Inscribed Quadrilaterals: Examples and Real-Life Applications

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.

Circle
Quadrilateral
Inscribed angle
Central angle
Arc measure

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.

c Calculate the sum of the arc measures.

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

e Use the Polygon Interior Angles Theorem to calculate the missing angle measure of

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.

In the diagram above, all four vertices of

lie on a circle. Therefore,

is a cyclic quadrilateral.

Rule

Inscribed Quadrilateral Theorem

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

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

Proof

This theorem will be proven in two parts.

If a quadrilateral can be inscribed in a circle, then its opposite angles are supplementary.
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

Notice that ${\color{#FF0000}{\overgroup{BCD}}}$ and ${\color{#0000FF}{\overgroup{BAD}}}$ 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 $\overgroup{BCD}$ and $\overgroup{BAD}$ are intercepted by

and

respectively. 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 $m\overgroup{BCD}$ and $m\overgroup{BAD},$ respectively, into the equation found earlier.

$m\overgroup{BCD}+m\overgroup{BAD}=360^\circ$

SubstituteII

$m\overgroup{BCD}={\color{#0000FF}{{\color{#FF0000}{2m\angle A}}}}$, $m\overgroup{BAD}={\color{#009600}{{\color{#0000FF}{2m\angle C}}}}$

FactorOut

Factor out

DivEqn

The sum of the measures of

and

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.

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

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

can not be the same as the measure of the interior angle

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

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	$99^\circ+74^\circ=173^\circ \ {\color{#FF0000}{\bm{\Large{\times }}}}$	$105^\circ+82^\circ=187^\circ \ {\color{#FF0000}{\bm{\Large{\times }}}}$

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.

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.

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.

Rule

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.

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

Proof

Consider an inscribed quadrilateral with one side extended to point

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. 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.

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.

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.

SubstituteII

SubEqn

The value of

Next, notice that

is the exterior angle of

while

is the opposite interior angle. Therefore, these angles are congruent.

The measure of

and the measure of

By substituting these values and solving the equation, the value of

can be found.

SubstituteII

DivEqn

RearrangeEqn

Rearrange equation

Therefore, the value of

Similarly,

and

are the exterior angle and the opposite interior angle, respectively. By the property mentioned earlier, these angles are congruent.

The measure of

and the measure of

which is equal to

This information can be used to determine the value of

SubstituteII

DivEqn

The value of

Discussion

Properties of Inscribed Quadrilaterals in a Circle

Consider an inscribed quadrilateral

Draw a perpendicular bisector to each side of the polygon.

As can be observed, all the perpendicular bisectors intersect at the center of the circle. This property is true for all cyclic quadrilaterals.

Closure

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.

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

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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Catch-Up and Review

Proof

Part

Part

Hint

Solution

Proof

Hint

Solution