# Chapter 5: Relationships with Triangles

**At Grade**Created by: CK-12

## Introduction

This chapter introduces different segments within triangles and how they relate to each other. We will explore the properties of midsegments, perpendicular bisectors, angle bisectors, medians, and altitudes. Next, we will look at the relationship of the sides of a triangle, how they relate to each other and how the sides of one triangle can compare to another.

- 5.1.
## Midsegment Theorem

- 5.2.
## Perpendicular Bisectors

- 5.3.
## Angle Bisectors in Triangles

- 5.4.
## Medians

- 5.5.
## Altitudes

- 5.6.
## Comparing Angles and Sides in Triangles

- 5.7.
## Triangle Inequality Theorem

- 5.8.
## Indirect Proof in Algebra and Geometry

### Chapter Summary

## Summary

This chapter begins with an introduction to the Midsegment Theorem. The definition of a perpendicular bisector is presented and the Perpendicular Bisector Theorem and its converse are explored. Now that the bisectors of segments have been discussed, the definition of an angle bisector is next and the Angle Bisector Theorem and its converse are presented. The properties of medians and altitudes of triangles are discussed in detail. The entire chapter builds to a discovery of the relationships between the angles and sides in triangles as a foundation for the Triangle Inequality Theorem. The chapter ends with a presentation of indirect proofs.

### Chapter Keywords

- Midsegment
- Midsegment Theorem
- Perpendicular Bisector Theorem
- Perpendicular Bisector Theorem Converse
- Point of Concurrency
- Circumcenter
- Concurrency of Perpendicular Bisectors Theorem
- Angle Bisector Theorem
- Angle Bisector Theorem Converse
- Incenter
- Concurrency of Angle Bisectors Theorem
- Median
- Centroid
- Concurrency of Medians Theorem
- Altitude
- Orthocenter
- Triangle Inequality Theorem
- SAS Inequality Theorem
- SSS Inequality Theorem
- Indirect Proof

### Chapter Review

If \begin{align*}C\end{align*} and \begin{align*}E\end{align*} are the midpoints of the sides they lie on, find:

- The perpendicular bisector of \begin{align*}\overline{FD}\end{align*}.
- The median of \begin{align*}\overline{FD}\end{align*}.
- The angle bisector of \begin{align*}\angle FAD\end{align*}.
- A midsegment.
- An altitude.
- Trace \begin{align*}\triangle FAD\end{align*} onto a piece of paper with the perpendicular bisector. Construct another perpendicular bisector. What is the point of concurrency called? Use this information to draw the appropriate circle.
- Trace \begin{align*}\triangle FAD\end{align*} onto a piece of paper with the angle bisector. Construct another angle bisector. What is the point of concurrency called? Use this information to draw the appropriate circle.
- Trace \begin{align*}\triangle FAD\end{align*} onto a piece of paper with the median. Construct another median. What is the point of concurrency called? What are its properties?
- Trace \begin{align*}\triangle FAD\end{align*} onto a piece of paper with the altitude. Construct another altitude. What is the point of concurrency called? Which points of concurrency can lie outside a triangle?
- A triangle has sides with length \begin{align*}x + 6\end{align*} and \begin{align*}2x - 1\end{align*}. Find the range of the third side.

### Texas Instruments Resources

*In the CK-12 Texas Instruments Geometry FlexBook, there are graphing calculator activities designed to supplement the objectives for some of the lessons in this chapter. See http://www.ck12.org/flexr/chapter/9690.*