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

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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? After completing this Concept, you'll be able to apply the Inscribed Quadrilateral Theorem to solve problems like this one.

### Guidance

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.

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

1. Draw a circle. Mark the center point A\begin{align*}A\end{align*}.
2. Place four points on the circle. Connect them to form a quadrilateral. Color the 4 angles of the quadrilateral 4 different colors.
3. Cut out the quadrilateral. Then cut the quadrilateral into two triangles, by cutting on a diagonal.
4. Line up B\begin{align*}\angle B\end{align*} and D\begin{align*}\angle D\end{align*} 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.

#### Example A

Find the value of the missing variable.

x+80=180\begin{align*}x+80^\circ=180^\circ\end{align*} by the Inscribed Quadrilateral Theorem. x=100\begin{align*}x=100^\circ\end{align*}.

y+71=180\begin{align*}y+71^\circ=180^\circ\end{align*} by the Inscribed Quadrilateral Theorem. y=109\begin{align*}y=109^\circ\end{align*}.

#### Example B

Find the value of the missing variable.

It is easiest to figure out z\begin{align*}z\end{align*} first. It is supplementary with 93\begin{align*}93^\circ\end{align*}, so z=87\begin{align*}z=87^\circ\end{align*}. Second, we can find x\begin{align*}x\end{align*}. x\begin{align*}x\end{align*} is an inscribed angle that intercepts the arc 58+106=164\begin{align*}58^\circ + 106^\circ = 164^\circ\end{align*}. Therefore, by the Inscribed Angle Theorem, x=82\begin{align*}x=82^\circ\end{align*}. y\begin{align*}y\end{align*} is supplementary with x\begin{align*}x\end{align*}, so y=98\begin{align*}y=98^\circ\end{align*}.Find the value of the missing variables.

#### Example C

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

The opposite angles are supplementary. Set up an equation for x\begin{align*}x\end{align*} and y\begin{align*}y\end{align*}.

(7x+1)+1057x+1067xx=180=180=84=12(4y+14)+(7y+1)=180 11y+15=18011y=165y=15\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 &= 84^\circ && \qquad \qquad \qquad \quad \quad 11y = 165^\circ\\ x &= 12^\circ && \qquad \qquad \qquad \qquad \quad y = 15^\circ\end{align*}

Watch this video for help with the Examples above.

### Vocabulary

A circle is the set of all points that are the same distance away from a specific point, called the center. A radius is the distance from the center to the circle. A chord is a line segment whose endpoints are on a circle. A diameter is a chord that passes through the center of the circle. The length of a diameter is two times the length of a radius. A central angle is an angle formed by two radii and whose vertex is at the center of the circle. An inscribed angle is an angle with its vertex on the circle and whose sides are chords. The intercepted arc is the arc that is inside the inscribed angle and whose endpoints are on the angle. An inscribed polygon is a polygon where every vertex is on the circle.

### Guided Practice

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

1. \begin{align*}m\angle A\end{align*}
2. \begin{align*}m\angle B\end{align*}
3. \begin{align*}m\angle C\end{align*}
4. \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*}.

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

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

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

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

### Practice

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*}.

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

1. \begin{align*}m \angle EBO\end{align*}
2. \begin{align*}m \angle EOB\end{align*}
3. \begin{align*}m \widehat{BC}\end{align*}
4. \begin{align*}m \angle ABO\end{align*}
5. \begin{align*}m \angle A\end{align*}
6. \begin{align*}m \angle EDC\end{align*}

### Vocabulary Language: English Spanish

central angle

central angle

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

chord

A line segment whose endpoints are on a circle.
diameter

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

inscribed angle

An angle with its vertex on the circle and whose sides are chords.
intercepted arc

intercepted arc

The arc that is inside an inscribed angle and whose endpoints are on the angle.
Inscribed Polygon

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.