1. **Problem statement:** Given that lines $r$ and $s$ are parallel, and line $u$ is perpendicular to line $t$, with $m \angle 3 = 50^\circ$, find $m \angle 5$. 2. **Key facts and formulas:** - When two lines are parallel, corresponding angles are equal. - Perpendicular lines form right angles ($90^\circ$). - Angles on a straight line sum to $180^\circ$. 3. **Step-by-step solution:** - Since $u$ is perpendicular to $t$, $m \angle 4 = 90^\circ$ because $\angle 4$ is formed by $u$ and $t$. - Angles $3$ and $4$ are adjacent and form a straight line along $t$, so: $$m \angle 3 + m \angle 4 = 180^\circ$$ - Substitute known values: $$50^\circ + m \angle 4 = 180^\circ$$ - Since $m \angle 4 = 90^\circ$, check consistency: $$50^\circ + 90^\circ = 140^\circ \neq 180^\circ$$ - This suggests $\angle 3$ and $\angle 4$ are not supplementary; instead, $\angle 3$ and $\angle 5$ are corresponding angles because $r \parallel s$ and $t$ is a transversal. - Therefore, $m \angle 5 = m \angle 3 = 50^\circ$ by the Corresponding Angles Postulate. 4. **Final answer:** $$m \angle 5 = 50^\circ$$