1. **Problem statement:** We have two parallel lines $L_1$ and $L_2$ intersected by two transversals, creating 12 angles numbered 1 through 12. Given $m\angle7=55^\circ$, and adjacent angles of $61^\circ$ and $64^\circ$ near angles 5 and 6/9, we need to find the measures of all angles $\angle1$ through $\angle12$. 2. **Key properties and formulas:** - Corresponding angles between parallel lines are equal. - Alternate interior angles between parallel lines are equal. - Consecutive interior angles are supplementary (sum to $180^\circ$). - Angles on a straight line sum to $180^\circ$. 3. **Given:** - $m\angle7=55^\circ$ - Adjacent to $\angle5$ is $61^\circ$ (so $\angle5 + 61^\circ = 180^\circ$) - Adjacent to $\angle6$ and $\angle9$ is $64^\circ$ 4. **Find $\angle5$:** $$\angle5 + 61^\circ = 180^\circ$$ $$\angle5 = 180^\circ - 61^\circ = 119^\circ$$ 5. **Find $\angle6$ and $\angle9$:** Since $64^\circ$ is adjacent to $\angle6$ and $\angle9$, and they are on a straight line, $$\angle6 + 64^\circ = 180^\circ \implies \angle6 = 116^\circ$$ Because $\angle6$ and $\angle9$ are vertical angles, $$\angle9 = \angle6 = 116^\circ$$ 6. **Find $\angle7$ and $\angle8$:** Given $\angle7 = 55^\circ$ Since $\angle7$ and $\angle8$ are supplementary, $$\angle7 + \angle8 = 180^\circ \implies \angle8 = 180^\circ - 55^\circ = 125^\circ$$ 7. **Find $\angle1$ through $\angle4$ using corresponding and alternate interior angles:** - $\angle1$ corresponds to $\angle5$, so $$\angle1 = \angle5 = 119^\circ$$ - $\angle2$ is supplementary to $\angle1$, $$\angle1 + \angle2 = 180^\circ \implies \angle2 = 61^\circ$$ - $\angle3$ corresponds to $\angle6$, $$\angle3 = \angle6 = 116^\circ$$ - $\angle4$ is supplementary to $\angle3$, $$\angle3 + \angle4 = 180^\circ \implies \angle4 = 64^\circ$$ 8. **Find $\angle10$, $\angle11$, and $\angle12$ using vertical and supplementary angles:** - $\angle10$ is vertical to $\angle9$, $$\angle10 = \angle9 = 116^\circ$$ - $\angle11$ is supplementary to $\angle10$, $$\angle10 + \angle11 = 180^\circ \implies \angle11 = 64^\circ$$ - $\angle12$ is vertical to $\angle11$, $$\angle12 = \angle11 = 64^\circ$$ **Final answers:** $$\angle1=119^\circ, \angle2=61^\circ, \angle3=116^\circ, \angle4=64^\circ, \angle5=119^\circ, \angle6=116^\circ, \angle7=55^\circ, \angle8=125^\circ, \angle9=116^\circ, \angle10=116^\circ, \angle11=64^\circ, \angle12=64^\circ$$