Quadrilateral Circles - Cyclic Quadrilaterals



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

Cyclic Quadrilateral

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.

Properties of a Cyclic Quadrilateral

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


a + c = 180 ˚, b + d = 180 ˚

Example:

cyclic quadrilateral

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 ∠ DCE is equal to the corresponding interior angle ∠ DAB.

Example:

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

cyclic quadrilateral


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



Proof for the Cyclic Quadrilateral

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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