# 6.15: Inscribed Quadrilaterals in Circles

- Page ID
- 5040

Quadrilaterals with every vertex on a circle and opposite angles that are supplementary.

An **inscribed polygon** is a polygon where every vertex is on the **circle**, as shown below.

For inscribed quadrilaterals in particular, the opposite angles will always be supplementary.

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

If \(ABCD\) is inscribed in \(\bigodot E\), then \(m\angle A+m\angle C=180^{\circ}\) and \(m\angle B+m\angle D=180^{\circ}\). Conversely, If \(m\angle A+m\angle C=180^{\circ}\) and \(m\angle B+m\angle D=180^{\circ}\), then \(ABCD\) is inscribed in \(\bigodot E\).

What if you were given a circle with a quadrilateral inscribed in it? How could you use information about the arcs formed by the quadrilateral and/or the quadrilateral's angle measures to find the measure of the unknown quadrilateral angles?

Example \(\PageIndex{1}\)

**Solution**

- \(\begin{aligned}

x+80^{\circ}&=180^{\circ} \qquad& y+71^{\circ}&=180^{\circ} \\

x&=100^{\circ} & y&=109^{\circ}

\end{aligned}\)

- \(\begin{aligned}

z+93^{\circ} &=180^{\circ} & x&=\frac{1}{2}\left(58^{\circ}+106^{\circ}\right) & y+82^{\circ}&=180^{\circ} \\

z &=87^{\circ} & x &=82^{\circ} & y&=98^{\circ}

\end{aligned}\)

Example \(\PageIndex{2}\)

Find \(x\) and \(y\) in the picture below.

**Solution**

\(\begin{array}{rlrl}

(7 x+1)^{\circ}+105^{\circ} & =180^{\circ} & (4 y+14)^{\circ}+(7 y+1)^{\circ} & =180^{\circ} \\

7 x+106^{\circ} & =180^{\circ} & 11 y+15^{\circ} & =180^{\circ} \\

7 x & =74 & 11 y & =165 \\

x & =10.57 & y&=15

\end{array}\)

Example \(\PageIndex{3}\)

Find the values of x and y in \(\bigodot A\).

**Solution**

Use the Inscribed Quadrilateral Theorem. \(x^{\circ}+108^{\circ}=180^{\circ}\) so \(x=72^{\circ}\). Similarly, \(y^{\circ}+88^{\circ}=180^{\circ}\) so \(y=92^{\circ}\).

Example \(\PageIndex{4}\)

Quadrilateral \(ABCD\) is inscribed in \(\bigodot E\). Find \(m\angle A\), \(m\angle B\), \(m\angle C\), and \(m\angle D\).

**Solution**

First, note that \(m\widehat{AD}=105^{\circ}\) because the complete circle must add up to \(360^{\circ}\).

\(\begin{aligned}m\angle A&=\dfrac{1}{2}m\widehat{BD}=12(115+86)=100.5^{\circ} \\ m\angle B&=\dfrac{1}{2}m\widehat{AC}=12(86+105)=95.5^{\circ} \\ m\angle C&=180^{\circ}−m\angle A=180^{\circ}−100.5^{\circ}=79.5^{\circ} \\ m\angle D&=180^{\circ}−m\angle B=180^{\circ}−95.5^{\circ}=84.5^{\circ}\end{aligned}\)

## Review

Fill in the blanks.

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

Quadrilateral \(ABCD\) is inscribed in \(\bigodot E\). Find:

- \(m\angle DBC\)
- \(m\widehat{BC}\)
- \(m\widehat{AB}\)
- \(m\angle ACD\)
- \(m\angle ADC\)
- \(m\angle ACB\)

Find the value of \(x\) and/or \(y\) in \(\bigodot A\).

Solve for \(x\).

## Vocabulary

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. |

circle |
The set of all points that are the same distance away from a specific point, called the .center |

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. |

radius |
The distance from the center to the outer rim of a circle. |

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. |

## Additional Resources

Interactive Element

Video: Inscribed Quadrilaterals in Circles Principles - Basic

Activities: Inscribed Quadrilaterals in Circles Discussion Questions

Study Aids: Inscribed in Circles Study Guide

Practice: Inscribed Quadrilaterals in Circles

Real World: Sunrise at Stonehenge