Proving AB is Not Equal to CD in Euclidean Geometry

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Discussion Overview

The discussion revolves around the concept of parallel lines in Euclidean geometry, specifically addressing whether line AB can be considered parallel to line CD under various conditions. Participants explore definitions of parallelism, the implications of lines having common points, and the relationship between line equality and parallelism.

Discussion Character

  • Debate/contested
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • Some participants argue that if line AB is parallel to BC and BC is parallel to CD, then AB must be parallel to CD unless AB equals CD, as a line cannot be parallel to itself.
  • Others propose that if AB and BC share a point (point B), they cannot be parallel if AB is not equal to BC, and the same reasoning applies to BC and CD.
  • One participant asserts that the definition of parallel lines includes the idea that a line can be parallel to itself, which contradicts other definitions presented.
  • Another participant emphasizes that parallel lines are defined as lines that do not intersect and are not in the same plane, arguing that if AB equals CD, they must intersect.
  • Some participants express confusion regarding the definitions of parallelism, particularly in relation to the concept of lines having endpoints versus line segments.
  • There is a discussion about whether curves can be considered parallel, with some suggesting that derivatives could be used to determine parallelism in curves.
  • Participants challenge each other's definitions and interpretations of parallel lines, leading to further clarification and debate about the nature of parallelism in both lines and curves.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the definitions and implications of parallel lines. Multiple competing views remain regarding whether a line can be parallel to itself and the conditions under which lines can be considered parallel.

Contextual Notes

There are limitations in the discussion regarding the definitions of parallel lines, the assumptions made about line equality, and the implications of lines having common points. The discussion also touches on the distinction between lines and line segments, which remains unresolved.

Imparcticle
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If you had line AB is parallel to BC and BC is parallel to CD, is AB parallel to CD?
----> Not if AB=CD since a line (at least in Euclidean Geometry) cannot be parallel to itself.
How would you prove that AB is not line CD?

PLEASE NOTE: Base all your input in the realm of Euclidean geometry.
 
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If you refer to the line AB as the line going through two (different) given points, point A and point B, then clearly AB and BC both pass through the same point, point B.
Therefore if AB is not equal to BC, they will not be parallel.

Same argument goes for BC and CD.
 
BTW, the definition of "parallel" with which I'm familiar says that a line is parallel to itself.
 
No, the definition of parallel lines is: Two lines are parallel if and only if they do not intersect and they are not in the same plane. This definition holds in Euclidean geometry.
Let line AB and CD be parallel lines. Let AB=CD. Then C=A and B=D. The two lines have points in common, therefore they intersect. By definition, parallel lines do not intersect. Therefore a line cannot be parallel to itself.

All of this I base on the laws of Euclidean Geometry.
 
Galileo said:
If you refer to the line AB as the line going through two (different) given points, point A and point B, then clearly AB and BC both pass through the same point, point B.
Therefore if AB is not equal to BC, they will not be parallel.

Same argument goes for BC and CD.


How about lines without points in common?
 
"AB is parallel to BC"

If both lines go trough a point B, they have at least one point in common. Therefore, they can't be parallel (that is, if a line cannot be parallel to itself).
 
I agree with you, koroljov.

now let us consider AC and GP and VL. Prove that AC and VL are two different lines and they are both parallel to GP.
 
Imparcticle said:
No, the definition of parallel lines is: Two lines are parallel if and only if they do not intersect and they are not in the same plane. This definition holds in Euclidean geometry.
Let line AB and CD be parallel lines. Let AB=CD. Then C=A and B=D. The two lines have points in common, therefore they intersect. By definition, parallel lines do not intersect. Therefore a line cannot be parallel to itself.

All of this I base on the laws of Euclidean Geometry.

Please be more careful in quoting definitions. That definition is completely wrong!

I would also point out that if AB and CD are lines such that AB= CD, it does NOT follow that "C= A and B= D". That does not follow even for line segments rather than lines.
 
Imparcticle said:
No, the definition of parallel lines is: Two lines are parallel if and only if they do not intersect and they are not in the same plane. This definition holds in Euclidean geometry.
...

I've always believed two parallel lines could be AT THE SAME PLANE...
 
  • #10
Imparcticle said:
If you had line AB is parallel to BC and BC is parallel to CD, is AB parallel to CD?
----> Not if AB=CD since a line (at least in Euclidean Geometry) cannot be parallel to itself.
How would you prove that AB is not line CD?

PLEASE NOTE: Base all your input in the realm of Euclidean geometry.

yeah "is parallel to" is even equivalence relation:
1. AB || AB
2. if AB || BC then BC || AB
3. if AB || BC & BC || CD then AB || CD
what's all the fuss about everybody? a line can always be parallel to itself
 
  • #11
http://dictionary.reference.com/search?q=parallel
dictionary.com said:
Of, relating to, or designating two or more straight coplanar lines that do not intersect.


I always thought that to be parrallel you only needed the same slope in two lines.

Also, when he said AB = CD, I don't think he was multiplying, instead he was saying the lines have the same end points. A = C and B = D.

So... is x^2 'parrallel' to x^2+1?
 
  • #12
Alkatran said:
http://dictionary.reference.com/search?q=parallel



I always thought that to be parrallel you only needed the same slope in two lines.
Yes, that's true but that statement, requiring trigonometry, is too complicated to be considered a definition.

Also, when he said AB = CD, I don't think he was multiplying, instead he was saying the lines have the same end points. A = C and B = D.
Yes, everyone understood that. My point was: first LINES don't have endpoints, line SEGMENTS do (and we were talking about "parallel" lines which doesn't apply to segments). Also, even for line segments AB, CD, in which A, B, C, D ARE the endpoints, AB= CD does not necessarily imply A= C and B= D. It might be that A= D and B= C!

So... is x^2 'parrallel' to x^2+1?
No, the concept of parallel we are talking about here (Euclidean geometry) only applies to straight lines.
Certainly, you could extend the concept of parallel (in several different ways, possibly) to curves. I suspect that in any reasonable definition of "parallel" for curves, the graphs given by y= x<sup>2</sup> and y= x<sup>2</sup>+ 1 would be parallel.
 
  • #13
HallsofIvy said:
Please be more careful in quoting definitions. That definition is completely wrong!

According to this source:
http://aleph0.clarku.edu/~djoyce/java/elements/bookI/defI23.html

Parallel lines are defined as follows:

Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction.

How is that different from what I stated? (I am inquiring with genuine curiosity, not to be defensive)
 
  • #14
I'm just throwing something out here, but regarding "parallel" curves (of which I've never heard any definition) such as x^2 and x^2 + 1, I would imagine that this could be solved by use of derivatives. Since the derivative is 2x in both cases, then they could be considered parallel.

However, since no actual definition exists, you could theoretically create any critieria you wish. ;)
 
  • #15
If the first curve were sin x and the second one were sin (x + 5) + 5, you'd probably say they were parallel even though their derivatives are different. You tend to say two things are "parallel" when one can be transformed into the other solely by translation.
 
  • #16
Imparticle said:
According to this source:
http://aleph0.clarku.edu/~djoyce/ja...okI/defI23.html

Parallel lines are defined as follows:

Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction.

How is that different from what I stated? (I am inquiring with genuine curiosity, not to be defensive)
Yes, but your original post was (I just "cut and paste" it):

"No, the definition of parallel lines is: Two lines are parallel if and only if they do not intersect and they are not in the same plane. This definition holds in Euclidean geometry."

Specifically, you said "they are not in the same plane" which is not what you intended.
 
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  • #17
oh, thanks!

I still don't understand how only one line, A can be parallel to itself.
 
  • #18
Depends entirely on the specific definition. I, personally, would be inclined to say that two lines are parallel if and only if they have no point in common so that a line is not parallel to itself, but if you are focusing on direction (two lines are parallel if and only if they have the same direction) you could certainly say a line is parallel to itself. It's not a definition I would use but I wouldn't argue with Hurkyl's use of it.
 
  • #19
HallsofIvy said:
Depends entirely on the specific definition. I, personally, would be inclined to say that two lines are parallel if and only if they have no point in common...

If that's true, then skew lines are parallel.

but if you are focusing on direction (two lines are parallel if and only if they have the same direction) you could certainly say a line is parallel to itself. It's not a definition I would use but I wouldn't argue with Hurkyl's use of it.

But that isn't the definition given by Euclid. I am focusing on Euclid's definition.

Anyway, saying that a line is parallel to itself does not satisfy the hypothesis of the definition (the second one) that you gave. The hypothesis says "two lines are parallel if...", but if a line is parallel to itself, then there is no need for a second line right? I'm probably mistaken though.

I don't challenge Hurkyl (among others including you HallsofIvy); I just ask how he arrives at conclusions or just ask him questions. I am doing the same here.
 
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