^\circ$$ **(9) Bottom-left triangle:** base 4 m (adjacent), hypotenuse 8 m, angle at top. $$\theta = \cos^{-1}\left(\frac{4}{8}\right) = \cos^{-1}(0.5) = 60^\circ$$ **(10) Bottom-right triangle:** vertical leg 6 m (opposite), hypotenuse 7 m, angle at top-right. $$\theta = \sin^{-1}\left(\frac{6}{7}\right) = \sin^{-1}(0.8571) \approx 59.0^\circ$$ 4. **Summary of angles rounded to nearest tenth:** 1) 60.3 2) 58.1 3) 45.0 4) 45.0 5) 36.9 6) 56.3 7) 28.1 8) 54.5 9) 60.0 10) 59.0 These are the measures of the indicated angles in degrees.