Since lines \( m \) and \( n \) are parallel and \( \angle 8 \) and \( \angle 1 \) are alternate exterior angles, they are equal. This means we can set up the equation:
\[ m\angle 1 = m\angle 8 \]
Given that:
\[ m\angle 1 = 9x - 12 \] \[ m\angle 8 = 105^\circ \]
We can substitute \( m\angle 8 \) into the equation:
\[ 9x - 12 = 105 \]
Now, we solve for \( x \).
- Add 12 to both sides:
\[ 9x = 105 + 12 \] \[ 9x = 117 \]
- Divide by 9:
\[ x = \frac{117}{9} \] \[ x = 13 \]
Thus, the value of \( x \) is \( \boxed{13} \).