Answer
m$\angle2$ = 57.7$^{\circ}$
m$\angle4$ = 41.5$^{\circ}$
m$\angle3$ = 80.8$^{\circ}$
Work Step by Step
$\angle1$ and $\angle2$ are supplementary angles, which means
m$\angle1$ + m$\angle2$ must equal 180$^{\circ}$
m$\angle1$ = 122.3$^{\circ}$ + m$\angle2$ = 180$^{\circ}$
Subtract m$\angle1$ from 180$^{\circ}$ to get m$\angle2$
180$^{\circ}$ - 122.3$^{\circ}$ = 57.7$^{\circ}$
Therefore, m$\angle2$ = 57.7$^{\circ}$
Since j $\parallel$ k we know that $\angle5$ and $\angle4$ are alternate interior angles.
Based off of the law of alternate interior angles know that since $\angle5$ and $\angle4$ are alternate interior angles, they are congruent.
So m$\angle5$ = m$\angle4$.
m$\angle5$ = 41.5$^{\circ}$
Therefore, m$\angle4$ = 41.5$^{\circ}$
Since we know the measures of two of the three $\angle$s which make up $\triangle ABC$ (m$\angle4$ & m$\angle2$) we can figure out the measure of the unknown angle, m$\angle3$.
We can find this out because of the properties of the interior angles of a triangle.
One of the properties of interior angles of a triangle is that they must add up to 180$^{\circ}$.
We know two of the interior angles measurements of $\triangle ABC$ (m$\angle4$ & m$\angle2$) we know the measure of the missing angle must bring their sum to equal 180$^{\circ}$.
m$\angle2$ + m$\angle4$ + m$\angle3$ = 180$^{\circ}$
57.7$^{\circ}$ + 41.5$^{\circ}$ + m$\angle3$ = 180$^{\circ}$
Subtract 99.2$^{\circ}$ from 180$^{\circ}$
Threrefore, m$\angle3$ = 80.8$^{\circ}$
