Since angles \( \angle 1 \) and \( \angle 2 \) are vertical angles, they are equal. This means we can set up the equation:
\[ \angle 1 = \angle 2 \]
Substituting the expressions for \( \angle 1 \) and \( \angle 2 \):
\[ x + 12 = 6x - 8 \]
Now, we'll solve for \( x \):
- Subtract \( x \) from both sides:
\[ 12 = 6x - x - 8 \]
This simplifies to:
\[ 12 = 5x - 8 \]
- Next, add 8 to both sides:
\[ 12 + 8 = 5x \]
\[ 20 = 5x \]
- Now, divide by 5:
\[ x = \frac{20}{5} = 4 \]
With \( x = 4 \), we can find the measure of \( \angle 2 \) by substituting \( x \) back into the expression for \( \angle 2 \):
\[ \angle 2 = 6x - 8 \]
Substitute \( x = 4 \):
\[ \angle 2 = 6(4) - 8 = 24 - 8 = 16 \]
Thus, the measure of \( \angle 2 \) is
\[ \boxed{16^\circ} \]