Question

in the figure below, m∠2 = 102°. find m∠1, m∠3, and m∠4.

Explanation:

Step1: Identify vertical - angle relationship

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.

Step2: Calculate $m\angle1$

Since $\angle1$ and $\angle2$ are supplementary (a linear - pair of angles), $m\angle1 + m\angle2=180^{\circ}$. Given $m\angle2 = 102^{\circ}$, then $m\angle1=180^{\circ}-102^{\circ}=78^{\circ}$.

Step3: Calculate $m\angle3$

$\angle1$ and $\angle3$ are vertical angles. Vertical angles are congruent, so $m\angle3 = m\angle1 = 78^{\circ}$.

Step4: Calculate $m\angle4$

$\angle2$ and $\angle4$ are vertical angles. Vertical angles are congruent, so $m\angle4 = m\angle2 = 102^{\circ}$.

Answer:

$m\angle1 = 78^{\circ}$
$m\angle3 = 78^{\circ}$
$m\angle4 = 102^{\circ}$