1. **State the problem:** We want to find the measure of angle $\angle QTN$ given two expressions related to angles involving $x$: $(2x + 55)^\circ$ and $(7x + 25)^\circ$. 2. **Analyze the given expressions:** Usually in such diagrams, if two angles are given in terms of $x$, they might be equal or supplementary depending on the configuration. Here, it's likely these two expressions represent angles around point $T$ or on a straight line, so they might be supplementary (sum to $180^\circ$). 3. **Set up the equation:** Assuming the angles are supplementary, $$ (2x + 55) + (7x + 25) = 180 $$ 4. **Simplify and solve for $x$:** $$ 2x + 55 + 7x + 25 = 180 $$ $$ 9x + 80 = 180 $$ $$ 9x = 100 $$ $$ x = \frac{100}{9} \approx 11.11 $$ 5. **Find $\angle QTN$:** The problem states $m\angle QTN$ is the measure of the desired angle. If $\angle QTN$ corresponds to one of the given expressions, say $(7x + 25)^\circ$, substitute $x$: $$ 7 \times \frac{100}{9} + 25 = \frac{700}{9} + 25 = \frac{700}{9} + \frac{225}{9} = \frac{925}{9} \approx 102.78^6 $$ 6. **Conclusion:** The measure of $\angle QTN$ is approximately $102.78^\circ$.