Question

two parallel lines are cut by a transversal as shown below. suppose m∠7 = 48°. find m∠2 and m∠4.

Explanation:

Step1: Identify vertical - angle relationship

$\angle7$ and $\angle5$ are vertical angles. Since vertical angles are congruent, $m\angle5 = m\angle7=48^{\circ}$.

Step2: Identify corresponding - angle relationship

$\angle5$ and $\angle1$ are corresponding angles. For parallel lines cut by a transversal, corresponding angles are congruent. So $m\angle1 = m\angle5 = 48^{\circ}$.

Step3: Identify linear - pair relationship for $\angle2$

$\angle1$ and $\angle2$ form a linear - pair. A linear - pair of angles is supplementary, so $m\angle1 + m\angle2=180^{\circ}$. Then $m\angle2 = 180^{\circ}-m\angle1=180 - 48=132^{\circ}$.

Step4: Identify vertical - angle relationship for $\angle4$

$\angle4$ and $\angle1$ are vertical angles. Since vertical angles are congruent, $m\angle4 = m\angle1 = 48^{\circ}$.

Answer:

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