Question

1. find the measures of the numbered angles.

Explanation:

Step1: Identify vertical - angle relationship

$\angle1$ and the given $61^{\circ}$ angle are vertical angles. Vertical angles are equal. So $\angle1 = 61^{\circ}$.

Step2: Identify linear - pair relationship

$\angle1$ and $\angle2$ form a linear - pair. The sum of angles in a linear - pair is $180^{\circ}$. So $\angle2=180^{\circ}-\angle1 = 180 - 61=119^{\circ}$.

Step3: Identify vertical - angle relationship for $\angle3$

$\angle3$ and $\angle1$ are vertical angles. So $\angle3=\angle1 = 61^{\circ}$.

Step4: Identify vertical - angle relationship for $\angle4$

$\angle4$ and $\angle2$ are vertical angles. So $\angle4=\angle2 = 119^{\circ}$.

Answer:

$\angle1 = 61^{\circ}$, $\angle2 = 119^{\circ}$, $\angle3 = 61^{\circ}$, $\angle4 = 119^{\circ}$