1. **State the problem:** We have an isosceles triangle with two equal angles each measuring $58.4^\circ$ and a perpendicular height of $16.3$ cm. We need to find the length of the base edge. 2. **Identify known values:** The two equal angles are $58.4^\circ$ each, so the vertex angle is $180^\circ - 2 \times 58.4^\circ = 63.2^\circ$. 3. **Use properties of isosceles triangle:** The perpendicular height bisects the base, so the base is split into two equal segments. We will find the length of one half of the base and then double it. 4. **Set up right triangle:** Consider one of the right triangles formed by the height. The angle adjacent to the height is $58.4^\circ$, the height is the opposite side to this angle, and half the base is the adjacent side. 5. **Use tangent function:** $$\tan(58.4^\circ) = \frac{\text{opposite}}{\text{adjacent}} = \frac{16.3}{\frac{b}{2}}$$ 6. **Solve for half the base:** $$\frac{b}{2} = \frac{16.3}{\tan(58.4^\circ)}$$ 7. **Calculate value:** First calculate $\tan(58.4^\circ) \approx 1.6003$. $$\frac{b}{2} = \frac{16.3}{1.6003} \approx 10.19$$ 8. **Find full base length:** $$b = 2 \times 10.19 = 20.38$$ **Final answer:** The length of the base edge is approximately $20.38$ cm.