In this lesson, we will learn

- what is a cyclic quadrilateral.
- the properties of a cyclic quadrilateral
- the opposite angles of a cyclic quadrilateral are supplementary.
- the exterior angle of a cyclic quadrilateral is equal to the interior opposite angle.

- to prove that the opposite angles of a cyclic quadrilaterals are supplementary.

Related Topics:

More Geometry Lessons

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

In the diagram shown below, *ABCD* is a cyclic quadrilateral because all its vertices lies on the circumference of the circle.

Cyclic quadrilaterals are also called inscribed quadrilaterals or chordal quadrilaterals.

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

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

* 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˚

This video shows how to use the properties of a cyclic quadrilateral to find missing angles. Opposite angles in a cyclic quadrilateral adds up to 180˚.

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 ∠

* 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˚

This video shows how to use the properties of a cyclic quadrilateral to find missing angles. Opposite angles in a cyclic quadrilateral adds up to 180˚. Interior opposite angles are equal to their corresponding exterior angles.

Circle theorems - Alternate segment theorem and Cyclic quadrilaterals

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."

Circle Geometry: Cyclic Quadrilateral (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

Arcs Between Parallel Lines

Using a computer model to show that the arcs between two parallel lines must be congruent.

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