# Similarity

Contents

Learn what it means for two figures to be similar, and how to determine whether two figures are similar or not. Use this concept to prove geometric theorems and solve some problems with polygons.

## Definitions of similarity

In this tutorial, you will learn how we determine whether two figures are similar.

2:11

Similar shapes & transformations

Sal is given pairs of polygons, and then he determines whether they are similar by trying to map one onto the other using angle-preserving transformations.

Exercise

Similarity & transformations

Given two polygons, try to map one onto the other using angle-preserving transformations, and determine whether they are similar. Transformations are done in "intuitive mode."

Exercise

Similarity & transformations (advanced)

Given two polygons, try to map one onto the other using angle-preserving transformations, and determine whether they are similar. Transformations are done in "formal mode."

6:40

Angles of similar quadrilaterals

Sal solves a problem about the conditions for two quadrilaterals to be similar, based on the definition that two polygons are similar if all their angles are congruent.

## Introduction to triangle similarity

In this tutorial, we will use what we learned about similarity of polygons in the specific case of triangles. We will come up with specific conditions for triangles to be similar, and use them to determine whether two triangles are similar.

9:17

Intro to triangle similarity

Sal explains what it means for triangles to be similar, and how this follows from the definition of similarity.

12:14

Triangle similarity postulates/criteria

Sal reviews all the different ways we can determine that two triangles are similar. This is similar to the congruence criteria, only for similarity!

Exercise

Determine similar triangles: AA

Given two triangles with some of their angle measures, determine whether the triangles are similar or not.

9:23

Determining similar triangles

Multiple examples where Sal analyzes the similarity of triangles.

Exercise

Determine similar triangles: SSS

Given two triangles with some of their side measures, determine whether the triangles are similar or not.

## Solving similar triangles

Learn how to find missing measures of triangles using the fact that they are similar to other triangles.

Exercise

Solve similar triangles (basic)

Given two similar triangles and some of their side lengths, find a missing side length.

7:37

Solving similar triangles

Sal solves two problems where a missing side length is found by proving that triangles are similar and using this to find the measure.

Exercise

Solve similar triangles (advanced)

Given two similar triangles and some of their side lengths, find a missing side length.

6:05

Solving similar triangles: same side plays different roles

Sal finds a missing side length ni a problem where the same side plays different roles in two similar triangles.

## Angle bisector theorem

Learn about the angle bisector theorem, which makes a clever use of triangle similarity in order to find a special proportion that holds for angle bisectors of triangles.

8:35

Intro to angle bisector theorem

Sal introduces the angle-bisector theorem and proves it.

3:04

Using the angle bisector theorem

Sal uses the angle bisector theorem to solve for sides of a triangle.

Exercise

Solve triangles: angle bisector theorem

Use the angle bisector theorem to find missing side lengths in triangles.

## Solving problems with similar and congruent triangles

You already saw how we can use similarity and congruence to find missing measures in triangles. Let's do the same now to solve more kinds of polygons, and more challenging problems!

10:07

Using similar & congruent triangles

Sal uses the similarity of triangles and the congruence of others in this multi-step problem to find the area of a polygon.

Exercise

Use similar & congruent triangles

Solve geometry problems with various polygons by using all you know about similarity and congruence.

9:53

Challenging similarity problem

Interesting similarity problem where we don't have a lot of information to work with

## Solving modeling problems with similar and congruent triangles

Now that you are an expert with solving geometry problems using congruence and similarity, you are ready to solve some real-world problems!

8:23

Geometry word problem: the golden ratio

Triangle similarity, the Golden ratio, and art, all converge in this inspiring video!

6:23

Geometry word problem: Earth & Moon radii

Similarity and the Golden ratio join forces again, this time to save the Earth! (Or maybe just finding its radius)

6:49

Geometry word problem: a perfect pool shot

Sal uses triangle similarity to plan the perfect shot in a pool game.