Find the value of x, if ∠B = ∠E.

Given, ∠B = ∠E ……………….(1)

$\Rightarrow \dfrac{\text{BC}}{\text{ED}} = \dfrac{6}{12} = \dfrac{1}{2}\\[1em] \Rightarrow \dfrac{\text{AB}}{\text{EF}} = \dfrac{7.5}{15} = \dfrac{1}{2}\\[1em] \therefore \dfrac{\text{BC}}{\text{ED}} = \dfrac{\text{AB}}{\text{EF}} \text{ ……….(2)}$

From (1) and (2), we get :

⇒ ∠B = ∠E

⇒ $\dfrac{\text{BC}}{\text{ED}} = \dfrac{\text{AB}}{\text{EF}}$

∴ Δ BAC ∼ Δ EFD (By SAS postulate)

We know that,

The corresponding sides of similar triangles are proportional.

$\Rightarrow \dfrac{\text{AC}}{\text{FD}} = \dfrac{\text{BC}}{\text{ED}}\\[1em] \Rightarrow \dfrac{9}{x + 3} = \dfrac{6}{12}\\[1em] \Rightarrow \dfrac{9}{x + 3} = \dfrac{1}{2}\\[1em] \Rightarrow 9 \times 2 = 1 \times (x + 3)\\[1em] \Rightarrow 18 = x + 3\\[1em] \Rightarrow x = 18 - 3\\[1em] \Rightarrow x = 15$

Hence, the value of x = 15.