Question

three students attempt to define what it means for two lines to be parallel.
can you match the teachers comments to the definitions?

definition of parallel lines	teacher’s comments
ori’s definition: two lines are parallel if they are close together but don’t intersect.	part of your definition is correct, but the other part is not. parallel lines don’t have to be close together.
kaori’s definition: two lines are parallel as long as they aren’t perpendicular.	nice definition! hip, hip, hurrah!

Explanation:

Step1: Analyze Daniela's definition

Daniela says two lines are parallel if one can be translated on top of the other. But this actually defines congruent and coincident (or maybe overlapping after translation) lines? Wait, no—parallel lines in a plane are lines that never meet, and translation can map one parallel line to another (since they have the same slope, translation preserves direction). Wait, maybe I made a mistake. Wait, no—if you translate a line onto another, they are either parallel (if the translation is along the direction perpendicular to the lines? No, translation is a vector. Wait, maybe the teacher's comment for Daniela: Wait, the correct definition of parallel lines (in Euclidean geometry) is two lines in a plane that do not intersect, or are coplanar and have the same slope (in coordinate geometry). Wait, Daniela's definition: "one can be translated on top of the other"—translation is a rigid motion that slides a figure. If two lines are parallel, translating one along the direction of the line (or perpendicular?) Wait, no—if you have two parallel lines, say \( y = x \) and \( y = x + 2 \), translating the first line by the vector \( (0,2) \) would map it to the second. But if you have two coincident lines (same line), translating by zero vector maps them. But Daniela's definition: "one can be translated on top of the other"—maybe the teacher thinks this is incorrect because it could include coincident lines? Wait, no, the problem is to match the teacher's comments. Let's look at the teacher's comments:

- Comment 1: "Sorry, your definition is incorrect."
- Comment 2: "Part of your definition is correct, but the other part is not. Parallel lines don't have to be close together."
- Comment 3: "Nice definition! Hip, hip, hurrah!"

Now, Ori's definition: "Two lines are parallel if they are close together but don't intersect." The teacher's comment "Part of your definition is correct, but the other part is not. Parallel lines don't have to be close together." So Ori's definition has "don't intersect" (correct part) and "close together" (incorrect part), so that comment matches Ori.

Kaori's definition: "Two lines are parallel as long as they aren't perpendicular." But lines can be neither parallel nor perpendicular (e.g., two lines with slopes 1 and 2—they aren't perpendicular, but they aren't parallel). So Kaori's definition is incorrect, so the comment "Sorry, your definition is incorrect." matches Kaori.

Daniela's definition: "Two lines are parallel if one can be translated on top of the other." In Euclidean geometry, parallel lines (in a plane) can be translated onto each other (since translation preserves direction and distance, so if two lines are parallel, translating one along the direction perpendicular to the lines? Wait, no—translation is a vector. If two lines are parallel, say \( y = x \) and \( y = x + c \), translating the first by \( (0,c) \) (if vertical translation) would map it to the second. Wait, maybe the teacher thinks this is a correct definition? Wait, no—the third comment "Nice definition!" should go to the correct one. Wait, maybe I messed up. Let's re-express:

Correct definition of parallel lines (in plane): two coplanar lines that do not intersect (or have the same slope, are equidistant, etc.).

Daniela: translation on top—if you can translate one to the other, they are parallel (since translation is a rigid motion, and parallel lines are "equidistant" and direction-preserving). So maybe Daniela's definition is correct? Wait, no—if two lines are coincident (same line), translating by zero vector maps them, but coincident lines are a special case of parallel lines (with distance zero). Wait, but maybe the teacher considers Daniela's definition correct? Wait, the third comment is "Nice definition!".

Ori: "close together but don't intersect"—the "close together" is wrong, "don't intersect" is right, so the comment "Part of your definition is correct..." matches Ori.

Kaori: "aren't perpendicular"—wrong, because lines can be neither parallel nor perpendicular, so her definition is incorrect, so comment "Sorry, your definition is incorrect." matches Kaori.

Daniela: translation on top—if you can translate one to the other, they are parallel (since translation preserves the direction, so the lines have the same slope, hence parallel). So Daniela's definition is correct, so the comment "Nice definition!" matches Daniela.

Wait, let's list:

- Daniela: definition about translation—correct (since parallel lines can be translated onto each other, as they have the same direction and are equidistant? Wait, in Euclidean geometry, parallel lines are lines that never meet, and translation is a way to map one to the other. So Daniela's definition is correct? Then the "Nice definition!" comment goes to Daniela.

- Ori: "close together but don't intersect"—"don't intersect" is correct, "close together" is wrong, so the comment "Part of your definition..." goes to Ori.

- Kaori: "aren't perpendicular"—incorrect, because non-perpendicular lines can be intersecting (not parallel) or parallel. So Kaori's definition is wrong, so the comment "Sorry, your definition is incorrect." goes to Kaori.

So matching:

Daniela's definition → "Nice definition! Hip, hip, hurrah!" (because translation can map one parallel line to another, which is a valid way to think about parallel lines—they are congruent and have the same direction, so translation preserves that)

Ori's definition → "Part of your definition is correct, but the other part is not. Parallel lines don't have to be close together." (because "don't intersect" is correct, "close together" is incorrect)

Kaori's definition → "Sorry, your definition is incorrect." (because lines can be neither parallel nor perpendicular)

Step2: Match each definition to the comment

- Daniela: "Two lines are parallel if one can be translated on top of the other." → "Nice definition! Hip, hip, hurrah!" (since translation is a valid way to characterize parallel lines—they are equidistant and direction-preserving, so translating one along the appropriate vector maps it to the other)
- Ori: "Two lines are parallel if they are close together but don't intersect." → "Part of your definition is correct, but the other part is not. Parallel lines don't have to be close together." (correct part: don't intersect; incorrect part: close together)
- Kaori: "Two lines are parallel as long as they aren't perpendicular." → "Sorry, your definition is incorrect." (because non-perpendicular lines can be intersecting and not parallel)

Answer:

• Daniela's definition ↔ "Nice definition! Hip, hip, hurrah!"
• Ori's definition ↔ "Part of your definition is correct, but the other part is not. Parallel lines don't have to be close together."
• Kaori's definition ↔ "Sorry, your definition is incorrect."