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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,

∠ M O N

measures

7 0^{\circ} .

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.

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 .

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

D E F G .

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

A B C D

lie on a circle. Therefore,

A B C D

is a cyclic quadrilateral.

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 $J K L M .$ 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 $J K L M .$

A quadrilateral with the given angle measures

Help Tiffaniqua determine whether it is possible to inscribe

J K L M

into a circle.

Hint

Compare the sums of the opposite angles' measures.

Solution

The Inscribed Quadrilateral Theorem can be used to determine whether $J K L M$ 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 $1$	Pair $2$
Opposite Angles	$∠ J$ and $∠ L$	$∠ K$ and $∠ M$
Sum	$9 9^{\circ} + 7 4^{\circ} = 17 3^{\circ} \times$	$10 5^{\circ} + 8 2^{\circ} = 18 7^{\circ} \times$

The sum of the angle measures in each pair is not equal to $18 0^{\circ} .$ Therefore, neither $∠ J$ and $∠ L$ nor $∠ K$ and $∠ M$ are supplementary. This finding implies that $J K L M$ is not a cyclic quadrilateral.

Pop Quiz

Practice Using the Inscribed Quadrilateral Theorem

Find the measure of $∠ D .$ Write your answer without the degree symbol.

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

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 $W X Y Z$ and the opposite interior angles. Then use the property that states these angles are congruent.

Solution

By observing the diagram,

∠ V X Y

and

∠ Y X W

can be recognized to form a linear pair, so they are supplementary angles.

m ∠ V X Y + m ∠ Y X W = 18 0^{\circ}

By substituting

8 4^{\circ}

for

m ∠ V X Y

and

a^{\circ}

for

m ∠ Y X W

into the equation, the value of

a

can be calculated.

m ∠ V X Y + m ∠ Y X W = 18 0^{\circ}

$m ∠ V X Y = 8 4^{\circ}$ , $m ∠ Y X W = a^{\circ}$

8 4^{\circ} + a^{\circ} = 18 0^{\circ}

$LHS - 8 4^{\circ} = RHS - 8 4^{\circ}$

a^{\circ} = 9 6^{\circ}

The value of

a

is

96 .

Next, notice that

∠ V X Y

is the exterior angle of

W X Y Z,

while

∠ W Z Y

is the opposite interior angle.

Therefore, these angles are congruent.

∠ V X Y ≅ ∠ W Z Y ⇓ m ∠ V X Y = m ∠ W Z Y

The measure of

V X Y

is

8 4^{\circ}

and the measure of

∠ W Z Y

is

3 b^{\circ} .

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

b

can be found.

m ∠ V X Y = m ∠ W Z Y

$m ∠ V X Y = 8 4^{\circ}$ , $m ∠ W Z Y = 3 b^{\circ}$

8 4^{\circ} = 3 b^{\circ}

$LHS / 3 = RHS / 3$

2 8^{\circ} = b^{\circ}

Rearrange equation

b^{\circ} = 2 8^{\circ}

Therefore, the value of

b

is

28 .

Similarly,

∠ Y Z U

and

∠ W X Y

are the exterior angle and the opposite interior angle, respectively.

By the property mentioned earlier, these angles are congruent.

∠ Y Z U ≅ ∠ W X Y ⇓ m ∠ Y Z U = m ∠ W X Y

The measure of

∠ Y Z U

is

2 c^{\circ}

and the measure of

∠ W X Y

is

a^{\circ},

which is equal to

9 6^{\circ} .

This information can be used to determine the value of

c .

m ∠ Y Z U = m ∠ W X Y

$m ∠ Y Z U = 2 c^{\circ}$ , $m ∠ W X Y = 9 6^{\circ}$

2 c^{\circ} = 9 6^{\circ}

$LHS / 2 = RHS / 2$

c^{\circ} = 4 8^{\circ}

The value of

c

is

48 .

Discussion

Properties of Inscribed Quadrilaterals in a Circle

Consider an inscribed quadrilateral

E F G H .

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.

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 $5000$ 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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