One angle of a rhombus is . Can this rhombus be inscribed in a circle?

### 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 and are supplementary.

First note that because these two arcs make a full circle. and because the measure of an inscribed angle is half the measure of its intercepted arc. By substitution, . Divide by 2 and you have . Therefore, and are supplementary.

#### Finding Contradictions

Consider the quadrilateral below. Assume that and are supplementary, but that does NOT lie on the circle. Find a contradiction. What does this prove?

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 and are supplementary, but is not on the circle. Find the point of intersection of and the circle and call it . Connect with .

is an exterior angle of , so its measure is equal to the sum of the measures of the remote interior angles of the triangle. This means that . Quadrilateral is cyclic, so and must be supplementary. This means that and must be congruent because they are both supplementary to the same angle.

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 and .

Opposite angles are supplementary, so and . This means and .

**Examples**

**Example 1**

Earlier, you were given a problem about a rhombus.

One angle of a rhombus is . Can this rhombus be inscribed in a circle?

Opposite angles of a rhombus are congruent. If a rhombus has a angle then it has one pair of opposite angles that are each and one pair of opposite angles that are each . Opposite angles are not supplementary so this rhombus cannot be inscribed in a circle.

#### Example 2

Find .

is the inscribed angle of . This means that the measure of the arc is twice the measure of the angle. . Since , .

#### Example 3

Find .

and are opposite angles of a cyclic quadrilateral so they are supplementary. .

#### 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 .

4. Find .

5. Find .

6. Find .

7. Find .

8. Find .

9. Find .

10. Solve for .

11. Solve for .

12. Solve for .

13. Solve for .

14. If a cyclic quadrilateral has a angle, must it be a square? If yes, explain. If no, give a counter example.

15. Use the picture below to prove that angles and must be supplementary.

### Review (Answers)

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