Similar Triangles

Definition

Similar triangles are triangles that have the same shape but may differ in size. They have corresponding angles that are congruent and corresponding sides that are proportional.

5 Must Know Facts For Your Next Test

1. The Angle-Angle-Angle (AAA) criterion states that if the corresponding angles of two triangles are equal, then the triangles are similar.
2. The Side-Angle-Side (SAS) criterion states that if two pairs of corresponding sides of two triangles are proportional and the included angles are equal, then the triangles are similar.
3. The Side-Side-Side (SSS) criterion states that if the corresponding sides of two triangles are proportional, then the triangles are similar.
4. Similar triangles can be used to solve problems involving scale, such as finding the height of an inaccessible object or the distance to a remote location.
5. The properties of similar triangles are often applied in the context of solving proportions and their applications, such as finding unknown side lengths or angle measures.

Review Questions

• Explain how the concept of similar triangles is applied in solving proportions.
  • The concept of similar triangles is central to solving proportions because the corresponding sides of similar triangles are proportional. By identifying pairs of similar triangles within a problem, you can set up proportions between the known and unknown side lengths to solve for the missing values. This allows you to find unknown side lengths or angle measures by using the proportional relationships between the similar triangles.
• Describe how the criteria for similar triangles (AAA, SAS, SSS) can be used to determine if two triangles are similar.
  • The three main criteria for determining if two triangles are similar are the Angle-Angle-Angle (AAA) criterion, the Side-Angle-Side (SAS) criterion, and the Side-Side-Side (SSS) criterion. The AAA criterion states that if the corresponding angles of two triangles are equal, then the triangles are similar. The SAS criterion states that if two pairs of corresponding sides are proportional and the included angles are equal, then the triangles are similar. The SSS criterion states that if the corresponding sides of two triangles are proportional, then the triangles are similar. Applying these criteria allows you to identify similar triangles, which is essential for solving proportions and their applications.
• Analyze how the properties of similar triangles can be used to solve real-world problems, such as finding the height of an inaccessible object or the distance to a remote location.
  • The properties of similar triangles, particularly the proportionality of corresponding sides, can be leveraged to solve a variety of real-world problems. For example, by setting up similar triangles formed by an inaccessible object, its shadow, and a known reference object, you can use the proportional relationships to calculate the height of the inaccessible object. Similarly, by identifying similar triangles formed by a remote location, a known reference point, and your position, you can use the proportional side lengths to determine the distance to the remote location. These applications of similar triangles are crucial for solving proportions and their practical uses in fields such as surveying, engineering, and navigation.

Related terms

Proportionality: The state of being proportional, where the ratios between corresponding sides of similar triangles are equal.

Congruent Angles: Angles that are equal in measure, which is a necessary condition for similar triangles.

Scale Factor: The ratio of the lengths of corresponding sides in similar triangles, which determines the relative size of the triangles.