Vertical angles are equal, so we can set the measures of the two angles equal to each other:
\[ m\angle 2 = m\angle 4 \]
Substituting the given expressions:
\[ 5x + 23 = 3x + 35 \]
Now, let's solve for \(x\):
- Subtract \(3x\) from both sides:
\[ 5x - 3x + 23 = 35 \]
This simplifies to:
\[ 2x + 23 = 35 \]
- Subtract 23 from both sides:
\[ 2x = 35 - 23 \]
This simplifies to:
\[ 2x = 12 \]
- Divide both sides by 2:
\[ x = 6 \]
Now that we have \(x\), we can find the measures of the angles:
- Substitute \(x\) back into the expression for \(m\angle 2\):
\[ m\angle 2 = 5(6) + 23 = 30 + 23 = 53 \]
- Substitute \(x\) back into the expression for \(m\angle 4\):
\[ m\angle 4 = 3(6) + 35 = 18 + 35 = 53 \]
Thus, the measure of each angle is:
\[ \boxed{53} \]