1. **State the problem:** We need to find the measure of angle $\angle RTN$ given two expressions for angles formed by two parallel lines cut by a transversal: one angle is $(55 - 2x)^\circ$ and another is $(63 - 6x)^\circ$. 2. **Identify the relationship:** Since the lines are parallel and cut by a transversal, corresponding angles or alternate interior angles are equal. The angles at points $S$ and $R$ are related by this property. 3. **Set up the equation:** Because the angles at $S$ and $R$ are alternate interior angles, they are equal: $$55 - 2x = 63 - 6x$$ 4. **Solve for $x$:** $$55 - 2x = 63 - 6x$$ $$55 - 2x + 6x = 63$$ $$55 + 4x = 63$$ $$4x = 63 - 55$$ $$4x = 8$$ $$x = \frac{8}{4}$$ $$x = 2$$ 5. **Find $\angle RTN$:** The angle $\angle RTN$ corresponds to the angle at $R$, which is $(63 - 6x)^\circ$. Substitute $x=2$: $$\angle RTN = 63 - 6(2) = 63 - 12 = 51^\circ$$ 6. **Final answer:** $$\boxed{51^\circ}$$