1. **Problem statement:** Find the angle $x$ in each right-angled triangle given the sides. 2. **Formula and rules:** Use trigonometric ratios: sine, cosine, or tangent depending on which sides are given relative to angle $x$. - $\sin x = \frac{\text{opposite}}{\text{hypotenuse}}$ - $\cos x = \frac{\text{adjacent}}{\text{hypotenuse}}$ - $\tan x = \frac{\text{opposite}}{\text{adjacent}}$ 3. **Calculations:** **Top-left:** Opposite = 5, Adjacent = 7, Hypotenuse = $\sqrt{5^2 + 7^2} = \sqrt{25 + 49} = \sqrt{74} \approx 8.6$ Use tangent: $x = \tan^{-1}\left(\frac{5}{7}\right)$ $x = \tan^{-1}(0.714) \approx 35^\circ$ **Top-center:** Adjacent = 4, Opposite = 9, Hypotenuse = $\sqrt{4^2 + 9^2} = \sqrt{16 + 81} = \sqrt{97} \approx 9.85$ Angle $x$ is adjacent to 4, so use cosine: $x = \cos^{-1}\left(\frac{4}{9.85}\right) = \cos^{-1}(0.406) \approx 66^\circ$ **Top-right:** Adjacent = 10, Opposite = 19, Hypotenuse = $\sqrt{10^2 + 19^2} = \sqrt{100 + 361} = \sqrt{461} \approx 21.47$ Angle $x$ adjacent to 10, use cosine: $x = \cos^{-1}\left(\frac{10}{21.47}\right) = \cos^{-1}(0.466) \approx 62^\circ$ **Bottom-left:** Opposite = 6.1, Adjacent = 7.5, Hypotenuse = $\sqrt{6.1^2 + 7.5^2} = \sqrt{37.21 + 56.25} = \sqrt{93.46} \approx 9.67$ Use tangent: $x = \tan^{-1}\left(\frac{6.1}{7.5}\right) = \tan^{-1}(0.813) \approx 39^\circ$ **Bottom-center:** Opposite = 5.6, Adjacent = 4.1, Hypotenuse = $\sqrt{5.6^2 + 4.1^2} = \sqrt{31.36 + 16.81} = \sqrt{48.17} \approx 6.94$ Use tangent: $x = \tan^{-1}\left(\frac{5.6}{4.1}\right) = \tan^{-1}(1.366) \approx 54^\circ$ **Bottom-right:** Adjacent = 4.5, Opposite = 4.2, Hypotenuse = $\sqrt{4.5^2 + 4.2^2} = \sqrt{20.25 + 17.64} = \sqrt{37.89} \approx 6.16$ Angle $x$ adjacent to 4.5, use cosine: $x = \cos^{-1}\left(\frac{4.5}{6.16}\right) = \cos^{-1}(0.730) \approx 43^\circ$ **Final answers:** - Top-left: $x = 35^\circ$ - Top-center: $x = 66^\circ$ - Top-right: $x = 62^\circ$ - Bottom-left: $x = 39^\circ$ - Bottom-center: $x = 54^\circ$ - Bottom-right: $x = 43^\circ$