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

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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\begin{align*}ABCD\end{align*} is inscribed in E\begin{align*}\bigodot E\end{align*}, then mA+mC=180\begin{align*}m\angle A+m\angle C=180^\circ\end{align*} and \begin{align*}m\angle B+m\angle D=180^\circ\end{align*}. Conversely, If \begin{align*}m\angle A+m\angle C=180^\circ\end{align*} and \begin{align*}m\angle B+m\angle D=180^\circ\end{align*}, then \begin{align*}ABCD\end{align*} is inscribed in \begin{align*}\bigodot E\end{align*}.

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?

Examples

Example 1

Find the values of the missing variables.

\begin{align*}x+80^\circ &= 180^\circ && y+71^\circ = 180^\circ\\ x &= 100^\circ && y=109^\circ\end{align*}

1.
\begin{align*}z+93^\circ &=180^\circ && x=\frac{1}{2} (58^\circ+106^\circ) && y+82^\circ=180^\circ\!\\ z &= 87^\circ && x=82^\circ && y = 98^\circ\end{align*}

Example 2

Find \begin{align*}x\end{align*} and \begin{align*}y\end{align*} in the picture below.

\begin{align*}(7x+1)^\circ+105^\circ& =180^\circ && (4y+14)^\circ+(7y+1)^\circ=180^\circ\\ 7x+106^\circ&=180^\circ && \qquad \qquad \quad \ 11y+15^\circ=180^\circ\\ 7x&=74 && \qquad \qquad \qquad \qquad 11y=165\\ x&=10.57 && \qquad \qquad \qquad \qquad \quad y=15\end{align*}

Example 3

Find the values of \begin{align*}x\end{align*} and \begin{align*}y\end{align*} in \begin{align*}\bigodot A\end{align*}.

Use the Inscribed Quadrilateral Theorem. \begin{align*}x^\circ + 108^\circ =180^\circ\end{align*} so \begin{align*} x=72^\circ\end{align*}. Similarly, \begin{align*}y^\circ + 88^\circ = 180^\circ\end{align*} so \begin{align*}y=92^\circ\end{align*}.

Example 4

Quadrilateral \begin{align*}ABCD\end{align*} is inscribed in \begin{align*}\bigodot E\end{align*}. Find \begin{align*}m\angle A\end{align*}\begin{align*}m\angle B\end{align*}\begin{align*}m\angle C\end{align*}, and \begin{align*}m\angle D\end{align*}.

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

\begin{align*}m\angle A=\frac{1}{2}m\widehat{BD}=\frac{1}{2}(115+86)=100.5^\circ\end{align*}

\begin{align*}m\angle B=\frac{1}{2}m\widehat{AC}=\frac{1}{2}(86+105)=95.5^\circ\end{align*}

\begin{align*}m\angle C=180^\circ-m\angle A=180^\circ-100.5^\circ=79.5^\circ\end{align*}

\begin{align*}m\angle D=180^\circ-m\angle B=180^\circ-95.5^\circ=84.5^\circ\end{align*}

Review

Fill in the blanks.

1. A\begin{align*}(n)\end{align*} _______________ polygon has all its vertices on a circle.
2. The _____________ angles of an inscribed quadrilateral are ________________.

Quadrilateral \begin{align*}ABCD\end{align*} is inscribed in \begin{align*}\bigodot E\end{align*}. Find:

1. \begin{align*}m\angle DBC\end{align*}
2. \begin{align*}m \widehat{BC}\end{align*}
3. \begin{align*}m \widehat{AB}\end{align*}
4. \begin{align*}m\angle ACD\end{align*}
5. \begin{align*}m\angle ADC\end{align*}
6. \begin{align*}m\angle ACB\end{align*}

Find the value of \begin{align*}x\end{align*} and/or \begin{align*}y\end{align*} in \begin{align*}\bigodot A\end{align*}.

Solve for \begin{align*}x\end{align*}.

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

Vocabulary Language: English Spanish

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.

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.

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.