1. **State the problem:** We need to find the measure of angle $\angle S$ given two expressions for angles related to chords intersecting inside a circle. 2. **Identify the relationship:** When two chords intersect inside a circle, the measure of the angle formed is half the sum of the measures of the arcs intercepted by the angle and its vertical angle. Here, the angles $8x + 45^\circ$ and $19x + 34^\circ$ are vertical angles formed by intersecting chords, so they are equal. 3. **Set up the equation:** $$8x + 45 = 19x + 34$$ 4. **Solve for $x$:** $$8x + 45 = 19x + 34$$ $$45 - 34 = 19x - 8x$$ $$11 = 11x$$ $$x = \cancel{\frac{11}{11}}1$$ 5. **Find $m\angle S$ by substituting $x=1$ into $8x + 45$:** $$m\angle S = 8(1) + 45 = 8 + 45 = 53^\circ$$ **Final answer:** $$m\angle S = 53^\circ$$