We know that when two adjacent angles form a straight line (i.e., they are supplementary), their measures add up to 180°. Therefore, we have:
\[ m\angle 21 + m\angle 22 = 135^\circ \]
Given that \( m\angle 21 = 2x \) and \( m\angle 22 = 2x + 7 \), we can set up the equation:
\[ 2x + (2x + 7) = 135 \]
Now, we solve for \( x \):
\[ 4x + 7 = 135 \]
Subtract 7 from both sides:
\[ 4x = 135 - 7 \]
\[ 4x = 128 \]
Divide both sides by 4:
\[ x = \frac{128}{4} = 32 \]
Now that we have found the value of \( x \), we can find the measures of \( m\angle 21 \) and \( m\angle 22 \):
\[ m\angle 21 = 2x = 2(32) = 64 \]
\[ m\angle 22 = 2x + 7 = 2(32) + 7 = 64 + 7 = 71 \]
So, \( m\angle 21 = 64 \) and \( m\angle 22 = 71 \).
