To solve for the measurement of angle \( J \) in the parallelogram, we will use the properties of parallelograms.
In a parallelogram, opposite angles are equal, and consecutive angles are supplementary (their measures add up to 180 degrees). From the details you've provided, we have the following angles:
- \( m \angle K = 10 + 11x \)
- \( m \angle J = 10x + 2 \)
Since angles \( K \) and \( J \) are consecutive angles in the parallelogram, they are supplementary:
\[ m \angle K + m \angle J = 180^\circ \]
Substituting the expressions for \( m \angle K \) and \( m \angle J \):
\[ (10 + 11x) + (10x + 2) = 180 \]
Combining like terms:
\[ 10 + 11x + 10x + 2 = 180 \] \[ 21x + 12 = 180 \]
Now, subtract 12 from both sides:
\[ 21x = 180 - 12 \] \[ 21x = 168 \]
Next, divide both sides by 21 to solve for \( x \):
\[ x = \frac{168}{21} = 8 \]
Now that we have \( x \), we can substitute back to find \( m \angle J \):
\[ m \angle J = 10x + 2 = 10(8) + 2 = 80 + 2 = 82 \]
Thus, the measurement of angle \( J \) is:
\[ \boxed{82} \]