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

#### Watch This

Watch the first part of this video that discuses cyclic quadrilaterals:

https://www.youtube.com/watch?v=vk0Td5MgW90 Brightstorm: Cyclic Quadrilaterals

#### Guidance

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
**
. You will prove this theorem in Examples A and B.

**
Example A
**

Consider the cyclic quadrilateral below. Prove that and are supplementary.

**
Solution:
**
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.

**
Example B
**

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

**
Solution:
**
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.

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!
*

**
Example C
**

Solve for and .

**
Solution:
**
Opposite angles are supplementary, so
and
. This means
and
.

**
Concept Problem Revisited
**

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.

#### Vocabulary

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*
A
**
proof by contradiction
**
assumes the opposite of what you are trying to prove and shows that a contradiction will exist with such an assumption. Therefore, what you are trying to prove must be true.

#### Guided Practice

1. Find .

2. Find .

3. Find .

**
Answers:
**

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

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

3. A full circle is . .

#### Practice

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.