1. **State the problem:** We have a right triangle with a hypotenuse of length 8 cm, a vertical side of length 5 cm, and we need to find the angle $z$ at the bottom-left corner between the base and the hypotenuse. 2. **Identify the sides relative to angle $z$:** The side opposite to angle $z$ is the vertical side (5 cm), and the hypotenuse is 8 cm. 3. **Formula used:** To find angle $z$, we use the sine function: $$\sin(z) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{5}{8}$$ 4. **Calculate $z$:** $$z = \sin^{-1}\left(\frac{5}{8}\right)$$ 5. **Evaluate the inverse sine:** $$z = \sin^{-1}(0.625)$$ 6. Using a calculator, $$z \approx 38.7^\circ$$ 7. **Final answer:** The angle $z$ is approximately $38.7^\circ$ to 1 decimal place.