One angle of a rhombus is

### Quadrilaterals Inscribed in Circles

A quadrilateral is said to be **inscribed in a circle** if all four vertices of the quadrilateral lie on the circle. Quadrilaterals that can be inscribed in circles are known as **cyclic quadrilaterals**. The quadrilateral below is a cyclic quadrilateral.

Not all quadrilaterals can be inscribed in circles and so not all quadrilaterals are cyclic quadrilaterals. **A** **quadrilateral** **is cyclic if and only if** **its** **opposite angles** **are** **supplementary**.

#### Proving Supplementary Angles

Consider the cyclic quadrilateral below. Prove that

First note that

#### Finding Contradictions

Consider the quadrilateral below. Assume that

One method of proof is called a proof by contradiction. With a proof by contradiction you prove that something cannot **not** be true. Therefore, it must **be** true. Here, you are attempting to prove that it is ** impossible** for a quadrilateral with opposite angles supplementary to

**not**be cyclic. Therefore, such a quadrilateral must

**be**cyclic.

#### Finding a Point of Intersection

Assume that

The two highlighted statements are a contradiction -- they cannot both be true. This means that your original assumption cannot exist. You cannot have a quadrilateral with opposite angles supplementary that is not cyclic. So, if opposite angles of a quadrilateral are supplementary then the quadrilateral must be cyclic.

*Note: You will learn more about proof by contradiction in future courses!*

#### Solving for Unknown Values

Solve for

Opposite angles are supplementary, so

**Examples**

**Example 1**

Earlier, you were given a problem about a rhombus.

One angle of a rhombus is

Opposite angles of a rhombus are congruent. If a rhombus has a

#### Example 2

Find

#### Example 3

Find

#### Example 4

Find

A full circle is

### Review

1. What is a cyclic quadrilateral?

2. A quadrilateral is cyclic if and only if its opposite angles are __________________.

3. Find \begin{align*}m \angle B\end{align*}.

4. Find \begin{align*}m \angle E\end{align*}.

5. Find \begin{align*}m \angle D\end{align*}.

6. Find \begin{align*}m \widehat{CD}\end{align*}.

7. Find \begin{align*}m \widehat{DE}\end{align*}.

8. Find \begin{align*}m \angle CBE\end{align*}.

9. Find \begin{align*}m \angle CEB\end{align*}.

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

11. Solve for \begin{align*}y\end{align*}.

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

13. Solve for \begin{align*}y\end{align*}.

14. If a cyclic quadrilateral has a \begin{align*}90^\circ\end{align*} angle, must it be a square? If yes, explain. If no, give a counter example.

15. Use the picture below to prove that angles \begin{align*}B\end{align*} and \begin{align*}D\end{align*} must be supplementary.

### Review (Answers)

To see the Review answers, open this PDF file and look for section 8.6.