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 quadrilatreral 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 quadrilatreral are supplementary.

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 circumeference 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:

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.

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.

Proof for the Cyclic Quadrilateral

This video shows how to prove that opposite angles in a cyclic quadrilateral are congruent; how to prove that parallel lines create congruent arcs in a circle.

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