These lessons cover the various angle properties of quadrilaterals which are inscribed within circles.

**Related Pages**

Circles

Tangents Of Circles

We will learn what a cyclic quadrilateral is and the related angle properties.

- Opposite angles of a cyclic quadrilateral are supplementary.
- Exterior angle of a cyclic quadrilateral is equal to the interior opposite angle.

We will also prove that the opposite angles of a cyclic quadrilaterals are supplementary.

A cyclic quadrilateral is a quadrilateral with its 4 vertices on the circumference of a circle.

The following diagram shows a cyclic quadrilateral and its properties. Scroll down the page for more examples and solutions.

Cyclic quadrilaterals are also called inscribed quadrilaterals or chordal quadrilaterals.

**Property 1: In a cyclic quadrilateral, the opposite angles are
supplementary i.e. they add up to 180˚. **

∠*a* + ∠*c* = 180˚, ∠*b* + ∠*d* = 180˚

**Property 2: The exterior angle of a cyclic quadrilateral is
equal to the interior opposite angle. **

**Example:**

The exterior angle ∠*ADF* is equal to the corresponding interior angle ∠*ABC*.

The exterior angle ∠*DCE* is equal to the corresponding interior angle ∠*DAB*.

Example:

*AC* is a diameter of the circle. Find the value of *x*.

Solution:

∠*ABC* = 90˚ (angle of semicircle)

∠*ABD* + ∠*DBC* = ∠*ABC*

∠*ABD* + 36˚ = 90˚

∠*ABD* = 54˚

∠*ABD* + *x* = 180˚ (interior opposite angles of a cyclic quadrilateral)

*x* = 180˚ – 54˚ = 126˚

Example:

Find the values of *x* and *y* in the following figure.

Solution:

*x* = 98˚ (Corresponding opposite angles of a cyclic quadrilateral )

*y* + 27˚ = 53˚ (Corresponding opposite angles of a cyclic quadrilateral)

*y* = 26˚

A quadrilateral is cyclic when its four vertices lie on a circle.

Opposite angles in a cyclic quadrilateral adds up to 180˚.

Opposite angles in a cyclic quadrilateral adds up to 180˚.

Interior opposite angles are equal to their corresponding exterior angles.

Find the missing angles using central and inscribed angle properties.

Can you find the relationship between the missing angles in each figure?

Exam Practice Question Example:

*ABCD* is a cyclic quadrilateral within a circle centre *O*. *XY* is the tangent to the circle at *A*.

Angle *XAB* = 58°, Angle *BAD* = 78°, Angle *DBC* = 34°. Prove that *AB* is parallel to *CD*.

This video shows how to prove that opposite angles in a cyclic quadrilateral are supplementary.

It is based on the theorem: Angle at the center is twice angle at the circumference.

What is the relationship between the angles of a quadrilateral that is inscribed in a circle?

This video shows how to prove that opposite angles in a cyclic quadrilateral are supplementary.

Try the free Mathway calculator and
problem solver below to practice various math topics. Try the given examples, or type in your own
problem and check your answer with the step-by-step explanations.

We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page.