<img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />

## Quadrilaterals with every vertex on a circle and opposite angles that are supplementary.

Estimated8 minsto complete
%
Progress

MEMORY METER
This indicates how strong in your memory this concept is
Progress
Estimated8 minsto complete
%

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

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.

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.

#### 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.

### Notes/Highlights Having trouble? Report an issue.

Color Highlighted Text Notes

### Vocabulary Language: English

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.