Question

in the figure below, m∠4 = 126°. find m∠1, m∠2, and m∠3.

Explanation:

Step1: Identify vertical - angle relationship

Vertical angles are equal. $\angle1$ and $\angle4$ are vertical angles.
So $m\angle1 = m\angle4$.
Since $m\angle4 = 126^{\circ}$, then $m\angle1=126^{\circ}$.

Step2: Identify supplementary - angle relationship

$\angle1$ and $\angle2$ are supplementary angles (a linear - pair), so $m\angle1 + m\angle2=180^{\circ}$.
$m\angle2=180^{\circ}-m\angle1$.
Substitute $m\angle1 = 126^{\circ}$ into the equation: $m\angle2 = 180 - 126=54^{\circ}$.

Step3: Identify vertical - angle relationship again

$\angle2$ and $\angle3$ are vertical angles.
So $m\angle3 = m\angle2$.
Since $m\angle2 = 54^{\circ}$, then $m\angle3 = 54^{\circ}$.

Answer:

$m\angle1 = 126^{\circ}$
$m\angle2 = 54^{\circ}$
$m\angle3 = 54^{\circ}$