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

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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:

#### 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.

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 .

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.

### Vocabulary Language: English

Inscribed Polygon

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.