# 9.6: Inscribed Quadrilaterals in Circles

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

**Practice**Inscribed Quadrilaterals in Circles

### Inscribed Quadrilaterals in Circles

An **inscribed polygon** is a polygon where every vertex is on a circle. Note, that not every quadrilateral or polygon can be inscribed in a circle. Inscribed quadrilaterals are also called *cyclic quadrilaterals.* For these types of quadrilaterals, they must have one special property. We will investigate it here.

#### Investigation: Inscribing Quadrilaterals

Tools Needed: pencil, paper, compass, ruler, colored pencils, scissors

- Draw a circle. Mark the center point
A . - Place four points on the circle. Connect them to form a quadrilateral. Color the 4 angles of the quadrilateral 4 different colors.
- Cut out the quadrilateral. Then cut the quadrilateral into two triangles, by cutting on a diagonal.
- Line up
∠B and∠D so that they are adjacent angles. What do you notice? What does this show?

This investigation shows that the opposite angles in an inscribed quadrilateral are supplementary. By cutting the quadrilateral in half, through the diagonal, we were able to show that the other two angles (that we did not cut through) formed a linear pair when matched up.

**Inscribed Quadrilateral Theorem:** A quadrilateral is inscribed in a circle if and only if the opposite angles are supplementary.

#### Solving for Unknown Values

1. Find the value of the missing variable.

2. Find the value of the missing variable.

It is easiest to figure out

3. Find

The opposite angles are supplementary. Set up an equation for

### Examples

Quadrilateral

First, note that

#### Example 1

Find

#### Example 2

Find

#### Example 3

Find

#### Example 4

Find

### Review

Fill in the blanks.

- A
(n) _______________ polygon has all its vertices on a circle. - The _____________ angles of an inscribed quadrilateral are ________________.

Quadrilateral

m∠DBC mBCˆ mABˆ m∠ACD m∠ADC m∠ACB

Find the value of

Solve for

Use the diagram below to find the measures of the indicated angles and arcs in problems 14-19.

m∠EBO m∠EOB mBCˆ - \begin{align*}m \angle ABO\end{align*}
- \begin{align*}m \angle A\end{align*}
- \begin{align*}m \angle EDC\end{align*}

### Review (Answers)

To view the Review answers, open this PDF file and look for section 9.6.

### Notes/Highlights Having trouble? Report an issue.

Color | Highlighted Text | Notes | |
---|---|---|---|

Please Sign In to create your own Highlights / Notes | |||

Show More |

Term | Definition |
---|---|

central angle |
An angle formed by two radii and whose vertex is at the center of the circle. |

chord |
A line segment whose endpoints are on a circle. |

diameter |
A chord that passes through the center of the circle. The length of a diameter is two times the length of a radius. |

inscribed angle |
An angle with its vertex on the circle and whose sides are chords. |

intercepted arc |
The arc that is inside an inscribed angle and whose endpoints are on the angle. |

Inscribed Polygon |
An inscribed polygon is a polygon with every vertex on a given circle. |

Inscribed Quadrilateral Theorem |
The Inscribed Quadrilateral Theorem states that a quadrilateral can be inscribed in a circle if and only if the opposite angles of the quadrilateral are supplementary. |

Cyclic Quadrilaterals |
A cyclic quadrilateral is a quadrilateral that can be inscribed in a circle. |

### Image Attributions

Here you'll learn properties of inscribed quadrilaterals in circles and how to apply them.

## Concept Nodes:

**Save or share your relevant files like activites, homework and worksheet.**

To add resources, you must be the owner of the Modality. Click Customize to make your own copy.