The angles \( A \), \( B \), and \( C \) are supplementary, meaning they add up to 180 degrees. We know:
- \( m<A = 67 \)
- \( m<B = 2x + 20 \)
- \( m<C = x \)
First, we can set up the equation based on the angle measures:
\[ m<A + m<B + m<C = 180 \]
Substituting the values we have:
\[ 67 + (2x + 20) + x = 180 \]
Now, combine like terms:
\[ 67 + 20 + 2x + x = 180 \]
\[ 87 + 3x = 180 \]
Next, isolate \( 3x \) by subtracting 87 from both sides:
\[ 3x = 180 - 87 \]
\[ 3x = 93 \]
Now, divide by 3:
\[ x = 31 \]
Now that we have \( x \), we can find \( m<B \):
\[ m<B = 2x + 20 = 2(31) + 20 \]
\[ m<B = 62 + 20 = 82 \]
Thus, the measure of angle \( B \) is \(\boxed{82}\).