To find the measure of angle \( B \), we first set up the equation based on the fact that the measures of angles \( A \), \( B \), and \( C \) add up to \( 180^\circ \):
\[ m\angle A + m\angle B + m\angle C = 180^\circ \]
Substituting the given expressions:
\[ 67^\circ + (2x + 20)^\circ + x^\circ = 180^\circ \]
Now, we combine the like terms:
\[ 67 + 2x + 20 + x = 180 \]
Simplifying the left side:
\[ 87 + 3x = 180 \]
Next, we isolate \( x \) by subtracting \( 87 \) from both sides:
\[ 3x = 180 - 87 \]
This simplifies to:
\[ 3x = 93 \]
Now, we divide both sides by \( 3 \):
\[ x = 31 \]
Now that we have the value of \( x \), we can find the measure of angle \( B \):
\[ m\angle B = 2x + 20 \]
Substituting the value of \( x \):
\[ m\angle B = 2(31) + 20 \]
Calculating this gives:
\[ m\angle B = 62 + 20 = 82^\circ \]
Thus, the measure of angle \( B \) is:
\[ \boxed{82^\circ} \]