1. **Problem statement:** Find the size of angle $XZY$ in a right triangle $XYZ$ where $\angle X = 90^\circ$, side $XY = 6$ cm, and side $ZY = 15$ cm. 2. **Identify the sides relative to angle $XZY$:** - $ZY$ is the hypotenuse (opposite the right angle). - $XY$ is one leg adjacent to angle $XZY$. - $XZ$ is the other leg opposite angle $XZY$ (unknown length). 3. **Use the Pythagorean theorem to find $XZ$:** $$XZ = \sqrt{ZY^2 - XY^2} = \sqrt{15^2 - 6^2} = \sqrt{225 - 36} = \sqrt{189}$$ 4. **Calculate $XZ$:** $$XZ = \sqrt{189} = 13.7477 \text{ cm (approx)}$$ 5. **Use trigonometric ratios to find angle $XZY$:** Since $XY$ is adjacent and $XZ$ is opposite to angle $XZY$, use tangent: $$\tan(\angle XZY) = \frac{XZ}{XY} = \frac{13.7477}{6}$$ 6. **Calculate the angle:** $$\angle XZY = \tan^{-1}\left(\frac{13.7477}{6}\right) = \tan^{-1}(2.2913)$$ 7. **Evaluate the inverse tangent:** $$\angle XZY \approx 66.0^\circ$$ **Final answer:** The size of angle $XZY$ is approximately $66.0^\circ$ to 1 decimal place.