1. **Problem statement:** Two parallel lines $u$ and $v$ are cut by a transversal $f$. Given $m \angle 1 = 70^\circ$, find the measures of angles 2 and 3 and identify their relationships. 2. **Step 1: Relationship between $\angle 1$ and $\angle 2$** - $\angle 1$ and $\angle 2$ are adjacent angles on a straight line formed by the transversal and line $u$. - Adjacent angles on a straight line are supplementary, so $$m \angle 1 + m \angle 2 = 180^\circ$$ - Substitute $m \angle 1 = 70^\circ$: $$70^\circ + m \angle 2 = 180^\circ$$ - Solve for $m \angle 2$: $$m \angle 2 = 180^\circ - 70^\circ = 110^\circ$$ 3. **Step 2: Relationship between $\angle 2$ and $\angle 3$** - $\angle 2$ and $\angle 3$ are corresponding angles formed by the transversal $f$ cutting parallel lines $u$ and $v$. - Corresponding angles are congruent when lines are parallel, so $$m \angle 2 = m \angle 3$$ - Therefore, $$m \angle 3 = 110^\circ$$ 4. **Step 3: Relationship between $\angle 1$ and $\angle 3$** - $\angle 1$ and $\angle 3$ are alternate interior angles formed by the transversal cutting parallel lines. - Alternate interior angles are congruent when lines are parallel. - So, $\angle 1$ and $\angle 3$ are congruent. 5. **Summary:** - $\angle 1$ and $\angle 2$ are supplementary. - $\angle 2$ and $\angle 3$ are corresponding angles and congruent. - $\angle 1$ and $\angle 3$ are alternate interior angles and congruent. **Final answers:** $$m \angle 2 = 110^\circ$$ $$m \angle 3 = 110^\circ$$