Theorem on the Exterior Angle in a Cyclic Quadrilateral

The theorem on the exterior angle of a cyclic quadrilateral is used to state the property of cyclic quadrilaterals and helps us in solving complex problems related to exterior angles in an easy and efficient manner. Cyclic quadrilaterals are the quadrilaterals that are formed when all the vertices of the quadrilateral lie on the circumference of the circle. In this article, we will discuss the detailed proof of the theorem related to exterior angles of the cyclic quadrilateral along with the application of the cyclic quadrilateral in our real life.

History of Euclid

Euclid

Name: Euclid

Born: Mid-4th century BC

Died: Mid-3rd century BC

Field: Mathematics

Nationality: Greek

Statement of Theorem on the Exterior Angle of a Cyclic Quadrilateral

The theorem on the exterior angle of a cyclic quadrilateral states that an exterior angle of a cyclic quadrilateral is equal to the interior angle at the opposite vertex.

Cyclic Quadrilateral ABCD

Proof of Theorem on the Exterior Angle of a Cyclic Quadrilateral

Proof of the Exterior Angle of a Cyclic Quadrilateral

Given: In a cyclic quadrilateral $A B C D$, side $A B$ is produced to a point $E$.

To prove: External $\angle C B E=\angle A D C$

Proof:

$\angle A B C+\angle C B E=180^{\circ}$ (Linear pair)

$\angle A B C+\angle A D C=180^{\circ}$ (Opposite angles of a cyclic quadrilateral)

Equating above both cases, we get

$\angle A B C+\angle C B E=\angle A B C+\angle A D C$

$\angle C B E=\angle A D C$

Hence proved.

Limitations of Theorem on the Exterior Angle of a Cyclic Quadrilateral

The theorem is applicable only in the case of cyclic quadrilaterals.

Applications of Theorem on the Exterior Angle of a Cyclic Quadrilateral

• It is used in making paintings, sculptures, etc.

• The Cyclic Quadrilateral Theorem is used in computer programming.

• It is used in graphic arts, logos, and packaging.

Solved Examples of Exterior Angles of Cyclic Quadrilateral

1. In the cyclic quadrilateral, side $\mathrm{BD}$ is produced to $\mathrm{E}$ and $\angle B A C=75^{\circ}$. What is the value of $\angle C D E$?

Angle A is given in ABCD is a Cyclic Quadrilateral

Ans:

$\angle B A C=\angle C D E$ (exterior angle of a cyclic quadrilateral is equal to the interior angle at the opposite vertex)

And we are given that $\angle B A C=75^{\circ}$

$\angle B A C=\angle C D E$

Therefore, $\angle C D E=75^{\circ}$

2. In the cyclic quadrilateral, side $\mathrm{BD}$ is produced to $\mathrm{E}$ and $\angle B A C=75^{\circ}$. What is the value of $\angle C D B$?

To Find Angle CDB in the Quad. ABCD

Ans:

$\angle B A C=\angle C D E$ (exterior angle of a cyclic quadrilateral is equal to the interior angle at the opposite vertex)

And we are given that $\angle B A C=75^0$

and $\angle B A C=\angle C D E$

Therefore, $\angle C D E=75^{\circ}$

$\angle C D B+\angle C B E=180^{\circ} \text { (Linear pair) } \\$

$\angle C D B+75^{\circ}=180^{\circ} \\$

$\angle C D B=105^{\circ}$

3. In the cyclic quadrilateral, side $\mathrm{BC}$ is produced to $\mathrm{E}$ and $\angle D C E=95^{\circ}$. What is the value of $\angle D A B$?

To Find Angle DAB

Ans:

$\angle B A D=\angle D C E$ (exterior angle of a cyclic quadrilateral is equal to the interior angle at the opposite vertex)

And we are given that $\angle D C E=95^{\circ}$

and $\angle B A C=\angle C D E$

Therefore, $\angle B A D=95^0$

Important Points to Remember

• The exterior angle of a cyclic quadrilateral is equal to the interior angle at the opposite vertex.

Important Formulas to Remember

• In a cyclic quadrilateral ABCD, side AB is produced to point E.
  Then, External $\angle C B E = \angle A D C$

Conclusion

Cyclic Quadrilaterals are connecting links between polygons and circles. In the article, we have discussed the detailed proof of the exterior angles of the Cyclic Quadrilateral Theorem. Applications of the Cyclic Quadrilaterals are also discussed in this article. And we can conclude that the theorem is a fundamental tool of Euclidean Geometry and is of great significance in solving problems related to Cyclic Quadrilaterals.