Question

in the figure below, m∠2 = 129°. find m∠1, m∠3, and m∠4.

Explanation:

Step1: Use vertical - angle property

Vertical angles are equal. $\angle1$ and $\angle2$ are vertical angles, and $\angle3$ and $\angle4$ are vertical angles. Given $m\angle2 = 129^{\circ}$, then $m\angle1=m\angle2 = 129^{\circ}$ since vertical angles are congruent.

Step2: Use linear - pair property

$\angle2$ and $\angle3$ form a linear pair. The sum of angles in a linear pair is $180^{\circ}$. So $m\angle3=180 - m\angle2$. Substitute $m\angle2 = 129^{\circ}$ into the formula: $m\angle3=180 - 129=51^{\circ}$.

Step3: Use vertical - angle property again

Since $\angle3$ and $\angle4$ are vertical angles, $m\angle4 = m\angle3=51^{\circ}$.

Answer:

$m\angle1 = 129^{\circ}$, $m\angle3 = 51^{\circ}$, $m\angle4 = 51^{\circ}$