### Unit 4: Triangle Relationships

In this unit, you will 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 9 hours.**

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

- 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.

### 4.1: Midsegment Theorem

Study this section which provides examples and definitions of the midsegments of a triangle. Then, complete the review questions. These questions will provide you with the opportunity to find the midsegments of a triangle.

### 4.2: Perpendicular Bisectors

Take notes as you read this section and watch the videos to learn about perpendicular bisectors and their properties. This section provides examples of find the missing sides and angles of an equilateral triangle. Then, complete the review exercises. These exercises will provide you with the opportunity to apply the properties of perpendicular bisectors.

### 4.3: Angle Bisectors in Triangles

Take notes as you read this section and watch the videos to learn about angle bisectors in triangles. This section provides examples of how to find the missing angles by using properties of an angle bisector. Then, complete the review exercises. These exercises will provide you with the opportunity to apply the properties of angle bisectors.

### 4.4: Medians

Study this section. This section provides examples and a definition of the median in a triangle. Then, complete the review questions, which will provide you with the opportunity to find the medians and the centroid of a triangle.

Take notes as you watch this video to determine the length of the median of a triangle. Watch the presentation carefully two or three times to understand the centroid and properties of a median.

### 4.5: Altitudes

Take notes as you watch the videos and read this section to learn about altitudes in triangles. Read the overview until you are able to understand altitudes within triangles. This reading also provides examples of how to find the altitude of a triangle. Then, complete the review questions and check your answers.

### 4.6: Triangle Inequalities

### 4.6.1: Triangle Inequality Theorem

Take notes as you watch the videos and read this section to learn about the triangle inequality theorem. This section provides applications of the triangle inequality theorem, which states that any side of a triangle must be equal to or less than the sum of the other two sides in order to form the triangle. Then, complete the practice problems. These exercises will provide you with the opportunity to apply the triangle inequality theorem.

### 4.6.2: Inequalities in Two Triangles

Watch the videos and read this section. This section provides examples and definitions of triangle inequality theorems. Then, complete the review questions. These questions will provide you with the opportunity to apply the inequality triangle theorems.

Complete this assessment, which tests your knowledge on the triangle inequality theorem. You can review the concepts associated with these questions with the content from sub-subunit 4.6.1 and 4.6.2.