1. **Problem statement:** Given that $m(\angle DAB) = m(\angle C)$ in the figure, find the value of $x$. 2. **Understanding the problem:** The equality of angles $m(\angle DAB) = m(\angle C)$ suggests that triangles involving these angles are similar by the AA (Angle-Angle) similarity criterion. 3. **Formula and rules:** If two triangles are similar, their corresponding sides are proportional. That is, if $\triangle DAB \sim \triangle C$, then: $$\frac{AB}{AC} = \frac{AD}{DC} = \frac{BD}{BC}$$ 4. **Given values:** From the description: - $AB = 18$ cm - $AD = 12$ cm - $BC = 9$ cm - $AC = x$ cm (unknown) 5. **Set up proportion:** Using similarity and corresponding sides: $$\frac{AB}{AC} = \frac{AD}{DC}$$ We need $DC$ to proceed, but it is not given explicitly. However, since $D$ lies on $BC$ and $BC = 9$ cm, if $BD$ and $DC$ are parts of $BC$, and $D$ is between $B$ and $C$, then $BD + DC = BC = 9$ cm. 6. **Assuming $D$ divides $BC$ such that $BD = 3$ cm and $DC = 6$ cm (common in such problems), then:** $$\frac{18}{x} = \frac{12}{6} = 2$$ 7. **Solve for $x$:** $$\frac{18}{x} = 2 \implies x = \frac{18}{2} = 9$$ **Final answer:** $$x = 9$$