Question

two parallel lines, m and n, are cut by the transversal w as shown. suppose m∠2 = 140°. image of parallel lines m, n and transversal w with angles 1, 2, 3 labeled 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 alternate exterior angles (or corresponding angles, depending on position; here, since lines m || n, and transversal w, ∠2 and ∠3 are alternate exterior angles or corresponding, so they are congruent). So \( m\angle3 = m\angle2 = 140^\circ \)? Wait, no, wait. Wait, ∠1 and ∠2: are they supplementary? Wait, let's re-examine.

Wait, first, ∠2 and ∠3: let's look at the diagram. Line m and n are parallel, transversal w. ∠2 is on line n, ∠3 is on line m. So ∠2 and ∠3: are they alternate exterior angles? Or maybe corresponding? Wait, maybe ∠2 and ∠3 are alternate interior? No, let's think again. Wait, ∠1 and ∠2: they form a linear pair, so they are supplementary. So ∠1 + ∠2 = 180°. Then ∠2 and ∠3: since m || n, ∠2 and ∠3 are alternate exterior angles (or corresponding), so they are congruent? Wait, no, maybe ∠3 is equal to ∠2? Wait, no, let's correct.

Wait, first part: ∠2 and ∠3. Let's see the positions. Line m (top), line n (bottom), transversal w. ∠2 is below line n, ∠3 is above line m. Wait, maybe ∠2 and ∠3 are corresponding angles? Or alternate exterior. So if lines are parallel, corresponding angles are congruent. So ∠2 and ∠3: if ∠2 is 140°, then ∠3 is also 140°? Wait, but then ∠1 and ∠2: linear pair, so ∠1 = 180 - 140 = 40°. Then ∠1 and ∠3: would they be equal? Wait, no, 40 and 140? No, that can't be. Wait, maybe I mixed up the angles.

Wait, let's start over.

1. ∠2 and ∠3: Let's see, ∠2 is adjacent to ∠1, forming a linear pair? No, ∠1 and ∠2 are adjacent, so they are supplementary (linear pair). So ∠1 + ∠2 = 180°. Then ∠2 and ∠3: since m || n, ∠2 and ∠3 are alternate exterior angles (or corresponding), so they are congruent. Wait, but if ∠2 is 140°, then ∠3 is 140°, and ∠1 is 40°, then ∠1 and ∠3 would be supplementary? No, 40 + 140 = 180. Wait, maybe ∠3 is equal to ∠1? No, that doesn't make sense. Wait, maybe the diagram is such that ∠2 and ∠3 are alternate interior angles? No, let's check the standard.

Wait, the first statement: "We see that ∠2 and ∠3 are [Choose one]". Options could be "corresponding angles", "alternate interior angles", "alternate exterior angles", "same-side interior angles", etc. Then "And since lines m and n are parallel, ∠2 and ∠3 are [congruent]". So \( m\angle3 = m\angle2 = 140^\circ \)? But then ∠1 and ∠2: linear pair, so ∠1 = 180 - 140 = 40°. Then ∠1 and ∠3: 40 and 140, which are supplementary? Wait, no, maybe ∠3 is equal to ∠1? No, that's conflicting. Wait, maybe I made a mistake.

Wait, let's look at the angles:

- ∠1 and ∠2: form a linear pair (adjacent, share a side, form a straight line), so they are supplementary (sum to 180°). So ∠1 + ∠2 = 180°, so ∠1 = 180 - 140 = 40°.

- ∠2 and ∠3: since m || n, ∠2 and ∠3 are alternate exterior angles (or corresponding angles), so they are congruent. So ∠3 = ∠2 = 140°? But then ∠1 (40°) and ∠3 (140°) would be supplementary. But then ∠1 and ∠3: are they equal? No, 40 ≠ 140. Wait, maybe ∠3 is equal to ∠1? No, that can't be. Wait, maybe the diagram is different. Maybe ∠3 is equal to ∠1. Wait, maybe I mixed up the angles.

Wait, perhaps ∠2 and ∠3 are alternate interior angles. Wait, no, let's think of the transversal. If line m is top, line n is bottom, transversal w crosses them. ∠2 is on the lower line (n) above the transversal? No, the problem says m∠2 = 140°. Let's assume:

- ∠1 and ∠2: linear pair, so supplementary. So ∠1 = 180 - 140 = 40°.

- ∠2 and ∠3: since m || n, ∠2 and ∠3 are corresponding angles (same position relative to transversal and parallel lines), so ∠3 = ∠2 = 140°? But then ∠1 (40°) and ∠3 (140°) are supplementary. But then ∠1 and ∠3: are they equal? No, but maybe ∠1 and ∠3 are alternate interior angles? Wait, no, 40 and 140 are not equal. Wait, maybe I got the angle positions wrong.

Wait, maybe ∠3 is equal to ∠1. Let's re-express:

If ∠1 and ∠2 are supplementary (linear pair), so ∠1 = 40°. Then ∠2 and ∠3: since m || n, ∠3 is equal to ∠1 (alternate interior angles). Wait, that makes sense. So ∠3 = ∠1 = 40°? But then ∠2 and ∠3 would be supplementary? No, 140 + 40 = 180. Oh! Wait, that's the key. So ∠2 and ∠3 are same-side interior angles? No, same-side interior angles are supplementary. Wait, maybe I mixed up the angle types.

Let's clarify the angle relationships:

- Linear pair: two angles adjacent, forming a straight line (sum to 180°). So ∠1 and ∠2 are a linear pair, so ∠1 + ∠2 = 180° ⇒ ∠1 = 180 - 140 = 40°.

- Corresponding angles: when parallel lines cut by transversal, corresponding angles are congruent. So ∠3 and ∠1: if ∠1 is 40°, then ∠3 is also 40° (corresponding angles). Then ∠2 and ∠3: 140 + 40 = 180°, so they are supplementary (same-side interior angles).

Ah, that makes sense. So let's correct the earlier mistake.

So first part: ∠2 and ∠3. Let's see their positions. ∠2 is on line n, ∠3 is on line m. If ∠1 is 40°, ∠3 is 40°, then ∠2 (140°) and ∠3 (40°) are supplementary (same-side interior angles). Wait, but the first question: "We see that ∠2 and ∠3 are [Choose one]". Options could be "same - side interior angles", "alternate interior angles", "corresponding angles", etc. Then "And since the lines m and n are parallel, ∠2 and ∠3 are [supplementary]" (if same - side interior). But then \( m\angle3 = 180 - 140 = 40^\circ \)? Wait, no, that contradicts. Wait, I'm getting confused. Let's use the linear pair first.

∠1 and ∠2: linear pair (adjacent, form a straight line) ⇒ supplementary ⇒ ∠1 = 180 - 140 = 40°.

∠2 and ∠3: since m || n, ∠3 is equal to ∠1 (alternate interior angles) ⇒ ∠3 = 40°. Then ∠2 and ∠3: 140 + 40 = 180 ⇒ supplementary (same - side interior angles).

So let's structure the steps:

1. ∠2 and ∠3: Let's say they are "same - side interior angles" (or "alternate exterior" no, alternate exterior would be congruent). Wait, no, alternate interior angles: ∠1 and ∠3. So ∠3 and ∠1 are alternate interior angles, so they are congruent (since m || n). So ∠3 = ∠1 = 40°. Then ∠2 and ∠3: 140 + 40 = 180 ⇒ supplementary (same - side interior).

So first part:

- ∠2 and ∠3: Let's assume the options are like "same - side interior angles" (or "linear pair" no, they are not adjacent). Then "And since the lines m and n are parallel, ∠2 and ∠3 are [supplementary]" (because same - side interior angles are supplementary when lines are parallel). Then \( m\angle3 = 180 - 140 = 40^\circ \).

But then ∠1 and ∠2: linear pair (adjacent, form a straight line) ⇒ supplementary ⇒ ∠1 = 180 - 140 = 40°.

Then ∠1 and ∠3: they are equal (40° each), so they are "congruent" (alternate interior angles or corresponding angles).

So let's proceed step by step:

Step1: Analyze ∠2 and ∠3

- ∠2 and ∠3: Let's say they are "same - side interior angles" (or another type, but let's use the correct relationship). Since m || n, same - side interior angles are supplementary. So \( m\angle2 + m\angle3 = 180^\circ \). Given \( m\angle2 = 140^\circ \), then \( m\angle3 = 180 - 140 = 40^\circ \). But wait, maybe ∠2 and ∠3 are alternate exterior angles? No, alternate exterior angles are congruent. So there must be a mistake in the diagram interpretation.

Wait, let's look at the standard angle relationships:

- Corresponding angles: congruent (parallel lines, transversal).

- Alternate interior angles: congruent.

- Alternate exterior angles: congruent.

- Same - side interior angles: supplementary.

- Linear pair: supplementary.

So let's assume the diagram is such that:

- ∠1 and ∠2: linear pair (supplementary) ⇒ ∠1 = 180 - 140 = 40°.

- ∠3 and ∠1: corresponding angles (since m || n) ⇒ ∠3 = ∠1 = 40°.

- ∠2 and ∠3: 140 + 40 = 180 ⇒ supplementary (same - side interior angles).

So now, let's answer the first part:

"We see that ∠2 and ∠3 are [same - side interior angles] (or another type, but based on the sum, they are supplementary). And since the lines m and n are parallel, ∠2 and ∠3 are [supplementary]. So, \( m\angle3 = 180 - 140 = 40^\circ \)." Wait, no, if ∠3 is equal to ∠1 (40°), then \( m\angle3 = 40^\circ \).

Then the second part: "We see that ∠1 and ∠2 are [a linear pair]. Thus, ∠1 and ∠2 are [supplementary]. So, \( m\angle1 = 180 - 140 = 40^\circ \)."

Third part: "Therefore, ∠1 and ∠3 are [equal] (congruent). We also see that ∠1 and ∠3 are [corresponding angles] (or alternate interior angles). The relationship between ∠1 and ∠3 is an example of the following rule: When parallel lines are cut by a transversal, [corresponding angles are congruent] (or alternate interior angles are congruent)."

Now, let's formalize the steps:

Step1: Analyze ∠2 and ∠3

- ∠2 and ∠3: Let's assume they are same - side interior angles (based on the sum). Since m || n, same - side interior angles are supplementary. So \( m\angle2 + m\angle3 = 180^\circ \). Given \( m\angle2 = 140^\circ \), then \( m\angle3 = 180 - 140 = 40^\circ \). But wait, if ∠3 is equal to ∠1 (40°), then ∠1 and ∠3 are corresponding angles.

Step2: Analyze ∠1 and ∠2

- ∠1 and ∠2: They form a linear pair (adjacent, form a straight line). So they are supplementary. Thus, \( m\angle1 + m\angle2 = 180^\circ \). So \( m\angle1 = 180 - 140 = 40^\circ \).

Step3: Analyze ∠1 and ∠3

- ∠1 and ∠3: Since \( m\angle1 = 40^\circ \) and \( m\angle3 = 40^\circ \), they are equal (congruent). They are corresponding angles (or alternate interior angles) because m || n and cut by transversal w.

Answer:

• For \( m\angle3 \): \( 40^\circ \)
• For \( m\angle1 \): \( 40^\circ \)

(Note: The choices for the dropdowns would be filled as follows:

• ∠2 and ∠3: same - side interior angles; supplementary; \( m\angle3 = 40^\circ \)
• ∠1 and ∠2: linear pair; supplementary; \( m\angle1 = 40^\circ \)
• ∠1 and ∠3: equal; corresponding angles (or alternate interior angles); the rule is corresponding angles are congruent (or alternate interior angles are congruent))