Question

in the figure below, $mangle2 = 42^{circ}$. find $mangle1$, $mangle3$, and $mangle4$.

Explanation:

Step1: Identify vertical - angle relationship

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

Step2: Find $m\angle1$

Since $\angle1$ and $\angle2$ are vertical angles, $m\angle1=m\angle2 = 42^{\circ}$.

Step3: Find $m\angle3$

$\angle2$ and $\angle3$ are supplementary (form a straight - line pair), so $m\angle2 + m\angle3=180^{\circ}$. Then $m\angle3 = 180^{\circ}-m\angle2=180 - 42=138^{\circ}$.

Step4: Find $m\angle4$

Since $\angle3$ and $\angle4$ are vertical angles, $m\angle4=m\angle3 = 138^{\circ}$.

Answer:

$m\angle1 = 42^{\circ}$
$m\angle3 = 138^{\circ}$
$m\angle4 = 138^{\circ}$