### Unit 6: Similarity

In this unit, we will review how ratios and proportions connect with the concept of similarity. We frequently consider objects to be similar to one another due to their design, size, or some other quality. In geometry, however, strict rules apply for determining whether two shapes are similar.

**Completing this unit should take you approximately 5 hours.**

Upon successful completion of this unit, you will be able to:

- determine whether two triangles are similar;
- write and identify accurate similarity statements; and
- use proportions to solve problems.

### 6.1: Similar Polygons

In this section, we explore the precise meaning of similarity when describing polygons. We will look at triangles and then larger polygons.

First, watch this video, which describes the meaning of similarity and applies its definition to triangles, the simplest polygon.

Next, read this article and watch the videos, which also define similarity for polygons. The first video offers an example of using similarity to solve for unknown values in a polygon. Closely read example 1.

Then, complete review questions 14–18 and check your answers.

### 6.2: Similarity Postulates

When describing similarity, we can use a series of postulates to determine if two polygons are similar.

Watch this video to learn how to apply similarity postulates to triangles.

### 6.3: Angle-Angle Similarity and Indirect Measurement

There are different ways to determine similarity between triangles. One method is called angle-angle (AA) similarity which tells us that if two triangles have two congruent angles, the two triangles must be similar. If triangles have AA similarity, we can use indirect measurement to determine unknown measurements within one of the triangles.

Read this article and watch the videos, which define AA similarity. Examples 1–5 show how to apply the AA similarity postulate to solve for unknowns in a triangle.

Then, complete review questions 6, 7, 8 and 10 and check your answers.

### 6.4: Side-Side-Side and Side-Angle-Side Similarity

Other types of similarity are side-side-side (SSS) and side-angle-side (SAS). In SSS similarity, we see that if the sides of two triangles are proportional, the triangles are similar. In SAS similarity, we see that if two sides and one angle are proportional between two triangles, the two triangles are similar.

Read this article and watch the four embedded videos. Pay attention to the Side-Side-Side (SSS) Similarity Theorem and the examples of determining if two triangles are similar and solving for unknown values.

After you have reviewed the material, complete review questions 11–16 and check your answers.Next, read this article and watch the embedded video. Pay attention to the Side-Angle-Side (SAS) criterion for similarity and Examples 1–3, which explain how to apply this criterion to solve for unknown values.

After you have reviewed the material, complete review questions 5–10 and check your answers.

### 6.5: Triangle Proportionality, Parallel Lines and Transversals

After learning the similarity criteria for triangles, we can apply our knowledge to solve more complex problems involving triangles.

Watch this video to see examples of problems where the similarity criteria are used to determine side lengths and angles of triangles.

After you have watched the video, complete these assessments to test what you have learned and check your answers.

Complete this second set of assessments and check your answers.