In a polygon, the sum of its interior angles is given by the n formula \((n - 2) \times 180^\circ\), where \(n\) is the number of sides of the polygon. Since we have a polygon with four angles, \(n = 4\).

The angles of the polygon are given by \(2x, 5x, x, \text{ and } 4x\) respectively. To find the value of \(x\), we sum up the angles and set them equal to the total sum of interior angles for a polygon with four sides:

\(2x + 5x + x + 4x = (4 - 2) \times 180^\circ\)

Combine like terms:

\(12x = 2 \times 180^\circ\)

Divide by 12:

\(x = \frac{2 \times 180^\circ}{12} = 30^\circ\)

Therefore, the value of \(x\) is \(30^\circ\).