Question

in the figure below, (mangle4 = 127^{circ}). find (mangle1), (mangle2), and (mangle3)

Explanation:

Step1: Use vertical - angle property

Vertical angles are equal. $\angle1$ and $\angle4$ are vertical angles. So $m\angle1 = m\angle4$.
$m\angle1=127^{\circ}$

Step2: Use linear - pair property

$\angle1$ and $\angle2$ form a linear pair. The sum of angles in a linear pair is $180^{\circ}$. So $m\angle2=180 - m\angle1$.
$m\angle2 = 180 - 127=53^{\circ}$

Step3: Use vertical - angle property again

$\angle2$ and $\angle3$ are vertical angles. So $m\angle3 = m\angle2$.
$m\angle3 = 53^{\circ}$

Answer:

$m\angle1 = 127^{\circ}$, $m\angle2 = 53^{\circ}$, $m\angle3 = 53^{\circ}$