1. **State the problem:** We need to find the measure of angle $\angle QTN$ given two expressions for angles formed by a transversal intersecting two parallel lines. 2. **Identify the angles and their relationship:** The angles given are $(2x + 55)^\circ$ and $(7x + 25)^\circ$. Since lines $OP$ and $QR$ are parallel and $MN$ is a transversal, these two angles are corresponding angles and therefore equal. 3. **Set up the equation:** $$2x + 55 = 7x + 25$$ 4. **Solve for $x$:** $$2x + 55 = 7x + 25$$ $$55 - 25 = 7x - 2x$$ $$30 = 5x$$ $$x = \frac{30}{5} = 6$$ 5. **Find $\angle QTN$ by substituting $x=6$ into one of the angle expressions:** $$\angle QTN = 7x + 25 = 7(6) + 25 = 42 + 25 = 67^\circ$$ 6. **Final answer:** $$\boxed{67^\circ}$$