1. **State the problem:** Given angles with expressions in terms of $x$, find $m\angle RT$ where $m\angle QT = (27x + 3)^\circ$, $m\angle RT = (9x - 5)^\circ$, and $m\angle RST = (10x - 2)^\circ$. 2. **Use the given relationship:** $$10x - 2 = \frac{1}{2} \big[(27x + 3) - (9x - 5)\big]$$ This formula states that $m\angle RST$ is half the difference of $m\angle QT$ and $m\angle RT$. 3. **Simplify inside the brackets:** $$ (27x + 3) - (9x - 5) = 27x + 3 - 9x + 5 = 18x + 8 $$ 4. **Rewrite the equation:** $$10x - 2 = \frac{1}{2} (18x + 8)$$ 5. **Multiply both sides by 2 to eliminate the fraction:** $$2(10x - 2) = 18x + 8$$ $$20x - 4 = 18x + 8$$ 6. **Subtract $18x$ from both sides:** $$20x - \cancel{18x} - 4 = \cancel{18x} + 8$$ $$2x - 4 = 8$$ 7. **Add 4 to both sides:** $$2x - 4 + 4 = 8 + 4$$ $$2x = 12$$ 8. **Divide both sides by 2:** $$\frac{2x}{\cancel{2}} = \frac{12}{\cancel{2}}$$ $$x = 6$$ 9. **Find $m\angle RT$ by substituting $x=6$ into $9x - 5$:** $$m\angle RT = 9(6) - 5 = 54 - 5 = 49^\circ$$ **Final answer:** $$m\angle RT = 49^\circ$$