To solve for the measures of the angles \( m\angle 1 \) and \( m\angle 2 \), we start with the information given:
- \( m\angle 1 = 2x \)
- \( m\angle 2 = 2x + 7 \)
- The sum of the two angles is given as \( 135^\circ \).
We can set up the equation:
\[ m\angle 1 + m\angle 2 = 135^\circ \]
Substituting the expressions for \( m\angle 1 \) and \( m\angle 2 \):
\[ 2x + (2x + 7) = 135 \]
Simplifying the equation:
\[ 2x + 2x + 7 = 135 \] \[ 4x + 7 = 135 \]
Now, we will isolate \( x \) by subtracting \( 7 \) from both sides:
\[ 4x = 135 - 7 \] \[ 4x = 128 \]
Next, we divide both sides by \( 4 \):
\[ x = \frac{128}{4} \] \[ x = 32 \]
Now that we have \( x \), we can find the measures of the angles:
- \( m\angle 1 = 2x = 2(32) = 64^\circ \)
- \( m\angle 2 = 2x + 7 = 2(32) + 7 = 64 + 7 = 71^\circ \)
Thus, the measures of the angles are:
\[ m\angle 1 = 64^\circ, \quad m\angle 2 = 71^\circ \]
Final answers:
- \( m\angle 1 = 64^\circ \)
- \( m\angle 2 = 71^\circ \)