To determine the value of \( x \) given that lines \( m \) and \( n \) are parallel and that \( \angle 8 \) and \( \angle 1 \) are alternate exterior angles, we can use the property that alternate exterior angles are equal when lines are parallel. This means that we can set the measure of angle \( 1 \) equal to the measure of angle \( 8 \).
Given that \( m \angle 1 = 9x - 12 \) and \( m \angle 8 = 105^\circ \), we can set up the equation:
\[ 9x - 12 = 105 \]
To solve for \( x \), we first isolate the term with \( x \) by adding 12 to both sides of the equation:
\[ 9x = 105 + 12 \] \[ 9x = 117 \]
Next, we divide both sides by 9 to find the value of \( x \):
\[ x = \frac{117}{9} \] \[ x = 13 \]
Thus, the solution leads us to conclude that the value of \( x \) is 13. This is supported by the property of alternate exterior angles in parallel lines, confirming that the relationships and calculations adhere to Euclidean geometry rules.
Therefore, the correct answer is 13, as it aligns with the calculations derived from the properties of the angles formed by the parallel lines and the transversal.
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