To find the measure of angle B, we start with the information given:
- \( m∠A = 67° \)
- \( m∠B = (2x + 20)° \)
- \( m∠C = x° \)
Since the angles add up to 180°, we can set up the following equation:
\[ m∠A + m∠B + m∠C = 180° \]
Substituting the values, the equation becomes:
\[ 67° + (2x + 20)° + x° = 180° \]
Now, combine the terms:
\[ 67° + 20° + 2x + x = 180° \]
This simplifies to:
\[ 87° + 3x = 180° \]
Next, subtract 87° from both sides:
\[ 3x = 180° - 87° \] \[ 3x = 93° \]
Now, divide by 3 to find \( x \):
\[ x = \frac{93°}{3} = 31° \]
Now we can find \( m∠B \):
\[ m∠B = 2x + 20 = 2(31°) + 20° \] \[ m∠B = 62° + 20° = 82° \]
Thus, the measure of \( ∠B \) is:
\[ \boxed{82°} \]