1. **Problem statement:** Given two parallel lines $c$ and $d$ cut by transversals $a$ and $b$, with $m\angle 12 = 60^\circ$ and $m\angle 14 = 80^\circ$, find $m\angle 3$ and $m\angle 6$. 2. **Key properties:** - Corresponding angles formed by a transversal with parallel lines are equal. - Vertical angles are equal. - Angles on a straight line sum to $180^\circ$. 3. **Find $m\angle 13$:** Since $\angle 12$ and $\angle 13$ are vertical angles, $$m\angle 13 = m\angle 12 = 60^\circ.$$ 4. **Find $m\angle 11$:** Angles $12$, $13$, $14$, and $11$ form a full circle around the intersection of transversals $a$ and $b$, summing to $360^\circ$. Calculate $m\angle 11$: $$m\angle 11 = 360^\circ - (m\angle 12 + m\angle 13 + m\angle 14) = 360^\circ - (60^\circ + 60^\circ + 80^\circ) = 160^\circ.$$ 5. **Find $m\angle 3$:** $\angle 3$ and $\angle 11$ are corresponding angles formed by transversal $b$ with parallel lines $c$ and $d$, so $$m\angle 3 = m\angle 11 = 160^\circ.$$ 6. **Find $m\angle 4$:** Given $m\angle 14 = 80^\circ$, and $\angle 4$ and $\angle 14$ are vertical angles, $$m\angle 4 = m\angle 14 = 80^\circ.$$ 7. **Find $m\angle 6$:** $\angle 4$ and $\angle 6$ are alternate interior angles formed by transversal $b$ with parallel lines $c$ and $d$, so $$m\angle 6 = m\angle 4 = 80^\circ.$$ **Final answers:** $$m\angle 3 = 160^\circ$$ $$m\angle 6 = 80^\circ$$