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.

### Watch This

Inscribed Quadrilaterals in Circles CK-12

### Guidance

An
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inscribed polygon
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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.

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Inscribed Quadrilateral Theorem:
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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 .

#### Example A

Find the values of the missing variables.

a)

b)

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Answers:
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a)

b)

#### Example B

Find and in the picture below.

#### Example C

Find the values of and in .

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

Inscribed Quadrilaterals in Circles CK-12

### Guided Practice

Quadrilateral is inscribed in . Find:

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

1.

2.

3.

4.

### Practice

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 .