Question

in the figure below, m∠2 = 46°. find m∠1, m∠3, and m∠4.

Explanation:

Step1: Use vertical - angle property

Vertical angles are equal. $\angle1$ and $\angle2$ are supplementary (form a linear pair), and $\angle1$ and $\angle3$ are vertical angles, $\angle2$ and $\angle4$ are vertical angles.
Since $\angle1+\angle2 = 180^{\circ}$ (linear - pair of angles), and $m\angle2 = 46^{\circ}$.
$m\angle1=180 - m\angle2$.
$m\angle1 = 180 - 46=134^{\circ}$.

Step2: Find $\angle3$

Because $\angle1$ and $\angle3$ are vertical angles, vertical angles are equal. So $m\angle3=m\angle1$.
$m\angle3 = 134^{\circ}$.

Step3: Find $\angle4$

Since $\angle2$ and $\angle4$ are vertical angles, vertical angles are equal. So $m\angle4=m\angle2$.
$m\angle4 = 46^{\circ}$.

Answer:

$m\angle1 = 134^{\circ}$
$m\angle3 = 134^{\circ}$
$m\angle4 = 46^{\circ}$