1. **Stating the problem:** We are given two parallel lines O-S-P and Q-T-R intersected by a transversal M-S-T-N. At point S, the angle is labeled as $(2x + 55)^\circ$, and at point T, the angle is labeled as $(7x + 25)^\circ$. We need to find the measure of angle $\angle QTN$. 2. **Understanding the relationship:** Since O-S-P and Q-T-R are parallel lines cut by a transversal, the angles at S and T 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 the measure of $\angle QTN$:** Substitute $x=6$ into the expression for the angle at T: $$ 7x + 25 = 7(6) + 25 = 42 + 25 = 67^\circ $$ 6. **Conclusion:** The measure of $\angle QTN$ is $67^\circ$.