Question

in the figure below, m∠4 = 38°. find m∠1, m∠2, and m∠3.

Explanation:

Step1: Identify vertical - angle relationship

Vertical angles are equal. $\angle1$ and $\angle4$ are vertical angles, and $\angle2$ and $\angle3$ are vertical angles.
Since $\angle1$ and $\angle4$ are vertical angles, $m\angle1=m\angle4$.
$m\angle1 = 38^{\circ}$

Step2: Use linear - pair property

$\angle1$ and $\angle2$ form a linear - pair. The sum of angles in a linear - pair is $180^{\circ}$. So $m\angle1 + m\angle2=180^{\circ}$.
$m\angle2=180^{\circ}-m\angle1$.
Substitute $m\angle1 = 38^{\circ}$ into the equation: $m\angle2=180 - 38=142^{\circ}$

Step3: Identify vertical - angle relationship for $\angle3$

Since $\angle2$ and $\angle3$ are vertical angles, $m\angle3=m\angle2$.
$m\angle3 = 142^{\circ}$

Answer:

$m\angle1 = 38^{\circ}$
$m\angle2 = 142^{\circ}$
$m\angle3 = 142^{\circ}$