1. **State the problem:** We need to find the value of $x$ given two angles in a circle: one angle is $x + 83^\circ$ and the other is $8x + 34^\circ$. 2. **Identify the relationship:** Since points $T$, $R$, and $S$ lie on the circle and chords $TR$ and $RS$ form triangle $TRS$, the angles at $U$ and $S$ are related by the property that the sum of angles in triangle $TRS$ is $180^\circ$. 3. **Write the equation:** The angles inside triangle $TRS$ are $x + 83^\circ$, $8x + 34^\circ$, and the third angle at $R$ (which we can call $y$). The sum of angles in a triangle is: $$ (x + 83) + (8x + 34) + y = 180 $$ 4. **Use the property of the circle:** Since $T$, $R$, and $S$ lie on the circle, the angle at $R$ is the angle subtended by the chord $TS$. The problem does not provide $y$, so we assume the two given angles are supplementary (sum to $180^\circ$) because they are opposite angles formed by intersecting chords or inscribed angles subtending the same arc. 5. **Set up the supplementary angle equation:** $$ (x + 83) + (8x + 34) = 180 $$ 6. **Simplify and solve for $x$:** $$ x + 83 + 8x + 34 = 180 $$ $$ 9x + 117 = 180 $$ $$ 9x = 180 - 117 $$ $$ 9x = 63 $$ $$ x = \frac{63}{9} = 7 $$ 7. **Final answer:** $$ \boxed{7^\circ} $$ Thus, the value of $x$ is $7$ degrees.