1. **Problem Statement:** Find the values of $x$, $y$, and $z$ in a right triangle with angles $45^\circ$ and $30^\circ$, and hypotenuse $14\sqrt{2}$. 2. **Relevant Formulas and Rules:** - In a right triangle, the sum of angles is $180^\circ$. - The third angle is $180^\circ - 45^\circ - 30^\circ = 105^\circ$, but since this is a right triangle, the right angle is $90^\circ$, so the triangle must be a right triangle with angles $45^\circ$, $45^\circ$, and $90^\circ$ or $30^\circ$, $60^\circ$, and $90^\circ$. Here, the problem states $45^\circ$ and $30^\circ$, so the right angle is $105^\circ$ which is impossible. Assuming the right angle is $90^\circ$, the angles are $45^\circ$, $45^\circ$, and $90^\circ$ or $30^\circ$, $60^\circ$, and $90^\circ$. Since both $45^\circ$ and $30^\circ$ are given, the triangle is likely a right triangle with angles $30^\circ$, $60^\circ$, and $90^\circ$. - For a $30^\circ$-$60^\circ$-$90^\circ$ triangle, the sides are in ratio $x : y : z = 1 : \sqrt{3} : 2$, where $z$ is the hypotenuse. - For a $45^\circ$-$45^\circ$-$90^\circ$ triangle, the sides are in ratio $x : y : z = 1 : 1 : \sqrt{2}$. 3. **Given:** Hypotenuse $z = 14\sqrt{2}$, angles $45^\circ$ and $30^\circ$. 4. **Assumption:** Since the hypotenuse is $14\sqrt{2}$, this matches the $45^\circ$-$45^\circ$-$90^\circ$ triangle ratio where hypotenuse $= x\sqrt{2}$. 5. **Calculate sides for $45^\circ$-$45^\circ$-$90^\circ$ triangle:** $$ z = x\sqrt{2} \implies x = \frac{z}{\sqrt{2}} = \frac{14\sqrt{2}}{\sqrt{2}} = 14 $$ 6. Since the triangle is isosceles right triangle, both legs are equal: $$ x = y = 14 $$ 7. **Check if $30^\circ$ is present:** The problem states $45^\circ$ and $30^\circ$, which is inconsistent for a right triangle. We proceed with the $45^\circ$-$45^\circ$-$90^\circ$ assumption. 8. **Final answers:** $$ x = 14 $$ $$ y = 14 $$ $$ z = 14\sqrt{2} $$