Showing segment congruence equivalent to having same length

In this video, we're going to talk a little bit about segment congruence and what we have here. Let's call this statement one. This is the definition of line segment congruence, or at least the one that we will use. Two segments are congruent; that means that we can map one segment onto the other using rigid transformations. Examples of rigid transformations are reflections, rotations, translations, and combinations of them.

Now, what we're going to see in this video is that statement 1 is actually equivalent to statement 2. Or another way of saying it is, if statement 1 is true, then statement 2 is true. And if statement 2 is true, then statement 1 is true. Or we can write it like this: we can map one segment onto another using rigid transformations if and only if the two segments have the same length.

So how do we go about proving it? Well, the first thing that we'd want to prove is that if statement 1 is true, then statement 2 is true. So how would we go about doing this? And like always, I encourage you to pause the video and have a go at it.

All right, now let's work through this together. Some proofs like this might be difficult because they feel so intuitive, but one way to prove this is to first say that by definition, rigid transformations preserve length. So by definition, rigid transformations— that's what makes them rigid— rigid transformations preserve length.

So if one segment can be mapped onto a second segment with rigid transformations, they must have had the same original length. They must have had the same original length. Or another way to say it is, then statement 2 is true.

Then we can try to do it the other way around. So let's see if we can prove that if statement 2 is true, then statement 1 is true. Why don't you pause this video and have a go at that as well?

So let's assume I have segment AB, and then I have another segment, let's call it CD, that have the same length. They meet statement two, the number two statement right over there. To map AB onto CD, all I have to do is this in two rotations.

First, I will translate so that A is on top of C. So I will translate segment AB so that point A is on top of point C. And then the next thing I would do is rotate segment AB so that point B is on top of point D.

And there you have it! For any two segments with the same length, I can always translate it so that I have one set of points overlap. Then to get the other points to overlap, I just have to rotate it.

I know that's going to work because they have the same length. So I've just shown you that if we can map one segment onto another using rigid transformations, then we know they have the same length. And if two segments have the same length, then we know that we can map one segment onto the other using rigid transformations.