Question

in the figure below, m∠2 = 106°. find m∠1, m∠3, and m∠4.

Explanation:

Step1: Use vertical - angle property

Vertical angles are equal. $\angle1$ and $\angle2$ are supplementary (a linear - pair), and $\angle1$ and $\angle3$ are vertical angles, $\angle2$ and $\angle4$ are vertical angles.
Since $\angle1$ and $\angle2$ form a linear pair, $m\angle1 + m\angle2=180^{\circ}$.
Given $m\angle2 = 106^{\circ}$, then $m\angle1=180 - 106=74^{\circ}$.

Step2: Find $m\angle3$

Because $\angle1$ and $\angle3$ are vertical angles, vertical angles are equal. So $m\angle3=m\angle1 = 74^{\circ}$.

Step3: Find $m\angle4$

Since $\angle2$ and $\angle4$ are vertical angles, vertical angles are equal. So $m\angle4=m\angle2 = 106^{\circ}$.

Answer:

$m\angle1 = 74^{\circ}$
$m\angle3 = 74^{\circ}$
$m\angle4 = 106^{\circ}$