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Inscribed Quadrilaterals in Circles

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

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Quadrilaterals Inscribed in Circles

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. 

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Vocabulary

Inscribed Polygon

An inscribed polygon is a polygon with every vertex on a given circle.

Inscribed Quadrilateral Theorem

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.

Cyclic Quadrilaterals

A cyclic quadrilateral is a quadrilateral that can be inscribed in a circle.

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