# Chapter 4: Triangles and Congruence

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

## Introduction

In this chapter, you will learn all about triangles. First, we will learn about the properties of triangles and the angles within a triangle. Second, we will use that information to determine if two different triangles are congruent. After proving two triangles are congruent, we will use that information to prove other parts of the triangles are congruent as well as the properties of equilateral and isosceles triangles.

## Chapter Outline

- 4.1. Triangle Sum Theorem
- 4.2. Exterior Angles Theorems
- 4.3. Congruent Triangles
- 4.4. Congruence Statements
- 4.5. Third Angle Theorem
- 4.6. SSS Triangle Congruence
- 4.7. SAS Triangle Congruence
- 4.8. ASA and AAS Triangle Congruence
- 4.9. HL Triangle Congruence
- 4.10. Isosceles Triangles
- 4.11. Equilateral Triangles

### Chapter Summary

## Summary

This chapter begins with the Triangle Sum Theorem, showing that the sum of the angles in a triangle is a constant. The definition of congruency is presented and from that foundation the chapter presents other important theorems related to congruent triangles, such as the Third Angle Theorem and the SSS, SAS, ASA, AAS and HL Triangle Congruency Theorems.

### Chapter Keywords

- Interior Angles
- Vertex
- Triangle Sum Theorem
- Exterior Angle
- Exterior Angle Sum Theorem
- Remote Interior Angles
- Exterior Angle Theorem
- Congruent Triangles
- Third Angle Theorem
- Reflexive Property of Congruence
- Symmetric Property of Congruence
- Transitive Property of Congruence
- Side-Side-Side (SSS) Triangle Congruence Postulate
- Included Angle
- Side-Angle-Side (SAS) Triangle Congruence Postulate
- Angle-Side-Angle (ASA) Congruence Postulate
- Angle-Angle-Side (AAS or SAA) Congruence Theorem
- HL Congruence Theorem
- Base Angles Theorem
- Isosceles Triangle Theorem
- Base Angles Theorem Converse
- Isosceles Triangle Theorem Converse
- Equilateral Triangles Theorem

### Chapter Review

For each pair of triangles, write what needs to be congruent in order for the triangles to be congruent. Then, write the congruence statement for the triangles.

- HL
- ASA
- AAS
- SSS
- SAS

Using the pictures below, determine which theorem, postulate or definition that supports each statement below.

- \begin{align*}m \angle 1+m \angle 2=180^\circ\end{align*}
- @$\begin{align*}\angle 5 \cong \angle 6\end{align*}@$
- @$\begin{align*}m \angle 1=m \angle 4+ m \angle 3\end{align*}@$
- @$\begin{align*}m \angle 8 = 60^\circ\end{align*}@$
- @$\begin{align*}m \angle 5+m \angle 6+m \angle 7=180^\circ\end{align*}@$
- @$\begin{align*}\angle 8 \cong \angle 9 \cong \angle 10\end{align*}@$
- If @$\begin{align*}m \angle 7 = 90^\circ\end{align*}@$, then @$\begin{align*}m \angle 5 = m \angle 6 = 45^\circ\end{align*}@$

### 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/9689.*