Question

in the figure, m is parallel to n and m∠4 = 115°. find the measures of the other angles.

Explanation:

Step1: Identify vertical - angle relationship

Vertical angles are equal. $\angle4$ and $\angle6$ are vertical angles. So $m\angle6 = m\angle4=115^{\circ}$.

Step2: Identify corresponding - angle relationship

Corresponding angles are equal when two lines are parallel. $\angle4$ and $\angle2$ are corresponding angles. So $m\angle2 = m\angle4 = 115^{\circ}$.

Step3: Identify vertical - angle relationship for $\angle8$

$\angle2$ and $\angle8$ are vertical angles. Since $m\angle2 = 115^{\circ}$, then $m\angle8=115^{\circ}$.

Step4: Identify linear - pair relationship for $\angle5$

$\angle4$ and $\angle5$ form a linear - pair. A linear - pair of angles is supplementary, so $m\angle5=180^{\circ}-m\angle4 = 180 - 115=65^{\circ}$.

Step5: Identify vertical - angle relationship for $\angle3$

$\angle4$ and $\angle3$ are vertical angles. So $m\angle3 = m\angle4=115^{\circ}$.

Step6: Identify corresponding - angle relationship for $\angle7$

$\angle5$ and $\angle7$ are corresponding angles. Since $m\angle5 = 65^{\circ}$, then $m\angle7 = 65^{\circ}$.

Step7: Identify vertical - angle relationship for $\angle1$

$\angle7$ and $\angle1$ are vertical angles. So $m\angle1 = m\angle7=65^{\circ}$.

Answer:

$m\angle6 = 115^{\circ}$
$m\angle2 = 115^{\circ}$
$m\angle8 = 115^{\circ}$
$m\angle5 = 65^{\circ}$
$m\angle3 = 115^{\circ}$
$m\angle7 = 65^{\circ}$
$m\angle1 = 65^{\circ}$