Question

in the figure below, m∠4 = 51°. find m∠1, m∠2, and m∠3.

Explanation:

Step1: Identify vertical - angle relationship

Vertical angles are equal. $\angle1$ and $\angle3$ are vertical angles, and $\angle2$ and $\angle4$ are vertical angles.

Step2: Find $m\angle2$

Since $\angle2$ and $\angle4$ are vertical angles, $m\angle2=m\angle4 = 51^{\circ}$.

Step3: Find $m\angle1$ and $m\angle3$

$\angle1$ and $\angle4$ are supplementary (linear - pair), so $m\angle1 = 180^{\circ}-m\angle4$. Also, since $\angle1$ and $\angle3$ are vertical angles, $m\angle1=m\angle3$.
$m\angle1 = 180 - 51=129^{\circ}$, and $m\angle3=m\angle1 = 129^{\circ}$.

Answer:

$m\angle1 = 129^{\circ}$
$m\angle2 = 51^{\circ}$
$m\angle3 = 129^{\circ}$