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More Geometry Lessons

In these lessons, 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.

A cyclic quadrilateral is a quadrilateral with 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.

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.

How to use the properties of a cyclic quadrilateral to find missing angles?

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˚

Using the properties of a cyclic quadrilateral to find missing angles
A quadrilateral is cyclic when its four vertices lie on a circle.

How to find missing angles using the properties of a cyclic quadrilateral?
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? Circle theorems - Alternate segment theorem and Cyclic quadrilaterals
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. 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.

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