1. **State the problem:** We have two parallel vertical lines $\ell, m, n$ and two parallel horizontal lines intersecting them, forming segments. The top horizontal line has segments of lengths 16 (between $\ell$ and $m$) and $x$ (between $m$ and $n$). The bottom horizontal line has segments of lengths $x$ (between $\ell$ and $m$) and 4 (between $m$ and $n$). We need to solve for $x$.\n\n2. **Use the property of parallel lines and proportional segments:** Since the vertical lines are parallel and the horizontal lines are parallel, the segments on the top and bottom lines are proportional. This means:\n$$\frac{\text{segment between } \ell \text{ and } m \text{ on top}}{\text{segment between } m \text{ and } n \text{ on top}} = \frac{\text{segment between } \ell \text{ and } m \text{ on bottom}}{\text{segment between } m \text{ and } n \text{ on bottom}}$$\n\n3. **Write the proportion with given lengths:**\n$$\frac{16}{x} = \frac{x}{4}$$\n\n4. **Cross multiply to solve for $x$:**\n$$16 \times 4 = x \times x$$\n$$64 = x^2$$\n\n5. **Take the square root of both sides:**\n$$x = \pm \sqrt{64}$$\n$$x = \pm 8$$\n\n6. **Interpret the solution:** Since $x$ represents a length, it must be positive. Therefore,\n$$x = 8$$\n\n**Final answer:**\n$$x = 8$$