We need to find the value of $x$ in the circle. The arc $FG$ has a measure of $116^\circ$. The angle $x$ is a central angle subtending arc $DE$. We need to find the measure of arc $DE$.

1. Problem Description

We need to find the value of $x$ in the circle. The arc $FG$ has a measure of $116^\circ$. The angle $x$ is a central angle subtending arc $DE$. We need to find the measure of arc $DE$.

2. Solution Steps

The measure of an inscribed angle is half the measure of its intercepted arc.

The inscribed angle that intercepts arc $FG$ measures half the arc $FG$ measure.

Let's find the measure of the inscribed angle that intercepts arc $FG$.

Inscribed angle $= \frac{1}{2} \times$ arc $FG$.

Inscribed angle $= \frac{1}{2} \times 116^\circ = 58^\circ$.

Since $DE$ and $FG$ are congruent chords (marked by pink lines), arcs $DE$ and $FG$ are congruent.

Therefore, arc $DE$ has the same measure as arc $FG$.

Measure of arc $DE$ = $116^\circ$.

The measure of a central angle is equal to the measure of its intercepted arc.

Therefore, $x =$ measure of arc $DE$.

$x = 116^\circ$.

3. Final Answer

116