Question

the measure of $angle 5$ is $110^\circ$. find the remaining angle measures.
$m\angle 6 = \boldsymbol{?}^\circ$
$m\angle 7 = \boldsymbol{?}^\circ$
$m\angle 8 = 110^\circ$

Explanation:

Step1: Find $m\angle6$ (supplementary to $\angle5$)

$\angle5$ and $\angle6$ form a linear pair, so their sum is $180^\circ$.
$m\angle6 = 180^\circ - m\angle5 = 180^\circ - 110^\circ = 70^\circ$

Step2: Find $m\angle7$ (vertical to $\angle6$)

$\angle7$ and $\angle6$ are vertical angles, so they are congruent.
$m\angle7 = m\angle6 = 70^\circ$

Answer:

$m\angle6 = 70^\circ$
$m\angle7 = 70^\circ$
$m\angle8 = 110^\circ$