To find the measure of angle \( B \), we first set up the equation based on the information given. The angles \( A \), \( B \), and \( C \) are supplementary, meaning their measures add up to 180 degrees:
\[ m<A + m<B + m<C = 180 \]
Given:
- \( m<A = 67 \)
- \( m<B = 2x + 20 \)
- \( m<C = x \)
Substituting these values into the equation gives us:
\[ 67 + (2x + 20) + x = 180 \]
Now, combine like terms:
\[ 67 + 20 + 3x = 180 \] \[ 87 + 3x = 180 \]
Next, we isolate \( 3x \) by subtracting 87 from both sides:
\[ 3x = 180 - 87 \] \[ 3x = 93 \]
Now, divide by 3 to find \( x \):
\[ x = 31 \]
Now we can substitute \( x \) back to find the measure of angle \( B \):
\[ m<B = 2x + 20 = 2(31) + 20 = 62 + 20 = 82 \]
Thus, the measure of angle \( B \) is:
\[ \boxed{82} \]
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