In the given scenario, we know that angles 1 and 3 are vertical angles. Vertical angles are always equal in measure, so we can set their measures equal to each other:
\[ m\angle 1 = m\angle 3 \]
From the problem, we have:
\[ m\angle 1 = 4x + 26 \] \[ m\angle 3 = 98 \]
Setting these equal gives us the equation:
\[ 4x + 26 = 98 \]
To solve for \(x\), first subtract 26 from both sides:
\[ 4x = 98 - 26 \] \[ 4x = 72 \]
Now, divide both sides by 4:
\[ x = \frac{72}{4} \] \[ x = 18 \]
The value of \(x\) is \(18\). Thus, the answer is:
\[ \boxed{18} \]