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Circles With and Without Coordinates

Quadrilaterals Inscribed in a Circle

Polygons with a different number of sides can be inscribed in a circle. In this lesson, inscribed quadrilaterals, or polygons with four sides, will be explored. Furthermore, three main properties of inscribed quadrilaterals will be investigated.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Try your knowledge on these topics.

a Pair each geometric object with its definition.
b In the circle, measures Find the measure of the corresponding inscribed angle.
A circle with a central angle and a corresponding inscribed angle
c Calculate the sum of the arc measures.
A circle with two central angles and intercepted arcs
Write the answer without the degree symbol.
d On the circle, the measures of all arcs except one are given. Find the measure of that arc and the measure of the inscribed angle
A circle with five arcs and one inscribed angle
e Use the Polygon Interior Angles Theorem to calculate the missing angle measure of
A quadrilateral with the three given angle measures

Explore

Investigating Quadrilaterals Inscribed in a Circle

Consider a quadrilateral inscribed in a circle. Move its vertices and analyze how the measures of the interior angles change.
What relationship between the angle measures can be observed?

Discussion

Inscribed Quadrilateral Theorem

An inscribed quadrilateral is a quadrilateral whose vertices all lie on a circle. It can also be called a cyclic quadrilateral.
An inscribed quadrilateral
In the diagram above, all four vertices of lie on a circle. Therefore, is a cyclic quadrilateral.

A quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary.

A quadrilateral inscribed in a circle

Based on the diagram above, the following relations hold true.


Proof

This theorem will be proven in two parts.

  1. If a quadrilateral can be inscribed in a circle, then its opposite angles are supplementary.
  2. If the opposite angles of a quadrilateral are supplementary, then it can be inscribed in a circle.

Part

Consider a circle and an inscribed quadrilateral

A quadrilateral inscribed in a circle with two colored arcs

Notice that and together span the entire circle. Therefore, by the Arc Addition Postulate, the sum of their measures is From the diagram, it can be seen that and are intercepted by and respectively.

A quadrilateral inscribed in a circle with two colored angles and intercepted arcs
By the Inscribed Angle Theorem, the measure of each of these inscribed angles is half the measure of its intercepted arc. The above equations can be simplified by multiplying both sides of each equation by Next, and can be substituted for and respectively, into the equation found earlier.
The sum of the measures of and is This means that these angles are supplementary. By the same logic, it can be also proven that and are supplementary. This concludes the proof of Part

Part

This part of the proof will be proven by contradiction. Suppose that is a quadrilateral that has supplementary opposite angles, but is not cyclic.

A quadrilateral whose three vertices lie on a circle

Since is not cyclic, the circle that passes through and does not pass through Let be the point of intersection of and the circle. Consider the quadrilateral

Colored quadrilateral ABCE

Because is inscribed in a circle, it can be concluded that the opposite angles and are supplementary. It was assumed that has supplementary opposite angles. Therefore, and are supplementary angles. By the Transitive Property of Equality, the above equations imply that and have equal measures. However, this is not possible. The reason is that the measure of the exterior angle of can not be the same as the measure of the interior angle

Colored triangle EDC with an exterior and interior angles

This contradiction proves that the initial assumption was false, and is a cyclic quadrilateral. Note that a similar argument can be used if lies inside the circle. The proof of Part is now complete.

Example

Using the Inscribed Quadrilateral Theorem

The Inscribed Quadrilateral Theorem can be used to identify whether a quadrilateral is cyclic.

Tiffaniqua is given a quadrilateral She wants to draw a circle that passes through all the vertices, but she does not know if it is possible. For that reason, she decided to measure the angles of

A quadrilateral with the given angle measures
Help Tiffaniqua determine whether it is possible to inscribe into a circle.

Hint

Compare the sums of the opposite angles' measures.

Solution

The Inscribed Quadrilateral Theorem can be used to determine whether is cyclic. According this theorem, the opposite angles of the quadrilateral need to be supplementary. Calculate the sum of opposite angles' measures and see if it is true.

Pair Pair
Opposite Angles and and
Sum

The sum of the angle measures in each pair is not equal to Therefore, neither and nor and are supplementary. This finding implies that is not a cyclic quadrilateral.

Not cyclic quadrilateral

Pop Quiz

Practice Using the Inscribed Quadrilateral Theorem

Find the measure of Write your answer without the degree symbol.

Quadrilateral ABCD inscribed in a circle with the given measure of angle B

Discussion

Cyclic Quadrilateral Exterior Angle Theorem

On the diagram below, one side of a cyclic quadrilateral is extended to As a result, — the exterior angle of — is formed.

A quadrilateral with one extended side

In this case, is said to be the opposite interior angle. The relationship between these angles is described by the Cyclic Quadrilateral Exterior Angle Theorem.

If a side of a cyclic quadrilateral is extended, then the exterior angle is congruent to the opposite interior angle.

A quadrilateral with one extended side

Based on the diagram above, the following relation holds true.

Proof

Consider an inscribed quadrilateral with one side extended to point

A quadrilateral with one extended side

From the diagram, it can be observed that and form a linear pair. Therefore, these angles are supplementary, which means that the sum of their measures is Also, by the Inscribed Quadrilateral Theorem, the opposite angles and are also supplementary.

A quadrilateral with one extended side

Therefore, the following relation is true. After analyzing the equations, and show to be supplementary to the same angle Therefore, by the Congruent Supplements Theorem, they are congruent.

This relation is illustrated on the diagram below.

A quadrilateral with one extended side

By the same logic, this theorem can be proven for any other extended side of The proof is now complete.

Example

Using the Cyclic Quadrilateral Exterior Angle Theorem

Davontay wants to go to a concert, but his parents say that he has to finish his homework first. In the last math exercise, he is asked to find the values of all variables.

A quadrilateral inscribed in a circle with two extended sides

Help Davontay solve the last exercise so that he can go to the concert.

Hint

Identify the exterior angles to the inscribed quadrilateral and the opposite interior angles. Then use the property that states these angles are congruent.

Solution

By observing the diagram, and can be recognized to form a linear pair, so they are supplementary angles. By substituting for and for into the equation, the value of can be calculated.
The value of is Next, notice that is the exterior angle of while is the opposite interior angle.
A quadrilateral inscribed in a circle with two extended sides
Therefore, these angles are congruent. The measure of is and the measure of is By substituting these values and solving the equation, the value of can be found.
Therefore, the value of is Similarly, and are the exterior angle and the opposite interior angle, respectively.
A quadrilateral inscribed in a circle with two extended sides
By the property mentioned earlier, these angles are congruent. The measure of is and the measure of is which is equal to This information can be used to determine the value of
The value of is

Discussion

Properties of Inscribed Quadrilaterals in a Circle

Consider an inscribed quadrilateral Draw a perpendicular bisector to each side of the polygon.
circle
As can be observed, all the perpendicular bisectors intersect at the center of the circle. This property is true for all cyclic quadrilaterals.

Example

Inscribed Quadrilaterals and Polygons in Real Life

It is worth mentioning that not only quadrilaterals can be inscribed in a circle. There can also be inscribed polygons with a different number of sides. Stonehenge is a real-world example of an inscribed polygon. Unfortunately, only some parts of it remain to this day.

Stonehenge now

However, when Stonehenge was built by ancient peoples about years ago, it had a cyclic polygon structure, as illustrated on the diagram below.

Stonehenge 5000 years ago
To sum up, inscribed quadrilaterals and polygons are not only interesting geometric objects — they can be seen and applied in real life.
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