# Ch 7: High School Geometry: Similar Polygons

### About This Chapter

## Similar Polygons - Chapter Summary and Learning Objectives

What do similar polygons have in common? You can learn all about the theorems, proportions, properties and constructions for working with triangles. Don't be surprised if you discover some applications for everyday life. In this chapter, you can meet the AAA postulate, the AA criterion and the transitive property of similar triangles while learning to prove theorems and converse theorems in addition to identifying special-case scenarios. After completing these lessons, you should be able to:

- Characterize proportion and ratio
- Provide examples of solutions involving proportions
- Identify similar polygons
- Establish the AA criterion while using the properties of similarity transformations
- Understand the transitive property of similar triangles
- Apply the Triangle Proportionality Theorem
- Give examples of similar polygon constructions
- Supply theorem proofs about median of right triangles
- Apply the Pythagorean Theorem
- Supply the proof of the converse Pythagorean Theorem and describe special instances of the Theorem

Video | Objective |
---|---|

Ratio, Proportion and Geometric Mean | Define ratio and proportion. Define and state theorems about proportions and geometric mean. |

Solving Problems Involving Proportions | Define proportional segments. Explain how to solve problems involving proportions. Provide examples. |

Similar Polygons: Definition and Examples | Define similar polygons. Provide examples. State the AAA postulate. |

Similar Triangles and the AA Criterion | Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. |

The Transitive Property of Similar Triangles | Discuss the transitive property of similar triangles. |

Triangle Proportionality Theorem | Define the Triangle Proportionality Theorem. Explain how to divide a triangle proportionally. Provide examples. Describe how to construct a segment proportional to three given line segments. |

Constructing Similar Polygons | Explain how to construct a polygon similar to a given polygon. Provide examples. |

Properties of Right Triangles | Use theorems involving an altitude of a right triangle. Prove theorem about median of right triangle. |

The Pythagorean Theorem: Practice and Application | Learn about the Pythagorean Theorem and its real-life applications. |

The Pythagorean Theorem: Converse and Special Cases | Prove the converse of the Pythagorean Theorem. Identify special cases of the Pythagorean Theorem. |

### 1. Ratios and Proportions: Definition and Examples

Watch this video lesson to learn how ratios and proportions are related. Also, learn how ratios and proportions are used in real life and how you can apply them to yours.

### 2. Geometric Mean: Definition and Formula

Watch this video lesson to learn how different the geometric mean is from the arithmetic mean, or average, that we are all so familiar with. Learn in what circumstance the geometric mean is preferred. Also, learn the formula and how to use it.

### 3. Angle Bisector Theorem: Definition and Example

Watch this video lesson and learn how you can use the angle bisector theorem to help you find the lengths of missing sides. Also learn how you can identify an angle bisector just by looking at the lengths of the sides of a triangle.

### 4. Solving Problems Involving Proportions: Definition and Examples

Learn how you can use proportions to find missing sides and how to increase or decrease the sizes of objects to create models of them in this video lesson.

### 5. Similar Polygons: Definition and Examples

Watch this video to learn what makes two polygons similar to each other. You'll also learn the one distinguishing aspect of similar polygons and see an example of how to compare two polygons.

### 6. The Transitive Property of Similar Triangles

Watch this video lesson and you will understand what the transitive property is and how it applies to similar triangles. Watch as we apply the transitive property to three similar triangles.

### 7. Triangle Proportionality Theorem

Watch this video lesson to learn all about the triangle proportionality theorem and how you can use this interesting theorem to help you solve problems. Learn how a parallel line can create sides that are proportional to each other.

### 8. Constructing Similar Polygons

This lesson will teach you how to construct similar polygons, which are very useful in many quantitative fields. You'll learn that all of this has something to do with, of all things, snowflakes!

### 9. Properties of Right Triangles: Theorems & Proofs

In this lesson, you will learn about the properties of and theorems associated with right triangles, which have a wide range of applications in math and science. Specifically, we will discuss and prove the Pythagorean theorem and the right triangle altitude theorem. Let's get started.

### 10. The Pythagorean Theorem: Practice and Application

The Pythagorean theorem is one of the most famous geometric theorems. Written by the Greek mathematician Pythagoras, this theorem makes it possible to find a missing side length of a right triangle. Learn more about the famous theorem here and test your understanding with a quiz.

### 11. The Pythagorean Theorem: Converse and Special Cases

The Pythagorean Theorem is a famous theorem for right triangles. Watch this video to learn how the Pythagorean Theorem relates to the law of cosines and how the converse of the Pythagorean Theorem can help you identify right triangles.

### 12. Similar Triangles & the AA Criterion

This video lesson explains the AA criterion and shows you why you only need two corresponding angles to be equal for two triangles to be similar. You will also see how you can use it to help you identify similar triangles.

### 13. What is a Polygon? - Definition, Shapes & Angles

Polygons are everywhere! In this lesson, you will learn what they are and what they look like. You will also learn about a special class of polygons and how to find its angles.

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### Other Chapters

Other chapters within the Geometry: High School course

- High School Geometry: Foundations of Geometry
- High School Geometry: Logic in Mathematics
- High School Geometry: Introduction to Geometric Figures
- High School Geometry: Properties of Triangles
- High School Geometry: Triangles, Theorems and Proofs
- High School Geometry: Parallel Lines and Polygons
- High School Geometry: Quadrilaterals
- High School Geometry: Circular Arcs and Circles
- High School Geometry: Conic Sections
- High School Geometry: Geometric Solids
- High School Geometry: Analytical Geometry
- High School Geometry: Probability
- High School Geometry: Introduction to Trigonometry
- Teacher Resources for High School Geometry