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,

∠ 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.

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.

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.

$m ∠ A + m ∠ C = 18 0^{\circ}$
$m ∠ B + m ∠ D = 18 0^{\circ}$

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 $1$

Consider a circle and an inscribed quadrilateral $A B C D .$

Notice that

B C D

and

B A D

together span the entire circle. Therefore, by the Arc Addition Postulate, the sum of their measures is

36 0^{\circ} .

m B C D + m B A D = 36 0^{\circ}

From the diagram, it can be seen that

B C D

and

B A D

are intercepted by

∠ A

and

∠ C,

respectively.

By the Inscribed Angle Theorem, the measure of each of these inscribed angles is half the measure of its intercepted arc.

m ∠ A = \frac{1}{2} m B C D m ∠ C = \frac{1}{2} m B A D

The above equations can be simplified by multiplying both sides of each equation by

2 .

2 m ∠ A = m B C D 2 m ∠ C = m B A D

Next,

2 m ∠ A

and

2 m ∠ C

can be substituted for

m B C D

and

m B A D,

respectively, into the equation found earlier.

m B C D + m B A D = 36 0^{\circ}

SubstituteII

$m B C D = 2 m ∠ A$ , $m B A D = 2 m ∠ C$

2 m ∠ A + 2 m ∠ C = 36 0^{\circ}

FactorOut

Factor out $2$

2 (m ∠ A + m ∠ C) = 36 0^{\circ}

DivEqn

$LHS / 2 = RHS / 2$

m ∠ A + m ∠ C = 18 0^{\circ}

The sum of the measures of

∠ A

and

∠ C

18 0^{\circ} .

This means that these angles are supplementary. By the same logic, it can be also proven that

∠ B

and

∠ D

are supplementary. This concludes the proof of Part

1 .

Part $2$

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

Since $A B C D$ is not cyclic, the circle that passes through $A,$ $B,$ and $C,$ does not pass through $D .$ Let $E$ be the point of intersection of $\overline{A D}$ and the circle. Consider the quadrilateral $A B C E .$

Because

A B C E

is inscribed in a circle, it can be concluded that the opposite angles

∠ A E C

and

∠ B

are supplementary.

m ∠ A E C + m ∠ B = 18 0^{\circ}

It was assumed that

A B C D

has supplementary opposite angles. Therefore,

∠ D

and

∠ B

are supplementary angles.

m ∠ D + m ∠ B = 18 0^{\circ}

By the Transitive Property of Equality, the above equations imply that

∠ A E C

and

∠ D

have equal measures.

m ∠ A E C + m ∠ B = 18 0^{\circ} m ∠ D + m ∠ B = 18 0^{\circ} ⇓ m ∠ A E C = m ∠ D

However, this is not possible. The reason is that the measure of the exterior angle

∠ A E C

△ E D C

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

∠ D .

This contradiction proves that the initial assumption was false, and $A B C D$ is a cyclic quadrilateral. Note that a similar argument can be used if $D$ lies inside the circle. The proof of Part $2$ 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 $J K L M .$ 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.

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