**Introduction to Geometry Ratio, Proportion, and Similarity**

**Lesson Summary**

In this lesson, you will learn how to write and simplify ratios. You will also learn how to determine whether two ratios are a proportion and how to use proportions to solve problems. In addition, you will learn how to determine whether two triangles are similar.

**R**atios and proportions have many applications. Architects use them when they make scale models of buildings. Interior designers use scale drawings of rooms to decide furniture size and placement. Similar triangles can be used to find indirect measurements. Measurements of distances such as heights of tall buildings and the widths of large bodies of water can be found using similar triangles and proportions. Let's begin by looking at ratios.

**What Is a Ratio?**

If you compare two quantities, then you have used a ratio. A *ratio* is the comparison of two numbers using division. The ratio of *x* to *y* can be written or x:y. Ratios are usually expressed in simplest form.

**What Is a Proportion?**

Since are both equal to , they are equal to each other. A statement that two ratios are equal is called a *proportion*. A proportion can be written in one of the following ways:

The first and last numbers in a proportion are called the *extremes*. The middle numbers are called the *means*.

**Means-Extremes Property**

In a proportion, the product of the means equals the product of the extremes.

**Solving Proportion Problems**

Proportions can also be used to solve problems. When three parts of a proportion are known, you can find the fourth part by using the *means-extremes property*.

**Triangle Similarity**

You can prove that two figures are similar by using the definition of *similar*. Two figures are similar if you can show that the following two statements are true:

- Corresponding angles are congruent.
- Corresponding sides are in proportion.

In addition to using the definition of similar, you can use three other methods for proving that two triangles are similar: the angle-angle postulate, the side-side-side postulate, and the side-angle-side postulate.

If you know the measurements of two angles of a triangle, can you find the measurement of the third angle? Yes, from Lesson 9, you know that the sum of the three angles of a triangle is 180°. Therefore, if two angles of one triangle are congruent to two angles of another triangle, then their third angles must also be congruent. This will help you understand the next postulate. You should know that the symbol used for similarity is ~.

Here are two more postulates you can use to prove that two triangles are similar:

Examples:Which postulate, if any, could you use to prove that each pair of triangles is similar?

Practice problems for these concepts can be found at: Geometry Ratio, Proportion, And Similarity Practice Questions.

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