Given that $m\angle 9 = 75^{\circ}$, find the measure of the following angles: $\angle 3$, $\angle 5$, $\angle 6$, $\angle 8$, $\angle 11$, and $\angle 12$. Lines $l$, $m$, and $n$ are parallel. Line $t$ is a transversal.

1. Problem Description

Given that $m\angle 9 = 75^{\circ}$, find the measure of the following angles: $\angle 3$, $\angle 5$, $\angle 6$, $\angle 8$, $\angle 11$, and $\angle 12$. Lines $l$, $m$, and $n$ are parallel. Line $t$ is a transversal.

2. Solution Steps

First, we find the measure of $\angle 3$. Since lines $l$, $m$ and $n$ are parallel, $\angle 3$ and $\angle 7$ are corresponding angles, and $\angle 7$ and $\angle 11$ are corresponding angles. Also $\angle 9$ and $\angle 11$ are vertical angles. Vertical angles are equal, so $m\angle 9 = m\angle 11 = 75^{\circ}$.

Also since lines $l$, $m$ and $n$ are parallel, $m\angle 3 = m\angle 7$.

Since $\angle 7$ and $\angle 11$ are corresponding angles, $m\angle 7=m\angle 11$. Therefore, $m\angle 3=m\angle 11$.

Thus, $m\angle 3 = 75^{\circ}$.

Next, we find the measure of $\angle 5$. Since $m\angle 9=75^{\circ}$, then $m\angle 10=180^{\circ}-75^{\circ}=105^{\circ}$.

Corresponding angles are congruent, so $m\angle 6=m\angle 10=105^{\circ}$ and $m\angle 5=m\angle 9 = 75^{\circ}$.

Therefore $m\angle 5 = 75^{\circ}$.

Next, we find the measure of $\angle 6$. As mentioned before, $m\angle 6 = 105^{\circ}$.

Next, we find the measure of $\angle 8$. Since $\angle 8$ and $\angle 5$ are vertical angles, $m\angle 8 = m\angle 5 = 75^{\circ}$.

Next, we find the measure of $\angle 11$. As mentioned before, since $\angle 9$ and $\angle 11$ are vertical angles, $m\angle 11 = m\angle 9 = 75^{\circ}$.

Finally, we find the measure of $\angle 12$. Since $\angle 11$ and $\angle 12$ form a straight line, they are supplementary. Therefore, $m\angle 11 + m\angle 12 = 180^{\circ}$.

Since $m\angle 11 = 75^{\circ}$, $m\angle 12 = 180^{\circ} - 75^{\circ} = 105^{\circ}$.

3. Final Answer

$m\angle 3 = 75^{\circ}$

$m\angle 5 = 75^{\circ}$

$m\angle 6 = 105^{\circ}$

$m\angle 8 = 75^{\circ}$

$m\angle 11 = 75^{\circ}$

$m\angle 12 = 105^{\circ}$