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

In this unit, we will explore different methods for comparing and identifying identical triangles depending on the information we have available. Whether we compare pieces of furniture, automobiles, or pieces of a candy bar, we will have to make sure that objects are the same size and shape.

We will also study triangles and their parts to predict their measurements and distances between them. For example, we can use the process of triangulation to calculate a distance when we have two known locations. This can be useful when we need to calculate precise measurements for a land survey on a building construction site, determine distances for a race, or calculate the correct angles for a rocket launch.

**Completing this unit should take you approximately 5 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;
- use theorems to calculate missing lengths of a triangle; and
- calculate the measure of a bisected angle.

### 3.1: Triangle Classification

Before we can begin comparing triangles, we need to be able to classify triangles based on their angles and sides.

Read this article and watch the videos to learn how to classify triangles based on their angles and sides. Pay attention to the list of different types of triangles. It may be helpful to make a list of these different classifications of triangles and their characteristics.

### 3.2: Triangle Sum Theorem

The triangle sum theorem states that the sum of the interior angles in a triangle add to 180 degrees. This theorem is useful when we need to solve for an unknown angle in a given triangle.

Watch this video to see why the triangle sum theorem works.

Then, complete this assessment and check your answers.

### 3.3: Exterior Angle Theorem

The exterior angle theorem states that an exterior angle of a triangle equals the sum of its nonadjacent interior angles. Like the triangle sum theorem, it allows us to determine unknown angles in a given triangle.

Watch this video to see an example of how to use the exterior angle theorem.

Then, complete this assessment and check your answers.

### 3.4: Congruent Figures

Congruent figures are figures that are exactly the same shape and size.

Read this article and watch the video. Pay attention to the steps you need to take to determine if two figures are congruent.

### 3.5: Congruent Triangles and SSS

We can use the side-side-side (SSS) postulate for triangles to determine whether triangles are congruent. The SSS postulate states that if the three sides of two triangles are congruent, then the triangles are congruent.

Watch this video to learn how to use the SSS postulate to determine if two triangles are congruent.

### 3.6: Other Triangle Congruence Postulates

While the SSS postulate is useful, sometimes we do not have enough information about the sides of two triangles to use it. We can use other postulates depending on the given information for a pair of triangles. These additional postulates include side-angle-side (SAS), angle-side-angle (ASA), and angle-angle-side (AAS).

Watch this video for definitions of these postulates and examples of how to use them.

### 3.7: Calculating Congruent Triangles

Now that we have explored the different ways to determine whether two triangles are congruent, we will use what we learned to solve congruence problems.

Watch this video for examples of how to use different postulates to determine the congruence between triangles.

Then, complete this assessment and check your answers.

Complete this assessment and check your answers.

### 3.8: Isosceles Triangles

Isosceles triangles are triangles that have at least two congruent sides.

Read this article and watch the videos. Pay attention to the terminology that it introduces about isosceles triangles. The base angle theorem states that the base angles of an isosceles triangle must be congruent. Do not focus on the proof. Pay attention to the examples of how to recognize congruent angles, measure vertex angles, and measure base angles.

Then, complete review questions 1–5 and check your answers.

### 3.9: Equilateral Triangles

Equilateral triangles are the last special type of triangle we explore in this unit. In an equilateral triangle, all sides and angles are congruent.

Read this article and watch the videos. Pay attention to the equilateral triangle theorem and examples 3–5.

Then, complete review questions 1–4 and check your answers.