TCS - Similarity and Congruence Lesson

Similarity and Congruence

What is the difference between similarity and congruence? At first glance, these words may seem very much the same. However, after a closer look, you'll see that there are some very specific differences and that these words cannot be used synonymously. Similarity means that two things are similar. Similar means that some things are the same and some things are not. For example, you may look similar to your brother or your father, but you don't look exactly like either of them in every way. The same is true when figures are similar. They have some characteristics about them that are the same, making them similar, but they are not exact duplicates of one another. In mathematics, we say that two things are similar if they have the same shape. They do not, however, have to be the same size.   Therefore, we can also say that similar shapes are proportional to each other. Here is an example of two squares that are similar. They are both squares, but they are not the same size, only the same shape.

Now, how does the word congruence differ from the word similar? Congruent means exactly the same. Figures that are congruent would have the same shape AND the same size. All measurements for one would be the exact same measurements for the other.

Here is an example of two congruent squares. The squares have the same size and the same shape. All measurements are equal.

Now that we understand the difference between similarity and congruence, let's take a look at transformations that yield congruency or similarity.

When figures are translated, rotated, or reflected, the resulting image is congruent to the preimage. View examples of each of these three in the activity below.

When an object undergoes a dilation, the resulting image is similar to the preimage. A dilation is essentially a shrinking or stretching of an object. The shape stays the same and the "new" figure is proportional to the original one, i.e. similar. This constant by which the figure is dilated is called the scale factor.

View THIS VIDEO Links to an external site. to gain a better understanding of reflections and translations. Now, you practice some: 

Let's experiment with sequences that exhibit similarity or congruence given two similar or congruent two ‐ dimensional figures. Remember, the only transformation we've discussed that does not lend itself to congruence is dilation. Any series of translations, reflections, and rotations will yield a congruent figure. Similarly, any series of transformations involving translations, reflections, rotations, and dilations will yield similar figures. Visit the three links in the More Resources sidebar and answer the questions given. You'll be a transformation mapping expert when you're done!

Assignment: Homework Set 1

Now that you have spent some time learning about transformations, you are ready to complete your first homework set.   Click here to download the Transformation, Congruence & Similarity: Lesson 1 Homework, Homework Set 1. Links to an external site.

Once you have completed Homework Set 1, make sure you ask your teacher if you have any questions.   When you feel confident in your work, you'll need to take the Similarity and Congruence Homework Quiz over Unit 1, Lesson 1: Similarity and Congruence.  This is a timed quiz so make sure you understand the homework practice questions before you begin.

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