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Chapter 7: Similarity

Difficulty Level: At Grade Created by: CK-12

Introduction

In this chapter, we will start with a review of ratios and proportions. Second, we will introduce the concept of similarity. Two figures are similar if they have the same shape, but not the same size. We will apply similarity to polygons, quadrilaterals and triangles. Then, we will extend this concept to proportionality with parallel lines and dilations. Finally, there is an extension about self-similarity, or fractals, at the end of the chapter.

Chapter Outline

Summary

This chapter is all about proportional relationships. It begins by introducing the concept of ratio and proportion and detailing properties of proportions. It then focuses on the geometric relationships of similar polygons. Applications of similar polygons and scale factors are covered. The AA, SSS, and SAS methods of determining similar triangles are presented and the Triangle Proportionality Theorem is explored. The chapter wraps up with the proportional relationships formed when parallel lines are cut by a transversal, similarity and dilated figures, and self-similarity.

Chapter Keywords

• Ratio
• Proportion
• Means
• Extremes
• Cross-Multiplication Theorem
• Similar Polygons
• Scale Factor
• AA Similarity Postulate
• Indirect Measurement
• SSS Similarity Theorem
• SAS Similarity Theorem
• Triangle Proportionality Theorem
• Triangle Proportionality Theorem Converse
• Transformation
• Rigid Transformation
• Non-rigid Transformation
• Dilation
• Self-Similar
• Fractal

Chapter Review

1. Solve the following proportions.
1. x+33=102\begin{align*}\frac{x+3}{3}=\frac{10}{2}\end{align*}
2. 85=2x1x+3\begin{align*}\frac{8}{5}=\frac{2x-1}{x+3}\end{align*}
2. The extended ratio of the angle in a triangle are 5:6:7. What is the measure of each angle?
3. Rewrite 15 quarts in terms of gallons.

Determine if the following pairs of polygons are similar. If it is two triangles, write why they are similar.

1. Draw a dilation of A(7,2),B(4,9),\begin{align*}A(7, 2), B(4, 9),\end{align*} and C(1,4)\begin{align*}C(-1, 4)\end{align*} with k=32\begin{align*}k=\frac{3}{2}\end{align*}.

Algebra Connection Find the value of the missing variable(s).

Texas Instruments Resources

In the CK-12 Texas Instruments Geometry FlexBook, there are graphing calculator activities designed to supplement the objectives for some of the lessons in this chapter. See http://www.ck12.org/flexr/chapter/9692.

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