Question

in the figure below, m∠3 = 74°. find m∠1, m∠2, and m∠4.

Explanation:

Step1: Use vertical - angle property

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

Step2: Use linear - pair property

$\angle1$ and $\angle2$ form a linear pair. The sum of angles in a linear pair is $180^{\circ}$. So $m\angle1 + m\angle2=180^{\circ}$.
Substitute $m\angle1 = 74^{\circ}$ into the equation: $74^{\circ}+m\angle2 = 180^{\circ}$. Then $m\angle2=180^{\circ}-74^{\circ}=106^{\circ}$.

Step3: Use vertical - angle property again

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

Answer:

$m\angle1 = 74^{\circ}$, $m\angle2 = 106^{\circ}$, $m\angle4 = 106^{\circ}$