1. **State the problem:** We are given two right triangles with legs labeled as follows: the first triangle has legs $2x - 4$ and $39$, and the second smaller triangle has legs $x + 6$ and $24$. We want to find the value of $x$ by using the similarity of triangles. 2. **Formula and rules:** Since the triangles are right triangles and appear similar, the ratios of corresponding sides are equal. This means: $$\frac{2x - 4}{x + 6} = \frac{39}{24}$$ 3. **Set up the equation:** $$\frac{2x - 4}{x + 6} = \frac{39}{24}$$ 4. **Cross multiply:** $$24(2x - 4) = 39(x + 6)$$ 5. **Expand both sides:** $$48x - 96 = 39x + 234$$ 6. **Bring all terms involving $x$ to one side and constants to the other:** $$48x - 39x = 234 + 96$$ 7. **Simplify:** $$9x = 330$$ 8. **Solve for $x$:** $$x = \frac{330}{9}$$ 9. **Simplify the fraction by canceling common factors:** $$x = \frac{\cancel{330}^{\ 30} \times 11}{\cancel{9}^{\ 3} \times 3} = \frac{110}{3}$$ 10. **Final answer:** $$x = \frac{110}{3} \approx 36.67$$ This value of $x$ satisfies the similarity ratio of the two triangles.