The problem asks us to find the value of $x$ given that the angle $\angle FHG$ is $57^{\circ}$, the angle $\angle DHE$ is $90^{\circ}$ (indicated by the square symbol), and the angle $\angle DHG$ is $(3x)^{\circ}$.

1. Problem Description

The problem asks us to find the value of $x$ given that the angle $\angle FHG$ is $57^{\circ}$, the angle $\angle DHE$ is $90^{\circ}$ (indicated by the square symbol), and the angle $\angle DHG$ is $(3x)^{\circ}$.

2. Solution Steps

We know that the sum of angles around a point on one side of a straight line is $180^{\circ}$.

Since $\angle FHG$, $\angle DHE$ (which is a right angle), and $\angle DHF$ make up a straight angle, their sum is $180^{\circ}$.

Thus, $\angle FHG + \angle DHF = \angle DHG + \angle DHE$

We have $\angle FHG = 57^{\circ}$, $\angle DHE = 90^{\circ}$, and $\angle DHG = (3x)^{\circ}$.

Therefore, we can set up the equation:

$57^{\circ} + 90^{\circ} + (3x)^{\circ} = 180^{\circ}$

$57 + 90 + 3x = 180$

$147 + 3x = 180$

$3x = 180 - 147$

$3x = 33$

$x = \frac{33}{3}$

$x = 11$

3. Final Answer

The value of $x$ is 11.