Congruence Transformation: Definition & Theorems

Instructor: Laura Pennington

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

Congruence transformations happen all around us. This lesson will define congruence transformations and explore a couple of interesting theorems about these types of transformations as well as look at examples.

Congruence Transformations

We're all aware that when we look in the mirror, we see ourselves, or an image of ourselves that looks exactly the same as we do. This activity could be compared to taking ourselves and reflecting ourselves over the mirror so that we are staring back at ourselves. In mathematics, this is called a reflection, and it's an example of a congruence transformation.


contrans1


We say that two objects are congruent if they have the same shape and size. For instance, our reflections in a mirror have the same shape and size as we do, so we would say that we are congruent to our reflection in a mirror.

As we said, our reflection in the mirror can be thought of as reflecting, or flipping, ourselves over the mirror so we are staring back at ourselves. In other words, we can move an original object, ourselves, in such a way that we would fit exactly over our image in the mirror. This is another way to define congruent objects. That is, two objects are congruent if we can move one of the objects, without changing its shape or size, in such a way that it fits exactly over the other image, and we call these movements congruence transformations.

Congruence transformations are transformations performed on an object that create a congruent object. There are three main types of congruence transformations:

  1. Translation (a slide)
  2. Rotation (a turn)
  3. Reflection (a flip)


contrans2


We can use these three transformations to determine if two objects are congruent by seeing if we can get from one of the objects to the other using only these congruence transformations.

For example, consider the two rectangles shown in the image.


contrans3


Notice that if we slide rectangle 1 in an up and right direction, it ends up exactly over rectangle 2. Therefore, we can obtain rectangle 2 by taking rectangle 1 through a congruence transformation (translation), so the two rectangles are congruent.

There are many theorems that have to do with congruence transformations. Let's take a look at a couple of really interesting ones!

Reflections in Parallel Lines Theorem

The first theorem that we're going to take a look at is the reflections in parallel lines theorem. This theorem states that if two lines, l1 and l2, are parallel and you reflect a shape over l1 and then over l2, the result is the same as a translation of the original shape. Furthermore, that translation is in the direction that is perpendicular to the parallel lines, and is two times the distance between the parallel lines.

For example, suppose two lines, l1 and l2, are parallel. Now consider taking a triangle and reflecting it over l1, and then taking that image and reflecting it over l2.


contran7capitalT


Notice that doing this could also be achieved by translating the original triangle to the right. As the reflections in parallel lines theorem states, this will always be the case.

Well, that's pretty neat! Let's consider another theorem involving congruence transformations.

Reflections in Intersecting Lines Theorem

This theorem is similar to the reflections in parallel lines theorem, but it has to do with reflecting over intersecting lines instead of parallel lines, so the result is different. The reflections in intersecting lines theorem states that if two lines, l1 and l2, intersect one another, and we reflect a shape over l1 and then over l2, the result is the same as a rotation of the original shape. This rotation is around the point of intersection of the two lines, and covers an angle that is two times the angle between the intersecting lines.

For instance, suppose two lines, l1 and l2, intersect at some point, and we reflect a triangle over l1, and then we reflect that image over l2.


contrans8intersection


We can see that we can also achieve the final image by rotating the original triangle around the intersection point of the two lines.

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