### Inscribed Quadrilaterals in Circles

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 is inscribed in , then and . Conversely, If and , then is inscribed in .

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.

#### Example 2

Find and in the picture below.

#### Example 3

Find the values of and in .

Use the Inscribed Quadrilateral Theorem. so . Similarly, so .

#### Example 4

Quadrilateral is inscribed in . Find , , , and .

First, note that because the complete circle must add up to .

### Review

Fill in the blanks.

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

Quadrilateral is inscribed in . Find:

Find the value of and/or in .

Solve for .

### Review (Answers)

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