# Chapter 6: Similarity

**At Grade**Created by: CK-12

Here you will start by reviewing another type of transformation, the dilation. You will learn that dilations combined with rigid transformations produce similar figures. You will then focus on similar triangles and learn three criteria for showing that two triangles are similar. Finally, you will use your knowledge of similar triangles to prove new theorems and solve application problems.

- 6.1.
## Dilations

- 6.2.
## Definition of Similarity

- 6.3.
## AA Triangle Similarity

- 6.4.
## SAS Triangle Similarity

- 6.5.
## SSS Triangle Similarity

- 6.6.
## Theorems Involving Similarity

- 6.7.
## Applications of Similar Triangles

### Chapter Summary

First you learned that dilations are an example of a non-rigid transformation. Dilations preserve angles, but not distance. You learned that dilations create similar figures, and similar figures have congruent corresponding angles and proportional corresponding sides. You learned three criteria for proving that two triangles are similar: AA, SAS, and SSS.

Using similar triangles, you proved the triangle proportionality theorem and the triangle angle bisector theorem. You also saw another proof of the Pythagorean Theorem that utilized similar triangles.

You looked at two special types of right triangles and learned that all triangles of each type will always be similar. All 30-60-90 triangles have sides in the ratio of \begin{align*}1:\sqrt{3}:2\end{align*} and all 45-45-90 triangles have sides in the ratio of \begin{align*}1:1:\sqrt{2}\end{align*}.

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