Since angles \( \angle 1 \) and \( \angle 2 \) are vertical angles, they are equal in measure. Therefore, we can set up the equation:
\[ 2x + 16 = 6x - 8 \]
To solve for \( x \), we first isolate the terms involving \( x \) on one side of the equation. We can start by subtracting \( 2x \) from both sides:
\[ 16 = 6x - 2x - 8 \]
This simplifies to:
\[ 16 = 4x - 8 \]
Next, we add \( 8 \) to both sides:
\[ 16 + 8 = 4x \]
So we have:
\[ 24 = 4x \]
Now, we divide both sides by \( 4 \):
\[ x = 6 \]
Now that we have \( x \), we can find the measure of \( \angle 2 \) using the expression for \( \angle 2 \):
\[ \angle 2 = 6x - 8 \]
Substituting \( x = 6 \):
\[ \angle 2 = 6(6) - 8 \]
Calculating this gives:
\[ \angle 2 = 36 - 8 = 28 \]
Thus, the measure of \( \angle 2 \) is:
\[ \boxed{28} \text{ degrees} \]