# Proof That Isometries Preserve Parallel Lines

1. Nov 22, 2012

### bistan

Hey guys, just wanted to get a quick check that this proof is sound.

1. The problem statement, all variables and given/known data
Let L, K be two parallel lines, and let F be an isometry. Prove that F(L) & F(K) are parallel.

2. Relevant Equations
If P and Q are points on the plane and F is an isometry, the distance PQ = the distance F(PQ).

3. The attempt at a solution

Let P be a point on L and Q a point on K. By definition of parallel lines, L & K have no point in common. Because F is an isometry, the distance PQ = the distance F(PQ). Therefore F(L) & F(K) must also have no point in common. Thus F(L) & F(K) are parallel.

Last edited: Nov 22, 2012
2. Nov 22, 2012

### haruspex

Looks fine to me. Maybe should mention use of the fact that the distance between two points is zero if and only if they are the same point.

3. Nov 22, 2012

### Dick

Maybe I'm missing something but I don't see anything in what bistan did that would be anything like a proof.

4. Nov 22, 2012

### haruspex

Maybe I'm reading too much between the lines
It appears to lean on some standard facts like F(K), F(L) will be straight lines; parallel lines don't intersect; non-parallel lines do intersect. And I assume this is Euclidean (otherwise I'm not sure how parallel lines are defined).

5. Nov 23, 2012

### bistan

It's not very formal I understand. I'm working through Serge Lang's Basic Mathematics and I'm using the facts that I've been given. Maybe I should've mentioned that.

6. Nov 23, 2012

### HallsofIvy

Staff Emeritus
I agree with Dick. Saying "Let P be a point on L and Q a point on K. By definition of parallel lines" means that P and Q are specific points on L and K. The fact that the distance between two specific points is the same as the distance between their images says nothing about the lines. You are trying to interpret that to say that the distance between any two points F(L) and F(K) is non-zero but that is not what you is means.

Better would be an indirect proof: if F(L) and F(K) are NOT parallel then there exist F(P) on F(L) and F(Q) on F(K) such that F(P)= F(Q)- that is they are the same point. That means the distance between them is 0. Now use the isometry to go back to L and K.

7. Nov 23, 2012

### bistan

Good thing I posted on here! That's more or less what I thought; that stating two arbitrary specific points would allow it to be made general for the whole line. Thanks for your help everyone.

So:

If F(L) and F(K) are not parallel then there exists point F(P) on line F(L) and point F(Q) on line F(K) such that F(P)= F(Q). Since F is an isometry and preserves distance, P = Q which is impossible since lines L and K are parallel. Thus F(L) and F(K) have no point in common and are parallel.

Last edited: Nov 23, 2012
8. Nov 23, 2012

### Dick

That looks better.