We are given a circle with center at some point. We have a triangle $\triangle RST$ inscribed in the circle. The segments $RS$ and $TS$ are congruent because they are both radii of the circle. The angle at $T$ is $58^{\circ}$, and we are asked to find the angle at $R$, which is $x^{\circ}$.

1. Problem Description

We are given a circle with center at some point. We have a triangle $\triangle RST$ inscribed in the circle. The segments $RS$ and $TS$ are congruent because they are both radii of the circle. The angle at $T$ is $58^{\circ}$, and we are asked to find the angle at $R$, which is $x^{\circ}$.

2. Solution Steps

Since $RS = TS$, the triangle $\triangle RST$ is an isosceles triangle. Therefore, the angles opposite to the equal sides are equal, which means $\angle SRT = \angle STR$.

We are given that $\angle STR = 58^{\circ}$. Thus, $\angle SRT = x^{\circ} = 58^{\circ}$.

The sum of the angles in a triangle is $180^{\circ}$. Therefore, $\angle SRT + \angle STR + \angle RST = 180^{\circ}$.

So, $58^{\circ} + 58^{\circ} + \angle RST = 180^{\circ}$.

$116^{\circ} + \angle RST = 180^{\circ}$.

$\angle RST = 180^{\circ} - 116^{\circ} = 64^{\circ}$.

However, we are looking for the value of $x$, which is the measure of $\angle SRT$. Since the triangle is isosceles with $RS = TS$, the base angles are congruent: $\angle SRT = \angle STR$. We are given $\angle STR = 58^{\circ}$, so $x = 58$.

3. Final Answer

$x = 58$