Parallel lines have equal corresponding angles, so $θ=α$.
By shared angles, $β=β$.
By $\text{AA}\sim$, the triangles are similar.

$\frac{12}{4}=\frac{8+x}{x}$

$12x = 4(8+x)$
$12x = 32 + 4x$
$8x=32$
$x=4$

Use the same method as for part (a).

Is there enough information to prove that the triangles are similar?

Vertical angles are equal, so $α=β$.
Parallel lines have equal alternate interior angles, so $θ=γ$.
Triangles are similar by $\text{AA}\sim$.

$\frac{6}{8}=\frac{8}{\textit{x}}$

$6x=8(8)$
$6x=64$
$x\approx10.67$