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## High school geometry

### Unit 3: Lesson 1

Transformations & congruence# Segment congruence equivalent to having same length

CCSS.Math:

Two line segments are congruent if and only if they have equal lengths.

## Video transcript

- [Instructor] So, what I have
here are a few definitions that will be useful for
a proof we're going to do that connects the worlds of
congruence of line segments to the idea of them
having the same length. So, first of all, there's this idea of rigid transformations,
which we've talked about in other videos, but just as a refresher, these are transformations that preserve distance between points. So, for example, if I have points A and B, a rigid transformation could
be something like a translation because after I've translated
them, notice the distance between the points is still the same. It could be like that. It includes rotation. Let's say I rotated about point
A as the center of rotation. That still would not change,
that still would not change my distance between points A and B. It could even be things like
taking the mirror image. Once again, that's not
going to change the distance between A and B. What's not a rigid transformation? Well, one thing you might
imagine is dilating, scaling it up or down. That is going to change the distance, so rigid transformation
are any transformations that preserve the distance between points. Now, another idea is congruence, and in the context of this
video, we're going to be viewing the definition of congruence
as two figures are congruent if and only if there exists a series of rigid transformations
which will map one figure onto the other. You might see other definitions
of congruence in your life, but this is the rigid
transformation definition of congruence that we will
use, and we're going to use these two definitions
to prove the following, to prove that saying two
segments are congruent is equivalent to saying that
they have the same length. So, let me get some
space here to do that in. So first, let me prove that if segment AB is congruent to segment CD, then the length of segment AB, which we'll just denote as
AB without the line over it, is equal to the length of segment CD. How do we do that? Well, the first thing to realize is if AB, if AB is congruent to CD, then AB can be mapped onto CD with rigid transformations,
rigid transformations. That comes out of the
definition of congruence. And then we could say, "Since "the transformations are rigid, "distance is preserved, "preserved," and so, that would imply that the distance between the points are going to be the same. AB, the distance between
points AB, or the length of segment AB, is equal to
the length of segment CD. That might almost seem
too intuitive for you, but that's all we're talking about. So, now, let's see if we
can prove the other way. Let's see if we can prove that
if the length of segment AB is equal to the length of
segment CD, then segment AB is congruent to segment
CD, and let me draw them right over here, just to, so,
let's say I have segment AB right over there, and
I'll draw another segment that has the same length,
so maybe it looks something like this, and this is
obviously hand-drawn. So, then, let's call this CD. So, in order to prove
this, I have to show, "Hey, if I have two segments
with the same length, "that there's always a set
of rigid transformations "that will map one segment onto the other, "which means, by definition,
they are congruent." So, let me just construct
those transformations. So, my first rigid
transformation that I could do is to translate, translate,
and I'll underline the name of the transformation,
segment AB, so that point A is on top of point C, or A is mapped onto C, and you could see that
there's always a translation to do that. It would be doing that, and
of course we would translate. B would end up like that, and
so, after this translation, it's going to be A right over there. A is going to be there,
and then B is going to be right over there. Now, the second step I
would do is then rotate AB about A, so A is the center of rotation, so I'm gonna rotate it so that point B lies on ray CD. Well, what does this transformation do? Well, since point A is
the center of rotation, A is going to stay mapped on top of C from our first translation,
but now B is rotated, so it sits on top of
the ray that starts at C and goes through D and keeps going, and where will B be on that ray? Well, since the distance
between B and A is the same as the distance between D and C, and A and C are the same point, and now B sits on that ray,
B will now sit right on top of D because AB is equal to CD. B will be mapped onto, onto D, and just like that, we've shown that if the
segment lengths are equal, there is always a set
of rigid transformations that will map one segment onto the other. Therefore, since A and B have
been mapped onto C and D, we know that A, that segment
AB is congruent to segment CD, and we are done. We have proven what we set
out to prove both ways.