Cyclic Quadrilateral: Definition, Properties & Rules

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  • 3:30 Cyclic Trapezoids
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Lesson Transcript
Instructor: Shaun Ault

Shaun is currently an Assistant Professor of Mathematics at Valdosta State University as well as an independent private tutor.

In this lesson, you will learn about a certain type of geometric shape called a cyclic quadrilateral and discover some properties and rules concerning these shapes.

Cyclic Quadrilateral: Definition

The word cyclic often means circular, just think of those two circular wheels on your bicycle. Quadrilateral means four-sided figure. Put them together, and we get the definition for cyclic quadrilateral: any four-sided figure (quadrilateral) whose four vertices (corners) lie on a circle.

Not every quadrilateral is cyclic, but I bet you can name a few familiar ones. Every rectangle, including the special case of a square, is a cyclic quadrilateral because a circle can be drawn around it touching all four vertices. However, no non-rectangular parallelogram is cyclic. There's no way to draw a circle around one that touches all four of the non-rectangular parallelogram's vertices.

Four different cyclic quadrilaterals
Four different cyclic quadrilaterals

Properties of Cyclic Quadrilaterals

There are more to cyclic quadrilaterals than circles. Here's a property of cyclic quadrilaterals that you'll soon see can help identify them:

  • The sum of opposite angles of a cyclic quadrilateral is 180 degrees.

In other words, angle A + angle C = 180, and angle B + angle D = 180.

Labeled cyclic quadrilateral
cyclic quadrilateral with labels

There are many ways to prove this property, but the quickest one has to do with arc measures and inscribed angles. To refresh your memory, an inscribed angle is an angle that has its vertex on the circle's circumference. You can see that all the angles of our cyclic quadrilateral are inscribed angles. We also know the measure of an inscribed angle is half the measure of its intercepted arc, from the interior angle theorem. Let's use this to prove that the sum of opposite angles of cyclic quadrilateral is 180 degrees.

In our figure, the arc BCD intercepted by angle A and the arc DAB intercepted by angle C together make up the entire circle. So, the measures of arcs BCD and DAB together add up to 360 degrees. Remember that every circle has 360 degrees.

We know the cyclic quadrilateral's opposite angles A and C are inscribed angles. From the inscribed angle theorem, we also know that the measure of angle A is half the measure of its arc BCD, and the measure of angle C is half the measure of its arc DAB.

So together, the sum of angles A and C is half the sum of arcs BCD and DAB. In other words, the sum of these angles is half of 360, or 180. We're done!

This property also works in reverse:

  • If a pair of opposite angles of a quadrilateral is supplementary, that is, the sum of the angles is 180 degrees, then the quadrilateral is cyclic.

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