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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 30. 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 DEB and DCB are supplementary.

First note that mDEBˆ+mDCBˆ=360 because these two arcs make a full circle. 2mDEB=mDEBˆ and 2mDCB=mDCBˆ because the measure of an inscribed angle is half the measure of its intercepted arc. By substitution, 2mDEB+2mDCB=360. Divide by 2 and you have mDEB+mDCB=180. Therefore, DEB and DCB are supplementary.

Finding Contradictions 

Consider the quadrilateral below. Assume that B and F are supplementary, but that F 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 B and F are supplementary, but F is not on the circle. Find the point of intersection of EF¯¯¯¯¯¯¯¯ and the circle and call it D. Connect D with C.

CDE is an exterior angle of ΔFCD, so its measure is equal to the sum of the measures of the remote interior angles of the triangle. This means that mCDE=mFCD+mF. Quadrilateral BCDE is cyclic, so CDE and B must be supplementary. This means that CDE and F 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 x and y.

Opposite angles are supplementary, so 90+x=180 and 100+y=180. This means x=90 and y=80.


Example 1

Earlier, you were given a problem about a rhombus. 

One angle of a rhombus is 30. Can this rhombus be inscribed in a circle?

Opposite angles of a rhombus are congruent. If a rhombus has a 30 angle then it has one pair of opposite angles that are each 30 and one pair of opposite angles that are each 150. Opposite angles are not supplementary so this rhombus cannot be inscribed in a circle.

Example 2

Find mDEˆ.

BCD is the inscribed angle of DEBˆ. This means that the measure of the arc is twice the measure of the angle. mDEBˆ=872=174. Since mBEˆ=76, mDEˆ=17476=98.

Example 3

Find mDEB.

BCD and DEB are opposite angles of a cyclic quadrilateral so they are supplementary. mDEB=18087=93.

Example 4

Find mCBˆ.

 A full circle is 360. \begin{align*}m \widehat{CB}=360^\circ - 60^\circ - 98^\circ - 76^\circ = 126^\circ\end{align*}.


1. What is a cyclic quadrilateral?

2. A quadrilateral is cyclic if and only if its opposite angles are __________________.

3. Find \begin{align*}m \angle B\end{align*}.

4. Find \begin{align*}m \angle E\end{align*}.

5. Find \begin{align*}m \angle D\end{align*}.

6. Find \begin{align*}m \widehat{CD}\end{align*}.

7. Find \begin{align*}m \widehat{DE}\end{align*}.

8. Find \begin{align*}m \angle CBE\end{align*}.

9. Find \begin{align*}m \angle CEB\end{align*}.

10. Solve for \begin{align*}x\end{align*}.

11. Solve for \begin{align*}y\end{align*}.

12. Solve for \begin{align*}x\end{align*}.

13. Solve for \begin{align*}y\end{align*}.

14. If a cyclic quadrilateral has a \begin{align*}90^\circ\end{align*} angle, must it be a square? If yes, explain. If no, give a counter example.

15. Use the picture below to prove that angles \begin{align*}B\end{align*} and \begin{align*}D\end{align*} must be supplementary.

Review (Answers)

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

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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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