Question

two parallel lines are cut by a transversal as shown below. suppose m∠7 = 142°. find m∠2 and m∠4.

Explanation:

Step1: Identify vertical - angle relationship

$\angle7$ and $\angle5$ are vertical angles. Since vertical angles are congruent, $m\angle5 = m\angle7=142^{\circ}$.

Step2: Identify corresponding - angle relationship

$\angle5$ and $\angle1$ are corresponding angles. For parallel lines cut by a transversal, corresponding angles are congruent. So $m\angle1 = m\angle5 = 142^{\circ}$.

Step3: Identify vertical - angle relationship for $\angle2$

$\angle1$ and $\angle2$ are a linear - pair. Since a linear - pair of angles are supplementary ($m\angle1 + m\angle2=180^{\circ}$), then $m\angle2=180 - m\angle1$. Substituting $m\angle1 = 142^{\circ}$, we get $m\angle2 = 180 - 142=38^{\circ}$.

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

$\angle4$ and $\angle8$ are corresponding angles. $\angle7$ and $\angle8$ are a linear - pair, so $m\angle8 = 180 - m\angle7=180 - 142 = 38^{\circ}$. Then $m\angle4=m\angle8 = 38^{\circ}$.

Answer:

$m\angle2 = 38^{\circ}$
$m\angle4 = 38^{\circ}$