### Unit 3: Triangles, Congruence, and Other Relationships

Making sure that objects are of equal size is important in life. Whether pieces of furniture, automobiles, or just pieces of a candy bar shared between siblings, we often have to make sure objects are the same size and shape.

Have you ever had to prove something to a child? Sometimes, the best way to do that is to give the child a series of logical statements and then show how it all summarizes to one conclusion. In this unit, you will focus on showing that two triangles are identical. There are different methods to doing this, and the one you use will depend upon the information you have available to you. It is nothing more than following a series of logical statements and then showing how the statements draw a conclusion.

In this unit, you will also learn more about triangles and their various parts. Triangles, believe it or not, are used more often in life than you might think. Triangles are often used to estimate distances. The process of triangulation has been used for over two thousand years; with this technique, we can use two known locations and determine the distance to a location that we can see but cannot necessarily get to, like a boat out in a lake. Today, we use triangulation for navigating boats, surveying land, launching model rockets, and other activities.

We can also use triangles to map things out. For example, city planners often use the circumcenter of a triangle, which is a point in the middle that is equally distant from all three sides. They might use this to determine the location of a major parking garage, so that it is the same distance from three different companies, or they might want to make sure that a city monument is in the middle of a town plaza.

If you live in a house, triangles are right over your heads. Construction workers use the midsegment of a triangle to help strengthen roof trusses when they build. If the truss is not supported properly, it could collapse, leading to lots of problems for the poor homeowner.

**Completing this unit should take you approximately 17 hours.**

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

- classify triangles by lengths and angles;
- determine the measures of a triangle's angles (interior and exterior);
- determine whether two triangles are congruent;
- find the length of a midsegment;
- use theorems to find missing lengths of a triangle;
- find the measure of a bisected angle; and
- determine if a given set of lengths makes a triangle.

### 3.1: Triangle Classification

Take notes as you watch the videos and read this section to learn about classifying triangles based on angles and sides. This section provides examples of triangle classification. Then, complete the interactive practice and "explore more" problems.

Complete this assessment, which tests your knowledge on triangle classification. You can review the concepts associated with these questions with the material in subunit 3.1.

### 3.2: Angle Properties of Triangles

### 3.2.1: Triangle Sum Theorem

Take notes as you watch the videos and read this section to learn about the sum of the interior angles of a triangle. This section provides examples of finding angles within a triangle. Then, complete exercises 1-15. These exercises will provide you with the opportunity to find the missing angle(s) of triangles. Once you have completed the exercises, check your answers against the linked answer key.

### 3.2.2: Exterior Angle Theorem

Take notes as you watch the videos and read this section to learn about the exterior angles and the relationship with triangles. This section provides examples of how to find the missing interior and exterior angles of a triangle. Then, complete exercises 1-12. These exercises will provide you with the opportunity to find the missing angles of triangles as well as exterior angles. Once you have completed the exercises, check your answers against the linked answer key.

### 3.2.3: Triangle Examples

Take notes as you watch these videos to learn how to find missing angles of triangles by applying various properties. Watch the presentations carefully to understand how to find the missing angles of multiple triangles.

Complete this assessment, which tests your knowledge on vertical angles. You can review the concepts associated with these questions with the material in sub-subunits 3.2.1 and 3.2.2.

### 3.3: Triangle Congruence

### 3.3.1: Congruent Figures

Study the "Congruent Figures" section. This section provides discussions on defining and writing congruence statements in triangles, the third angle theorem, and congruence properties. Then, complete the review questions. These exercises will provide you with the opportunity to identify congruent figures and explain why certain triangles are congruent.

### 3.3.2: Congruent Triangles and SSS

Take notes as you watch this video to review the properties of congruent triangles as well as the Side-Side-Side (SSS) Congruent Postulate for Triangles. Watch the presentation carefully two or three times to understand how to triangles are congruent by side-side-side.

Study this section. This section provides the reasons why two triangles are congruent by using SSS. Then, complete the practice questions, which will provide you with the opportunity to identify congruent triangles by using SSS.

### 3.3.3: Other Triangle Congruent Postulates

Take notes as you watch this video to prove triangles are congruent by using the following postulates: Side-Angle-Side (SAS), Angle-Side-Angle (ASA), and Angle-Angle-Side (AAS). Watch the presentation carefully two or three times to understand how two triangles are congruent by these three congruence properties.

Study this section. This section provides the reasons why two triangles are congruent by using ASA and AAS. Then, complete the review questions, which will provide you with the opportunity to identify congruent triangles by using ASA and AAS.

Read this section. This section provides you with the reasons why two triangles are congruent by using HL. Then, complete the review questions.

### 3.3.4: More on Why SSA Is Not a Postulate

Take notes as you watch this video to show why Side-Side-Angle cannot prove two triangles to be congruent. Watch the presentation carefully two or three times to understand how two triangles are not congruent by side-side-angle.

Read this section. This section provides the reasons why two triangles are not congruent by using AAA and SSA.

### 3.3.5: Finding Congruent Triangles

Take notes as you watch this video to determine which congruent triangle postulate proves two triangles to be congruent. Watch the presentation carefully two or three times to understand if two triangles are congruent by SSS, ASA, SAS, or AAS.

Complete this assessment, which tests your knowledge on congruent triangles using the triangle congruence postulates. You can review the concepts associated with these questions with the Khan videos in sub-subunits 3.3.2-3.3.5.

Complete this assessment, which tests your knowledge on congruent triangles using the triangle congruence postulates. You can review the concepts associated with these questions with the Khan videos in sub-subunits 3.3.2-3.3.5.

### 3.4: Isosceles Triangles

Take notes as you watch the videos and read this section to learn about the isosceles triangles angles and their properties. This section provides examples of how to find the missing sides and angles of an isosceles triangle. Then, complete the review questions.

Take notes as you watch this video to determine the missing sides and angles of an isosceles triangle. Watch the presentation carefully two or three times to understand on how to write algebraic equations to find the missing angles and sides of an isosceles triangle.

### 3.5: Equilateral Triangles

Take notes as you watch the videos and read this section to learn about the equilateral triangles angles and their properties. This section provides examples of find the missing sides and angles of an equilateral triangle. Then, complete exercises 1-15. These exercises will provide you with the opportunity to find the missing angles and sides of isosceles triangles. Once you have completed the exercises, check your answers against the linked answer key.