1. **Problem Statement:** Given two parallel lines $\ell \parallel m$ cut by a transversal, find the measure of angle $\angle QTP$ given the angles $3x^\circ$, $89^\circ$, and $(5x-32)^\circ$ as shown. 2. **Key Concept:** When two parallel lines are cut by a transversal, alternate interior angles are equal, and corresponding angles are equal. 3. **Identify Angles:** The angle $3x^\circ$ and the angle $89^\circ$ are on the upper intersection near $P$ and $Q$ respectively, and $(5x-32)^\circ$ is at the lower intersection near $T$. 4. **Use the fact that the sum of angles around point $Q$ on a straight line is $180^\circ$:** $$3x + 89 = 180$$ 5. **Solve for $x$:** $$3x = 180 - 89$$ $$3x = 91$$ $$x = \frac{91}{3}$$ 6. **Calculate $x$ value:** $$x = 30.333...$$ 7. **Find $\angle QTP = (5x - 32)^\circ$:** $$5x - 32 = 5 \times 30.333... - 32$$ $$= 151.666... - 32$$ $$= 119.666...$$ 8. **Final answer:** $$m\angle QTP = 119.67^\circ$$ (rounded to two decimal places)