To solve the problem, we need to use the property of similar triangles: the corresponding sides of similar triangles are proportional.

From the figure:

• In the smaller triangle:
  • Corresponding sides are $x$, $2$, and $3$.
• In the larger triangle:
  • Corresponding sides are $4$, $3$, and $x + 2$ (the base).

The similarity ratio can be written as:

$\frac{x}{4} = \frac{2}{3}$

Solve for $x$:

$x = \frac{2}{3} \cdot 4 = \frac{8}{3}$

Thus, the value of $x$ is $\frac{8}{3}$.

Let me know if you need more details or have questions! Here are some questions you can explore further:

1. What are the properties of similar triangles?
2. How do we identify corresponding sides in similar triangles?
3. Can the ratio of perimeters of two similar triangles be derived from side ratios?
4. How do proportions help solve real-world problems involving similar figures?
5. What happens when one of the triangle sides has a variable expression?

Tip: Always ensure to align corresponding sides correctly when working with similar triangles to avoid errors!