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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 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 $\bigodot E$ .

#### Example A

Find the values of the missing variables.

a)

b)

a) $x+80^\circ &= 180^\circ && y+71^\circ = 180^\circ\\x &= 100^\circ && y=109^\circ$

b) $z+93^\circ &=180^\circ && x=\frac{1}{2} (58^\circ+106^\circ) && y+82^\circ=180^\circ\!\\z &= 87^\circ && x=82^\circ && y = 98^\circ$

#### Example B

Find $x$ and $y$ in the picture below.

$(7x+1)^\circ+105^\circ& =180^\circ && (4y+14)^\circ+(7y+1)^\circ=180^\circ\\7x+106^\circ&=180^\circ && \qquad \qquad \quad \ 11y+15^\circ=180^\circ\\7x&=74 && \qquad \qquad \qquad \qquad 11y=165\\x&=10.57 && \qquad \qquad \qquad \qquad \quad y=15$

#### Example C

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

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

### Guided Practice

Quadrilateral $ABCD$ is inscribed in $\bigodot E$ . Find:

1. $m\angle A$
2. $m\angle B$
3. $m\angle C$
4. $m\angle D$

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

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

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

3. $m\angle C=180^\circ-m\angle A=180^\circ-100.5^\circ=79.5^\circ$

4. $m\angle D=180^\circ-m\angle B=180^\circ-95.5^\circ=84.5^\circ$

### Practice

Fill in the blanks.

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

Quadrilateral $ABCD$ is inscribed in $\bigodot E$ . Find:

1. $m\angle DBC$
2. $m \widehat{BC}$
3. $m \widehat{AB}$
4. $m\angle ACD$
5. $m\angle ADC$
6. $m\angle ACB$

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

Solve for $x$ .

### 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.
circle

circle

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

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

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