21 in the diagram, \overleftrightarrow{mn} is parallel to \overleftrightarrow{pq}. find the following angle measures. hint: draw this transversal on your paper, label the angles and the measurements for each angle. diagram with lines mn, pq and transversal st, angles labeled 1 - 8, angle 6 is 128° m\angle 1 = \boxed{} degrees m\angle 2 = \boxed{} degrees m\angle 3 = \boxed{} degrees

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# Explanation:
## Step1: Find $ m\angle1 $
$\angle1$ and the $128^\circ$ angle ( $\angle6$ ) are same - side interior angles? No, wait, $\angle1$ and $\angle5$ are corresponding angles? Wait, $\angle6 = 128^\circ$, and $\angle6$ and $\angle5$ are supplementary (linear pair), so $m\angle5=180 - 128=52^\circ$? No, wait, $\overline{MN}\parallel\overline{PQ}$, and the transversal is $ST$. $\angle1$ and $\angle5$ are corresponding angles? Wait, no, $\angle2$ and $\angle6$ are alternate interior angles? Wait, let's correct. $\angle6 = 128^\circ$, $\angle6$ and $\angle2$: since $\overline{MN}\parallel\overline{PQ}$, $\angle2$ and $\angle6$ are same - side interior angles? No, same - side interior angles are supplementary. Wait, $\angle6$ and $\angle7$ are vertical angles? No, $\angle6$ and $\angle8$ are supplementary. Wait, let's start with $\angle1$. $\angle1$ and $\angle5$: since $\overline{MN}\parallel\overline{PQ}$, and transversal $ST$, $\angle1$ and $\angle5$ are corresponding angles? Wait, $\angle5$ and $\angle6$ are supplementary (linear pair), so $m\angle5 = 180 - 128=52^\circ$? No, that's wrong. Wait, $\angle6 = 128^\circ$, $\angle6$ and $\angle2$: if $\overline{MN}\parallel\overline{PQ}$, then $\angle2$ and $\angle6$ are same - side interior angles, so they are supplementary? Wait, no, same - side interior angles are on the same side of the transversal and inside the two parallel lines. $\angle2$ is between $MN$ and $ST$, $\angle6$ is between $PQ$ and $ST$, and they are on the same side of $ST$. So $m\angle2 + m\angle6=180^\circ$? Then $m\angle2 = 180 - 128 = 52^\circ$? No, that can't be. Wait, maybe $\angle2$ and $\angle6$ are alternate interior angles. Wait, alternate interior angles are equal. Wait, I think I mixed up. Let's use linear pairs first. $\angle6 = 128^\circ$, so the angle adjacent to $\angle6$ (linear pair) is $180 - 128 = 52^\circ$. Now, $\angle1$: since $\overline{MN}\parallel\overline{PQ}$, $\angle1$ and the angle equal to $52^\circ$ (let's say $\angle5$) are corresponding angles, so $m\angle1 = 128^\circ$? Wait, no, I'm getting confused. Let's start over.

1. For $m\angle1$:
$\angle1$ and $\angle6$: Wait, $\overline{MN}\parallel\overline{PQ}$, transversal $ST$. $\angle1$ and $\angle5$ are corresponding angles. $\angle5$ and $\angle6$ are supplementary (linear pair), so $m\angle5=180 - 128 = 52^\circ$? No, that's incorrect. Wait, $\angle6 = 128^\circ$, $\angle6$ and $\angle2$: $\angle2$ and $\angle6$ are alternate interior angles? If $\overline{MN}\parallel\overline{PQ}$, then alternate interior angles are equal. So $m\angle2=m\angle6 = 128^\circ$? No, that can't be. Wait, no, alternate interior angles are inside the two parallel lines and on opposite sides of the transversal. $\angle2$ is inside $MN$ and $ST$, $\angle6$ is inside $PQ$ and $ST$, and on opposite sides of $ST$. So $\angle2$ and $\angle6$ are alternate interior angles, so $m\angle2 = m\angle6=128^\circ$? Then $\angle1$ and $\angle2$ are supplementary (linear pair), so $m\angle1=180 - 128 = 52^\circ$. Ah, that makes sense.

So Step 1: Find $m\angle1$
$\angle1$ and $\angle2$ are linear pair (supplementary), and $\angle2=\angle6 = 128^\circ$ (alternate interior angles, since $\overline{MN}\parallel\overline{PQ}$). So $m\angle1=180 - 128 = 52^\circ$? Wait, no, if $\angle2 = 128^\circ$, then $m\angle1=180 - 128 = 52^\circ$.

## Step2: Find $m\angle2$
$\angle2$ and $\angle6$ are alternate interior angles (since $\overline{MN}\parallel\overline{PQ}$ and transversal $ST$). So $m\angle2=m\angle6 = 128^\circ$.

## Step3: Find $m\angle3$
$\angle3$ and $\angle1$ are vertical angles? No, $\angle3$ and $\angle1$: $\angle3$ and $\angle2$ are supplementary, and $\angle1$ and $\angle2$ are supplementary, so $\angle3=\angle1$ (vertical angles or equal - supplementary to the same angle). Wait, $\angle3$ and $\angle1$ are corresponding angles? No, $\angle3$ and $\angle2$ are linear pair, $\angle1$ and $\angle2$ are linear pair, so $\angle3=\angle1 = 52^\circ$? Wait, no, $\angle3$ and $\angle2$ are supplementary, so $m\angle3=180 - m\angle2=180 - 128 = 52^\circ$. Wait, let's check:

- $m\angle1$: Since $\angle1$ and $\angle2$ are linear pair, $m\angle1 + m\angle2=180$. $\angle2 = 128^\circ$ (alternate interior with $\angle6$), so $m\angle1=180 - 128 = 52^\circ$.
- $m\angle2$: Alternate interior angles with $\angle6$ ( $\overline{MN}\parallel\overline{PQ}$ ), so $m\angle2 = m\angle6=128^\circ$.
- $m\angle3$: Vertical angle with $\angle1$ (or supplementary to $\angle2$), so $m\angle3=m\angle1 = 52^\circ$ (or $180 - 128 = 52^\circ$).

# Answer:
$m\angle1=\boxed{52}$ degrees, $m\angle2=\boxed{128}$ degrees, $m\angle3=\boxed{52}$ degrees

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Explanation:

Step1: Find \( m\angle1 \)

Step2: Find \(m\angle2\)

Answer: