# Chapter 9: Circles

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

## Introduction

Finally, we dive into a different shape, circles. First, we will define all the parts of circles and then explore the properties of tangent lines, arcs, inscribed angles, and chords. Next, we will learn about the properties of angles within circles that are formed by chords, tangents and secants. Lastly, we will place circles in the coordinate plane, find the equations of, and graph circles.

- 9.1.
## Parts of Circles

- 9.2.
## Tangent Lines

- 9.3.
## Arcs in Circles

- 9.4.
## Chords in Circles

- 9.5.
## Inscribed Angles in Circles

- 9.6.
## Inscribed Quadrilaterals in Circles

- 9.7.
## Angles On and Inside a Circle

- 9.8.
## Angles Outside a Circle

- 9.9.
## Segments from Chords

- 9.10.
## Segments from Secants

- 9.11.
## Segments from Secants and Tangents

- 9.12.
## Circles in the Coordinate Plane

### Chapter Summary

## Summary

This chapter begins with vocabulary associated with the parts of circles. It then branches into theorems about tangent lines; properties of arcs and central angles; and theorems about chords and how to apply them. Inscribed angles and inscribed quadrilaterals and their properties are explored. Angles on, inside, and outside a circle are presented in detail and the subsequent relationships are used in problem solving. Relationships among chords, secants, and tangents are discovered and applied. The chapter ends with the connection between algebra and geometry as the equations of circles are discussed.

### Chapter Keywords

- Circle
- Center
- Radius
- Chord
- Diameter
- Secant
- Tangent
- Point of Tangency
- Congruent Circles
- Concentric Circles
- Tangent to a Circle Theorem
- Central Angle
- Arc
- Semicircle
- Minor Arc
- Major Arc
- Congruent Arcs
- Arc Addition Postulate
- Inscribed Angle
- Intercepted Arc
- Inscribed Angle Theorem
- Inscribed Polygon
- Standard Equation of a Circle

### Chapter Review

Match the description with the correct label.

- minor arc - A. \begin{align*}\overline{CD}\end{align*}
- chord - B. \begin{align*}\overline{AD}\end{align*}
- tangent line - C. \begin{align*}\overleftrightarrow{CB}\end{align*}
- central angle - D. \begin{align*}\overleftrightarrow{EF}\end{align*}
- secant - E. \begin{align*}A\end{align*}
- radius - F. \begin{align*}D\end{align*}
- inscribed angle - G. \begin{align*}\angle BAD\end{align*}
- center - H. \begin{align*}\angle BCD\end{align*}
- major arc - I. \begin{align*}\widehat{BD}\end{align*}
- point of tangency - J. \begin{align*}\widehat{BCD}\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/9694.*