Question

in the figure below, m∠2 = 42°. find m∠1, m∠3, and m∠4.

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, $m\angle1 = m\angle2$. Given $m\angle2 = 42^{\circ}$, so $m\angle1=42^{\circ}$.

Step3: Find $m\angle3$

$\angle1$ and $\angle3$ are supplementary (a linear - pair, sum to $180^{\circ}$). So $m\angle3=180^{\circ}-m\angle1$. Substituting $m\angle1 = 42^{\circ}$, we get $m\angle3 = 180 - 42=138^{\circ}$.

Step4: Find $m\angle4$

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

Answer:

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