Question

question 15 of 20 (1 point) | question attempt: 1 of 1
complete the statements below.
we see that ∠2 and ∠3 are (choose one)
and since the lines m and n are parallel, ∠2 and ∠3 are (choose one)
so, m∠3 = □°.
we see that ∠1 and ∠2 are (choose one)
thus, ∠1 and ∠2 are (choose one)
so, m∠1 = □°.
therefore, ∠1 and ∠3 are (choose one)
we also see that ∠1 and ∠3 are (choose one)
the relationship between ∠1 and ∠3 is an example of the following rule.
when parallel lines are cut by a transversal, (choose one)

Explanation:

Step1: Analyze ∠2 and ∠3

∠2 and ∠3 are corresponding angles (since lines \( m \) and \( n \) are parallel, cut by transversal \( f \)). Corresponding angles are equal when lines are parallel. But first, identify ∠1 and ∠2: ∠1 and ∠2 are vertical angles? No, ∠1 and ∠2 are supplementary? Wait, ∠1 and ∠2 form a linear pair, so they are supplementary (sum to \( 180^\circ \))? Wait, no, ∠1 and ∠2: if ∠2 is adjacent to ∠1, they are a linear pair, so \( m\angle1 + m\angle2 = 180^\circ \). But maybe ∠2 is equal to ∠3 (corresponding angles). Then, if ∠1 and ∠2 are supplementary (linear pair), ∠1 and ∠3 would be supplementary? Wait, no, let's correct.

First, ∠2 and ∠3: lines \( m \) and \( n \) are parallel, transversal \( f \). So ∠2 and ∠3 are corresponding angles (same position relative to parallel lines and transversal). So corresponding angles are equal when lines are parallel.

Then, ∠1 and ∠2: they form a linear pair (adjacent, form a straight line), so they are supplementary (\( m\angle1 + m\angle2 = 180^\circ \))? Wait, no, if ∠2 and ∠3 are equal (corresponding angles), and ∠1 and ∠2 are vertical angles? Wait, no, ∠1 and ∠2: looking at the diagram, ∠1 and ∠2 are adjacent, forming a linear pair, so they are supplementary. Wait, maybe ∠1 is vertical to the angle adjacent to ∠3? Wait, maybe the diagram has ∠2 as, say, \( 120^\circ \), but since the problem is about angle relationships, let's proceed with definitions.

Step2: ∠1 and ∠2 relationship

∠1 and ∠2 are a linear pair (adjacent, form a straight line), so they are supplementary (\( m\angle1 + m\angle2 = 180^\circ \)).

Step3: ∠2 and ∠3 relationship

∠2 and ∠3 are corresponding angles (parallel lines \( m \), \( n \), transversal \( f \)), so \( m\angle2 = m\angle3 \) (corresponding angles postulate).

Step4: ∠1 and ∠3 relationship

Since \( m\angle1 + m\angle2 = 180^\circ \) (linear pair) and \( m\angle2 = m\angle3 \) (corresponding angles), then \( m\angle1 + m\angle3 = 180^\circ \), so ∠1 and ∠3 are supplementary. Also, since ∠2 = ∠3 (corresponding) and ∠1 + ∠2 = 180 (linear pair), ∠1 + ∠3 = 180, so they are supplementary. Also, ∠1 and ∠3 could be same - side interior angles? Wait, no, same - side interior angles are between the parallel lines. Wait, maybe ∠1 and ∠3 are supplementary (because ∠2 = ∠3, ∠1 + ∠2 = 180, so ∠1 + ∠3 = 180).

But since the problem is about completing the statements, let's assume typical angle measures. For example, if ∠2 is \( 120^\circ \), then ∠3 is \( 120^\circ \) (corresponding angles), ∠1 is \( 60^\circ \) (linear pair with ∠2: \( 180 - 120 = 60 \)), then ∠1 and ∠3 are supplementary (\( 60 + 120 = 180 \)).

But the key is the relationships:

- ∠2 and ∠3: corresponding angles, equal (if lines are parallel).
- ∠1 and ∠2: linear pair, supplementary.
- ∠1 and ∠3: supplementary (since ∠2 = ∠3, ∠1 + ∠2 = 180 ⇒ ∠1 + ∠3 = 180).

So, for the first part:

We see that ∠2 and ∠3 are **corresponding angles** (choose one: corresponding angles).

And since the lines \( m \) and \( n \) are parallel, ∠2 and ∠3 are **equal** (corresponding angles postulate).

Then, if we assume ∠2 is, say, \( 120^\circ \) (common example), then \( m\angle3 = 120^\circ \).

We see that ∠1 and ∠2 are **a linear pair** (choose one: linear pair), thus, ∠1 and ∠2 are **supplementary** (\( m\angle1 + m\angle2 = 180^\circ \)). So, if \( m\angle2 = 120^\circ \), \( m\angle1 = 60^\circ \).

Therefore, ∠1 and ∠3 are **supplementary** (since \( 60 + 120 = 180 \)). We also see that ∠1 and ∠3 are **supplementary** (same as above). The relationship between ∠1 and ∠3 is an example of: when parallel lines are cut by a transversal, **same - side interior angles are supplementary**? Wait, no, ∠1 and ∠3: if ∠3 is on line \( m \), ∠1 on line \( n \), transversal \( f \), maybe they are same - side interior? Wait, no, same - side interior are between the lines. Wait, maybe ∠1 and ∠3 are supplementary because ∠2 (corresponding to ∠3) and ∠1 are supplementary (linear pair). So the rule is: when parallel lines are cut by a transversal, **corresponding angles are equal, and linear pairs are supplementary, leading to supplementary angles for ∠1 and ∠3**? Wait, maybe the correct rule is that same - side interior angles are supplementary, but in this case, ∠1 and ∠3 might be same - side interior? Wait, no, let's re - examine the diagram.

Assuming line \( m \) is top, line \( n \) is bottom, transversal \( f \) crossing them. ∠3 is on line \( m \), below the transversal? Wait, no, the diagram: line \( m \) (top), line \( n \) (bottom), transversal \( f \) going from top left to bottom right. ∠3 is on line \( m \), below the transversal (between \( m \) and \( n \)), ∠2 is on line \( n \), below the transversal (same side as ∠3), so ∠2 and ∠3 are alternate interior angles? Wait, alternate interior angles are equal when lines are parallel. Oh! I made a mistake earlier. ∠2 and ∠3: lines \( m \) and \( n \) are parallel, transversal \( f \). ∠3 is on \( m \), ∠2 is on \( n \), both between the lines (interior) and on alternate sides of the transversal? No, if ∠3 is on the left of transversal \( f \) on line \( m \), and ∠2 is on the left of transversal \( f \) on line \( n \), then they are corresponding angles. Wait, no, alternate interior angles are on opposite sides of the transversal, between the lines.

Let's correct:

- Corresponding angles: same position relative to parallel lines and transversal (e.g., above the line, left of transversal).
- Alternate interior angles: between the lines, opposite sides of transversal.
- Linear pair: adjacent, form a straight line.

So, in the diagram, ∠2 and ∠3: if line \( m \) and \( n \) are parallel, transversal \( f \). ∠3 is on line \( m \), ∠2 is on line \( n \), same side of transversal, same position (e.g., below the transversal, left of transversal), so corresponding angles. Thus, \( m\angle2 = m\angle3 \) (corresponding angles postulate).

∠1 and ∠2: adjacent, form a straight line (linear pair), so \( m\angle1 + m\angle2 = 180^\circ \) (supplementary).

Thus, \( m\angle1 + m\angle3 = 180^\circ \) (since \( m\angle2 = m\angle3 \)), so ∠1 and ∠3 are supplementary.

So, to fill in:

1. We see that ∠2 and ∠3 are **corresponding angles** (choose one: corresponding angles).
2. And since the lines \( m \) and \( n \) are parallel, ∠2 and ∠3 are **equal** (corresponding angles are equal).
3. So, if we assume \( m\angle2 = 120^\circ \) (for example), then \( m\angle3 = 120^\circ \).
4. We see that ∠1 and ∠2 are **a linear pair** (choose one: linear pair).
5. Thus, ∠1 and ∠2 are **supplementary** (sum to \( 180^\circ \)).
6. So, if \( m\angle2 = 120^\circ \), \( m\angle1 = 60^\circ \).
7. Therefore, ∠1 and ∠3 are **supplementary** (sum to \( 180^\circ \)).
8. We also see that ∠1 and ∠3 are **supplementary** (same as above).
9. The relationship between ∠1 and ∠3 is an example of the following rule: When parallel lines are cut by a transversal, **same - side interior angles are supplementary**? Wait, no, ∠1 and ∠3: if ∠3 is on line \( m \), ∠1 on line \( n \), and they are on the same side of the transversal, between the lines (interior), then they are same - side interior angles, which are supplementary. Yes, same - side interior angles are supplementary when lines are parallel.

So, putting it all together (assuming typical angle measures, say \( m\angle2 = 120^\circ \)):

- ∠2 and ∠3: corresponding angles, equal (\( m\angle3 = 120^\circ \))
- ∠1 and ∠2: linear pair, supplementary (\( m\angle1 = 60^\circ \))
- ∠1 and ∠3: supplementary (same - side interior angles, supplementary)

Answer:

(For the angle measures, assuming \( m\angle2 = 120^\circ \)):

• \( m\angle3 = \boldsymbol{120}^\circ \)
• \( m\angle1 = \boldsymbol{60}^\circ \)

(For the relationships):

• ∠2 and ∠3: corresponding angles, equal
• ∠1 and ∠2: linear pair, supplementary
• ∠1 and ∠3: supplementary, same - side interior angles are supplementary (when parallel lines cut by transversal)