1. **Problem Statement:** Given two parallel lines L1 and L2 intersected by two transversals, find the measures of angles $\angle 1$ through $\angle 12$ given $\angle 4 = 61^\circ$, $\angle 6 = 64^\circ$, and $\angle 7 = 55^\circ$. 2. **Key Properties:** - Corresponding angles between parallel lines are equal. - Alternate interior angles are equal. - Angles on a straight line sum to $180^\circ$. - Vertically opposite angles are equal. 3. **Find $\angle 1$:** - $\angle 1$ and $\angle 4$ are corresponding angles (since L1 and L2 are parallel). - Therefore, $m\angle 1 = m\angle 4 = 61^\circ$. 4. **Find $\angle 2$ and $\angle 3$:** - $\angle 2$ and $\angle 3$ are adjacent to $\angle 4$ on line L1. - Since $\angle 3$ and $\angle 4$ are supplementary (linear pair), $$m\angle 3 = 180^\circ - m\angle 4 = 180^\circ - 61^\circ = 119^\circ$$ - $\angle 2$ is vertically opposite to $\angle 3$, so $$m\angle 2 = m\angle 3 = 119^\circ$$ 5. **Find $\angle 5$:** - $\angle 5$ and $\angle 4$ are adjacent on a straight line, so $$m\angle 5 = 180^\circ - m\angle 4 = 119^\circ$$ 6. **Find $\angle 6$ and $\angle 7$:** - Given $m\angle 6 = 64^\circ$ and $m\angle 7 = 55^\circ$. 7. **Find $\angle 8$:** - $\angle 7$ and $\angle 8$ are supplementary (linear pair), so $$m\angle 8 = 180^\circ - m\angle 7 = 180^\circ - 55^\circ = 125^\circ$$ 8. **Find $\angle 9$ and $\angle 10$:** - $\angle 9$ is vertically opposite to $\angle 8$, so $$m\angle 9 = 125^\circ$$ - $\angle 10$ is supplementary to $\angle 9$, so $$m\angle 10 = 180^\circ - 125^\circ = 55^\circ$$ 9. **Find $\angle 11$ and $\angle 12$:** - $\angle 11$ is vertically opposite to $\angle 6$, so $$m\angle 11 = 64^\circ$$ - $\angle 12$ is supplementary to $\angle 11$, so $$m\angle 12 = 180^\circ - 64^\circ = 116^\circ$$ **Final answers:** - $m\angle 1 = 61^\circ$ - $m\angle 2 = 119^\circ$ - $m\angle 3 = 119^\circ$ - $m\angle 4 = 61^\circ$ - $m\angle 5 = 119^\circ$ - $m\angle 6 = 64^\circ$ - $m\angle 7 = 55^\circ$ - $m\angle 8 = 125^\circ$ - $m\angle 9 = 125^\circ$ - $m\angle 10 = 55^\circ$ - $m\angle 11 = 64^\circ$ - $m\angle 12 = 116^\circ$