1. **Stating the problem:** We have two geometry problems involving triangles and angles. 2. **Problem 1:** Triangle XYZ has a right angle at Z, angle $\angle X = 62^\circ$, and sides marked 1 and 2. We want to find the missing angles and side lengths. 3. **Formula and rules:** In a triangle, the sum of interior angles is $180^\circ$. For right triangles, the Pythagorean theorem applies: $$c^2 = a^2 + b^2$$ where $c$ is the hypotenuse. 4. **Step 1:** Find the missing angle $\angle Y$ in triangle XYZ. $$\angle Y = 180^\circ - 90^\circ - 62^\circ = 28^\circ$$ 5. **Step 2:** Use trigonometric ratios to find the unknown side. Assume side opposite $\angle X$ is 1, adjacent is 2, or vice versa (clarify which side is which). Since the problem states sides 1 and 2, let's assume side $XZ = 1$ and side $YZ = 2$. 6. **Step 3:** Check consistency with trigonometric ratios. Using $\sin 62^\circ = \frac{\text{opposite}}{\text{hypotenuse}}$, if $XZ=1$ is adjacent to $\angle X$, then hypotenuse $XY$ is: $$XY = \frac{XZ}{\cos 62^\circ} = \frac{1}{\cos 62^\circ}$$ Calculate: $$\cos 62^\circ \approx 0.4695$$ $$XY \approx \frac{1}{0.4695} \approx 2.13$$ 7. **Step 4:** Verify side $YZ$: $$YZ = XY \sin 62^\circ = 2.13 \times 0.8829 \approx 1.88$$ This is close to 2, so the assumption is consistent. --- 8. **Problem 2:** Lines $\overrightarrow{UY}$, $\overrightarrow{VX}$, and $\overrightarrow{XZ}$ are straight lines with angles 38°, 28°, and 76° given in the figure involving triangles UXW and WVY. 9. **Step 1:** Since $\overrightarrow{UY}$, $\overrightarrow{VX}$, and $\overrightarrow{XZ}$ are straight lines, angles on a straight line sum to $180^\circ$. 10. **Step 2:** Check angle sums for adjacent angles: - $38^\circ + 28^\circ + 76^\circ = 142^\circ$ which is less than $180^\circ$, so these angles are parts of different lines or triangles. 11. **Step 3:** Use triangle angle sum rule for triangles UXW and WVY to find missing angles if needed. Since no specific question is asked for problem 2, we conclude here. **Final answers:** - Triangle XYZ: $\angle Y = 28^\circ$, hypotenuse $XY \approx 2.13$, side $YZ \approx 1.88$. - Lines $\overrightarrow{UY}$, $\overrightarrow{VX}$, and $\overrightarrow{XZ}$ are straight lines with given angles consistent with the figure.