Copyright

Quadrilaterals Inscribed in a Circle: Opposite Angles Theorem

An error occurred trying to load this video.

Try refreshing the page, or contact customer support.

Coming up next: Tangent of a Circle: Definition & Theorems

You're on a roll. Keep up the good work!

Take Quiz Watch Next Lesson
 Replay
Your next lesson will play in 10 seconds
  • 0:30 Cyclic Quadrilaterals
  • 1:00 Opposite Angles Theorem
  • 1:35 Example Problem
  • 3:45 Lesson Summary
Save Save Save

Want to watch this again later?

Log in or sign up to add this lesson to a Custom Course.

Log in or Sign up

Timeline
Autoplay
Autoplay
Speed Speed

Recommended Lessons and Courses for You

Lesson Transcript
Instructor: Elizabeth Foster

Elizabeth has been involved with tutoring since high school and has a B.A. in Classics.

Cyclic quadrilaterals can be inscribed in a circle, and their angles follow a special rule that can help you solve problems more quickly. Learn more about the properties of this particular type of quadrilateral.

Cyclic Quadrilaterals

Have you ever looked at your geometry book and thought, 'Hey, you know what these pictures need? More shapes!' No? Well, even if you never thought it, that's what you're going to get in this lesson: double-shape action with circles and quadrilaterals.

A quadrilateral is any shape with four sides and four angles. Quadrilaterals include squares, rectangles, trapezoids, and just random shapes that happen to have four sides and four angles.

Now imagine trying to draw a circle around each of these shapes. See how some of them touch the circle with all four corners, while other ones have one or two corners that are just kind of floating around? If you can draw a circle around the quadrilateral to touch all four corners of the shape to the outside of the circle, it's called inscribing the quadrilateral in the circle.

A cyclic quadrilateral is a quadrilateral inscribed in a circle. Not all quadrilaterals are cyclic. But quite a few of them are.

Opposite Angles Theorem

But why would anyone want to draw a circle around a quadrilateral? Why does it matter whether the quadrilateral is cyclic or not? Because if you can inscribe it in a circle, you know something about the quadrilateral. In a cyclic quadrilateral, opposite angles are supplementary.

If a pair of angles are supplementary, that means they add up to 180 degrees. So if you have any quadrilateral inscribed in a circle, you can use that to help you figure out the angle measures.

Example Problem

Now let's look at an example problem where that knowledge can help you get the answer.

If D = 3E and triangle AFB is equilateral, what is the measure of angle C?

We can see that this is basically a quadrilateral with a line through the middle dividing it into two triangles. Since it's a cyclic quadrilateral, the opposite angles must be supplementary. In this drawing, we have two pairs of opposite angles:

  • A + D = 180
  • (F + E) + (B + C) = 180

We also know that if the triangle AFB is equilateral, then all the angles are equal to 60 degrees. By plugging these values into the supplementary angles equation, we can say that 120 + C + E = 180.

We also know that C + E + D = 180 because all the angles in a triangle add up to 180. The problem tells us that D = 3E. If we substitute 3E for D in the triangle equation, we get C + E + 3E = 180, or C + 4E = 180.

Rearrange that just a little, so it reads C = 180 - 4E. Now we can plug in for C in our original equation: 120 + (180 - 4E) + E = 180. We'll simplify: 120 - 3E = 0. 3E = 120. E = 40.

To unlock this lesson you must be a Study.com Member.
Create your account

Register to view this lesson

Are you a student or a teacher?

Unlock Your Education

See for yourself why 30 million people use Study.com

Become a Study.com member and start learning now.
Become a Member  Back
What teachers are saying about Study.com
Try it risk-free for 30 days

Earning College Credit

Did you know… We have over 200 college courses that prepare you to earn credit by exam that is accepted by over 1,500 colleges and universities. You can test out of the first two years of college and save thousands off your degree. Anyone can earn credit-by-exam regardless of age or education level.

To learn more, visit our Earning Credit Page

Transferring credit to the school of your choice

Not sure what college you want to attend yet? Study.com has thousands of articles about every imaginable degree, area of study and career path that can help you find the school that's right for you.

Create an account to start this course today
Try it risk-free for 30 days!
Create an account
Support