Question

in the figure below, (mangle3 = 48^{circ}). find (mangle1), (mangle2), and (mangle4).

Explanation:

Step1: Identify vertical - angle relationship

Vertical angles are equal. $\angle1$ and $\angle3$ are vertical angles.
$m\angle1 = m\angle3$

Step2: Substitute the value of $\angle3$

Since $m\angle3 = 48^{\circ}$, then $m\angle1=48^{\circ}$

Step3: Identify linear - pair relationship

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

Step4: Solve for $m\angle2$

$m\angle2 = 180^{\circ}-m\angle1$. Substituting $m\angle1 = 48^{\circ}$, we get $m\angle2=180 - 48=132^{\circ}$

Step5: Identify vertical - angle relationship for $\angle2$ and $\angle4$

$\angle2$ and $\angle4$ are vertical angles. So $m\angle4 = m\angle2$

Step6: Determine $m\angle4$

Since $m\angle2 = 132^{\circ}$, then $m\angle4 = 132^{\circ}$

Answer:

$m\angle1 = 48^{\circ}$
$m\angle2 = 132^{\circ}$
$m\angle4 = 132^{\circ}$