two parallel lines, m and n, are cut by the transversal k as shown. suppose m∠1 = 65°. image of parallel lines m, n cut by transversal k with angles 1, 2, 3 labeled complete the statements below. we see that ∠1 and ∠2 are select thus, ∠1 and ∠2 are select so, m∠2 = °. we see that ∠2 and ∠3 are select and since the lines m and n are parallel, ∠2 and ∠3 are select so, m∠3 = °. therefore, ∠1 and ∠3 are select we also see that ∠1 and ∠3 are select the relationship between ∠1 and ∠3 is an example of the following rule. when parallel lines are cut by a transversal, select

Question

two parallel lines, m and n, are cut by the transversal k as shown. suppose m∠1 = 65°. image of parallel lines m, n cut by transversal k with angles 1, 2, 3 labeled complete the statements below. we see that ∠1 and ∠2 are select thus, ∠1 and ∠2 are select so, m∠2 =  °. we see that ∠2 and ∠3 are select and since the lines m and n are parallel, ∠2 and ∠3 are select so, m∠3 =  °. therefore, ∠1 and ∠3 are select we also see that ∠1 and ∠3 are select the relationship between ∠1 and ∠3 is an example of the following rule. when parallel lines are cut by a transversal, select

Accepted Answer

### For ∠1 and ∠2:
# Explanation:
## Step1: Identify angle relationship
∠1 and ∠2 are adjacent and form a linear pair (supplementary angles).
## Step2: Calculate m∠2
Since linear pairs sum to $180^\circ$, $m\angle2 = 180^\circ - m\angle1$. Given $m\angle1 = 65^\circ$, so $m\angle2 = 180 - 65 = 115^\circ$.

### For ∠2 and ∠3:
# Explanation:
## Step1: Identify angle relationship
∠2 and ∠3 are alternate interior angles (since $m \parallel n$ and $k$ is transversal).
## Step2: Determine angle measure
Alternate interior angles are equal, so $m\angle3 = m\angle2 = 115^\circ$? Wait, no, wait. Wait, ∠1 and ∠3: Wait, let's re - examine. Wait, ∠1 and ∠2: linear pair, so $m\angle2 = 180 - 65 = 115^\circ$. Then ∠2 and ∠3: since $m\parallel n$, ∠2 and ∠3 are alternate interior angles? Wait, no, ∠3 and ∠1: let's see the diagram. Line $m$ and $n$ are parallel, transversal $k$. ∠1 and ∠3: maybe corresponding angles? Wait, let's start over.

1. ∠1 and ∠2: They are adjacent and form a linear pair (they are supplementary). So first blank for ∠1 and ∠2: "linear pair", second blank: "supplementary", $m\angle2 = 180 - 65 = 115^\circ$.

2. ∠2 and ∠3: Since $m\parallel n$, ∠2 and ∠3 are alternate interior angles (or corresponding? Wait, ∠3 and ∠1: let's see. ∠1 and ∠3: if we consider the transversal, ∠1 and ∠3 are corresponding angles? Wait, no, ∠2 and ∠3: ∠2 is on line $n$, ∠3 is on line $m$, and the transversal $k$. So ∠2 and ∠3 are alternate interior angles. So alternate interior angles are equal, so $m\angle3 = m\angle2 = 115^\circ$? Wait, but ∠1 is $65^\circ$, ∠3: wait, maybe I made a mistake. Wait, ∠1 and ∠3: let's check the vertical angles or corresponding. Wait, ∠1 and ∠3: if ∠2 is $115^\circ$, and ∠3 is equal to ∠1? Wait, no, let's use the properties.

Wait, the first part: ∠1 and ∠2 are a linear pair (adjacent angles forming a straight line), so they are supplementary. So $m\angle2 = 180 - 65 = 115^\circ$.

Then ∠2 and ∠3: since $m\parallel n$, ∠2 and ∠3 are alternate interior angles (because they are on opposite sides of the transversal and inside the two parallel lines). Alternate interior angles are congruent, so $m\angle3 = m\angle2 = 115^\circ$? Wait, but then ∠1 and ∠3: ∠1 is $65^\circ$, ∠3 is $115^\circ$? No, that can't be. Wait, maybe ∠2 and ∠3 are corresponding angles? Wait, no, let's look at the diagram again. Line $m$ is top, line $n$ is bottom. Transversal $k$ crosses them. ∠1 is below line $n$, ∠2 is above line $n$ and to the left of transversal, ∠3 is above line $m$ and to the left of transversal. So ∠2 and ∠3 are corresponding angles (same position relative to parallel lines and transversal), so they are equal. So $m\angle3 = m\angle2 = 115^\circ$. But then ∠1 and ∠3: ∠1 is $65^\circ$, ∠3 is $115^\circ$, which are supplementary? Wait, no, maybe I mixed up ∠1 and ∠3. Wait, ∠1 and ∠3: are they vertical angles? No. Wait, maybe ∠1 and ∠3 are equal? Wait, no, let's use the properties step by step.

- ∠1 and ∠2: linear pair (adjacent, form a straight line) → supplementary → $m\angle2 = 180 - 65 = 115^\circ$.

- ∠2 and ∠3: alternate interior angles (since $m\parallel n$) → congruent → $m\angle3 = m\angle2 = 115^\circ$. Wait, but then ∠1 and ∠3: ∠1 is $65^\circ$, ∠3 is $115^\circ$, which are supplementary? But that seems off. Wait, maybe the diagram is different. Wait, maybe ∠1 and ∠3 are corresponding angles. Wait, no, let's check the standard angle relationships.

Wait, another approach: ∠1 and ∠3: if we consider that ∠1 and ∠3 are vertical angles? No. Wait, maybe the first angle ∠1 is $65^\circ$, ∠2 is supplementary (115°), ∠3 is equal to ∠1 (65°)? Wait, that would mean ∠2 and ∠3 are supplementary. Wait, I think I made a mistake in the angle relationship between ∠2 and ∠3. Let's re - examine the diagram description: two parallel lines $m$ (top) and $n$ (bottom), transversal $k$. ∠1 is below $n$, ∠2 is above $n$ (left of $k$), ∠3 is above $m$ (left of $k$). So ∠2 and ∠3 are corresponding angles (same position: above the parallel lines, left of transversal), so they are equal. So $m\angle3 = m\angle2$. ∠1 and ∠2 are linear pair, so $m\angle2 = 180 - 65 = 115^\circ$, so $m\angle3 = 115^\circ$. But then ∠1 and ∠3: ∠1 is $65^\circ$, ∠3 is $115^\circ$, which are supplementary. But maybe the question is about ∠1 and ∠3 being equal? No, that can't be. Wait, maybe I misidentified the angles.

Wait, let's start with the first part:

- ∠1 and ∠2: They are adjacent and form a straight line (linear pair), so they are supplementary. So first blank: "linear pair", second blank: "supplementary", $m\angle2 = 180 - 65 = 115^\circ$.

- ∠2 and ∠3: Since $m\parallel n$, ∠2 and ∠3 are alternate interior angles (or corresponding angles). Wait, alternate interior angles are equal. So $m\angle3 = m\angle2 = 115^\circ$? But then ∠1 and ∠3: ∠1 is $65^\circ$, ∠3 is $115^\circ$, so they are supplementary. But maybe the diagram has ∠3 equal to ∠1. Wait, maybe the angle ∠1 and ∠3 are corresponding angles. Wait, no, ∠1 is below $n$, ∠3 is above $m$. Maybe the transversal is crossing the lines, and ∠1 and ∠3 are vertical angles? No. I think the key is:

1. ∠1 and ∠2: linear pair (supplementary), so $m\angle2 = 180 - 65 = 115^\circ$.

2. ∠2 and ∠3: alternate interior angles (since $m\parallel n$), so they are equal, so $m\angle3 = 115^\circ$. But then ∠1 and ∠3: ∠1 is $65^\circ$, ∠3 is $115^\circ$, so they are supplementary. But maybe the question has a different diagram. Alternatively, maybe ∠1 and ∠3 are equal, which would mean that ∠2 and ∠3 are supplementary. Wait, I think I made a mistake in the angle relationship between ∠2 and ∠3. Let's assume that ∠1 and ∠3 are corresponding angles, so they are equal. Then $m\angle3 = 65^\circ$, and ∠2 and ∠3 are supplementary (since ∠2 and ∠1 are linear pair). So ∠2 would be $115^\circ$, and ∠3 would be $65^\circ$. That makes more sense. So let's correct:

- ∠1 and ∠2: linear pair (supplementary), so $m\angle2 = 180 - 65 = 115^\circ$.

- ∠2 and ∠3: same - side interior angles? No, alternate interior angles: if ∠3 is equal to ∠1, then ∠2 and ∠3 are supplementary. Wait, maybe the diagram shows that ∠3 and ∠1 are corresponding angles. So:

1. ∠1 and ∠2: linear pair (supplementary), $m\angle2 = 115^\circ$.

2. ∠2 and ∠3: supplementary? No, alternate interior angles: if $m\parallel n$, alternate interior angles are equal. So if ∠3 is equal to ∠1, then ∠2 and ∠3 are supplementary. So maybe the first angle relationship for ∠2 and ∠3 is "same - side interior angles" (supplementary), but that would mean $m\angle3 = 180 - m\angle2 = 65^\circ$. Ah! That's the mistake. ∠2 and ∠3 are same - side interior angles? No, same - side interior angles are supplementary. Wait, no, alternate interior angles are equal, same - side interior angles are supplementary. Let's look at the positions:

- Line $m$ (top), line $n$ (bottom), transversal $k$.

- ∠2 is above $n$, left of $k$.

- ∠3 is above $m$, left of $k$.

So ∠2 and ∠3 are alternate interior angles (between the two parallel lines, on opposite sides of the transversal? No, same side. Wait, alternate interior angles are on opposite sides of the transversal and inside the two parallel lines. ∠2 is inside (between $m$ and $n$)? Wait, line $m$ is top, line $n$ is bottom, so the area between $m$ and $n$ is the interior. ∠2 is above $n$ (so below $m$), left of $k$: interior. ∠3 is above $m$: no, ∠3 is above $m$, so exterior. Wait, I think I misidentified the interior. The interior of two parallel lines is between them. So ∠2 is between $m$ and $n$ (above $n$, below $m$), left of $k$. ∠3 is above $m$ (exterior), left of $k$. So ∠2 and ∠3: ∠3 is corresponding to ∠1? Wait, ∠1 is below $n$ (exterior), right of $k$. ∠3 is above $m$ (exterior), left of $k$. Maybe ∠1 and ∠3 are vertical angles? No.

Let's use the correct angle properties:

- Linear pair: sum to $180^\circ$.

- Corresponding angles: equal (when parallel lines cut by transversal).

- Alternate interior angles: equal.

- Same - side interior angles: supplementary.

Given $m\parallel n$, transversal $k$:

- ∠1 and ∠2: linear pair (adjacent, form straight line) → supplementary → $m\angle2 = 180 - 65 = 115^\circ$.

- ∠2 and ∠3: alternate interior angles (since $m\parallel n$, and they are between the lines, on opposite sides of transversal? Wait, no, ∠2 is between $m$ and $n$, left of $k$; ∠3 is between $m$ and $n$, left of $k$? No, ∠3 is above $m$, so outside. I think the diagram is such that ∠3 is equal to ∠1. So ∠1 and ∠3 are corresponding angles, so they are equal. Then ∠2 and ∠3 are supplementary (since ∠2 and ∠1 are supplementary). So:

1. ∠1 and ∠2: linear pair (supplementary), $m\angle2 = 115^\circ$.

2. ∠2 and ∠3: supplementary (since ∠3 = ∠1 = 65^\circ, and $115 + 65 = 180$), so $m\angle3 = 65^\circ$.

3. ∠1 and ∠3: corresponding angles (equal), and also vertical angles? No, corresponding angles.

So to summarize:

- ∠1 and ∠2: linear pair, supplementary, $m\angle2 = 115^\circ$.

- ∠2 and ∠3: supplementary (or same - side interior angles), so $m\angle3 = 65^\circ$.

- ∠1 and ∠3: equal (corresponding angles), and also vertical angles? No, corresponding angles.

Now, let's fill in the blanks:

1. We see that ∠1 and ∠2 are $\boldsymbol{\text{linear pair}}$. Thus, ∠1 and ∠2 are $\boldsymbol{\text{supplementary}}$. So, $m\angle2=\boldsymbol{115}^\circ$.

2. We see that ∠2 and ∠3 are $\boldsymbol{\text{supplementary (or same - side interior angles)}}$ (since $m\parallel n$, same - side interior angles are supplementary). And since the lines $m$ and $n$ are parallel, ∠2 and ∠3 are $\boldsymbol{\text{supplementary}}$. So, $m\angle3=\boldsymbol{65}^\circ$.

3. Therefore, ∠1 and ∠3 are $\boldsymbol{\text{equal (corresponding angles)}}$. We also see that ∠1 and ∠3 are $\boldsymbol{\text{vertical angles? No, corresponding angles}}$ (wait, maybe corresponding angles or alternate interior angles? Wait, if ∠1 and ∠3 are equal, and $m\parallel n$, they are corresponding angles).

The relationship between ∠1 and ∠3 is an example of the rule: When parallel lines are cut by a transversal, $\boldsymbol{\text{corresponding angles are equal}}$ (or alternate interior angles, but in this case, corresponding angles).

# Answers:
- For ∠1 and ∠2: First select: linear pair; Second select: supplementary; $m\angle2 = 115^\circ$.

- For ∠2 and ∠3: First select: supplementary (or same - side interior angles); Second select: supplementary; $m\angle3 = 65^\circ$.

- For ∠1 and ∠3: First select: equal; Second select: corresponding angles (or alternate interior angles); The rule: corresponding angles are equal (when parallel lines cut by transversal).

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QUESTION IMAGE

Question

Explanation:

For ∠1 and ∠2:

Step1: Identify angle relationship

Step2: Calculate m∠2

For ∠2 and ∠3:

Step1: Identify angle relationship

Step2: Determine angle measure

Answer: