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24
0
Sophia + Charlotte
02/09/2025
Maths
Maths- Circle Theorems
556
•
2 Sept 2025
•
Fun Circle Theorems: Learn Triangles, Cyclic Quadrilaterals, and Chords!
Cyclic Quadrilaterals and Tangents
This page focuses on cyclic quadrilateral properties and tangent lines to circles. It provides several important theorems and their applications.
Definition: A cyclic quadrilateral is a four-sided figure whose vertices all lie on a circle.
The page explains that the sum of opposite angles in a cyclic quadrilateral is always 180°, which is a key property used in many geometric proofs.
Formula: For a cyclic quadrilateral ABCD, a + b + c + d = 360°, where a, b, c, and d are the angles of the quadrilateral.
This formula is essential for solving problems involving cyclic quadrilaterals and is derived from the properties of angles in a circle.
Highlight: A radius of a circle is always perpendicular to a tangent line at the point of tangency.
This property is crucial for problems involving tangent lines and is often used in conjunction with other circle theorems.
The page also mentions that tangents drawn from an external point to a circle are equal in length, which is another important property for solving circle-related problems.
Radius and Chord Relationships
This page delves into the relationships between radii and chords in circles, focusing on perpendicular bisectors and the alternate segment theorem.
Theorem: If a radius cuts a chord at 90°, it bisects the chord.
This theorem is fundamental in understanding the relationship between radii and chords and is often used in circle geometry problems.
Theorem: If a radius bisects a chord, it does so at a 90° angle.
This is the converse of the previous theorem and is equally important in solving circle-related problems.
Definition: The alternate segment theorem states that the angle between a tangent and a chord at the point of contact is equal to the angle in the alternate segment.
This theorem is illustrated with a diagram and a mathematical expression: 180° - (90° - x) - (90° - x) = 2x.
These theorems provide powerful tools for solving complex geometry problems involving circles, chords, and tangents.
Application of Circle Theorems
This page demonstrates the practical application of circle theorems through two detailed examples.
Example: In a circle with a tangent line, given that angle OAB = 90° and angle BOC = 52°, the problem asks to find angle x.
The solution uses the property that a radius is perpendicular to a tangent at the point of contact, and applies the theorem that angles in a triangle sum to 180°.
Example: Another problem involves finding angles x and y in a circle configuration with given angle measures.
This example utilizes multiple circle theorems, including:
- Base angles in an isosceles triangle are equal
- Angles on a straight line sum to 180°
- Angles in a triangle sum to 180°
These examples demonstrate how to combine multiple circle theorems to solve complex geometry problems.
Advanced Circle Theorem Applications
The final page presents more advanced applications of circle theorems, incorporating multiple concepts in a single problem.
Example: A complex diagram is presented with various angles and segments, asking to find angle x.
This problem combines several key concepts:
- Base angles in an isosceles triangle are equal
- The angle at the center is twice the angle at the circumference when subtended by the same arc
- Tangents from a point to a circle are equal in length
Highlight: The solution demonstrates how to break down a complex problem into smaller, manageable steps using known circle theorems.
The page emphasizes the importance of recognizing and applying multiple theorems in conjunction to solve advanced geometry problems involving circles.
Vocabulary: Subtended angle - An angle formed by two lines or planes that intersect at a point on the circumference of a circle.
Understanding this concept is crucial for applying many circle theorems correctly.
Circle Theorems and Angles in Triangles
This page introduces fundamental circle theorems and their applications to angles in triangles. It provides visual examples and explanations of key concepts.
Example: In a circle where AB is a diameter and angle COB is 60°, the value of angle OAC is calculated to be 30°.
The example demonstrates the relationship between angles in a circle and an equilateral triangle. It also introduces the concept of isosceles triangles within circles.
Highlight: Angles subtended from the center of a circle are twice those at the circumference when subtended by the same arc.
This principle is crucial for solving many circle-related problems and is illustrated through the given example.
Vocabulary: Isosceles triangle - A triangle with two equal sides and two equal angles.
The page uses an isosceles triangle to explain how base angles are equal and how this property can be used to solve for unknown angles in circle problems.
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4.8/5
Google Play
The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.
Stefan S
iOS user
This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.
Samantha Klich
Android user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
Anna
iOS user
Best app on earth! no words because it’s too good
Thomas R
iOS user
Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.
Basil
Android user
This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.
Rohan U
Android user
I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.
Xander S
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now
Paul T
iOS user
The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.
Stefan S
iOS user
This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.
Samantha Klich
Android user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
Anna
iOS user
Best app on earth! no words because it’s too good
Thomas R
iOS user
Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.
Basil
Android user
This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.
Rohan U
Android user
I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.
Xander S
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now
Paul T
iOS user
Fun Circle Theorems: Learn Triangles, Cyclic Quadrilaterals, and Chords!
Circle theorems and angles in triangles are fundamental concepts in geometry. This summary covers key theorems, properties, and examples related to circles, cyclic quadrilaterals, and angles in various geometric configurations.
- Circle theorems angles in a triangleconcepts are explored through... Show more
Cyclic Quadrilaterals and Tangents
This page focuses on cyclic quadrilateral properties and tangent lines to circles. It provides several important theorems and their applications.
Definition: A cyclic quadrilateral is a four-sided figure whose vertices all lie on a circle.
The page explains that the sum of opposite angles in a cyclic quadrilateral is always 180°, which is a key property used in many geometric proofs.
Formula: For a cyclic quadrilateral ABCD, a + b + c + d = 360°, where a, b, c, and d are the angles of the quadrilateral.
This formula is essential for solving problems involving cyclic quadrilaterals and is derived from the properties of angles in a circle.
Highlight: A radius of a circle is always perpendicular to a tangent line at the point of tangency.
This property is crucial for problems involving tangent lines and is often used in conjunction with other circle theorems.
The page also mentions that tangents drawn from an external point to a circle are equal in length, which is another important property for solving circle-related problems.
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Radius and Chord Relationships
This page delves into the relationships between radii and chords in circles, focusing on perpendicular bisectors and the alternate segment theorem.
Theorem: If a radius cuts a chord at 90°, it bisects the chord.
This theorem is fundamental in understanding the relationship between radii and chords and is often used in circle geometry problems.
Theorem: If a radius bisects a chord, it does so at a 90° angle.
This is the converse of the previous theorem and is equally important in solving circle-related problems.
Definition: The alternate segment theorem states that the angle between a tangent and a chord at the point of contact is equal to the angle in the alternate segment.
This theorem is illustrated with a diagram and a mathematical expression: 180° - (90° - x) - (90° - x) = 2x.
These theorems provide powerful tools for solving complex geometry problems involving circles, chords, and tangents.
Sign up to see the contentIt's free!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Application of Circle Theorems
This page demonstrates the practical application of circle theorems through two detailed examples.
Example: In a circle with a tangent line, given that angle OAB = 90° and angle BOC = 52°, the problem asks to find angle x.
The solution uses the property that a radius is perpendicular to a tangent at the point of contact, and applies the theorem that angles in a triangle sum to 180°.
Example: Another problem involves finding angles x and y in a circle configuration with given angle measures.
This example utilizes multiple circle theorems, including:
- Base angles in an isosceles triangle are equal
- Angles on a straight line sum to 180°
- Angles in a triangle sum to 180°
These examples demonstrate how to combine multiple circle theorems to solve complex geometry problems.
Sign up to see the contentIt's free!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Advanced Circle Theorem Applications
The final page presents more advanced applications of circle theorems, incorporating multiple concepts in a single problem.
Example: A complex diagram is presented with various angles and segments, asking to find angle x.
This problem combines several key concepts:
- Base angles in an isosceles triangle are equal
- The angle at the center is twice the angle at the circumference when subtended by the same arc
- Tangents from a point to a circle are equal in length
Highlight: The solution demonstrates how to break down a complex problem into smaller, manageable steps using known circle theorems.
The page emphasizes the importance of recognizing and applying multiple theorems in conjunction to solve advanced geometry problems involving circles.
Vocabulary: Subtended angle - An angle formed by two lines or planes that intersect at a point on the circumference of a circle.
Understanding this concept is crucial for applying many circle theorems correctly.
Sign up to see the contentIt's free!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Circle Theorems and Angles in Triangles
This page introduces fundamental circle theorems and their applications to angles in triangles. It provides visual examples and explanations of key concepts.
Example: In a circle where AB is a diameter and angle COB is 60°, the value of angle OAC is calculated to be 30°.
The example demonstrates the relationship between angles in a circle and an equilateral triangle. It also introduces the concept of isosceles triangles within circles.
Highlight: Angles subtended from the center of a circle are twice those at the circumference when subtended by the same arc.
This principle is crucial for solving many circle-related problems and is illustrated through the given example.
Vocabulary: Isosceles triangle - A triangle with two equal sides and two equal angles.
The page uses an isosceles triangle to explain how base angles are equal and how this property can be used to solve for unknown angles in circle problems.
We thought you’d never ask...
What is the Knowunity AI companion?
Our AI Companion is a student-focused AI tool that offers more than just answers. Built on millions of Knowunity resources, it provides relevant information, personalised study plans, quizzes, and content directly in the chat, adapting to your individual learning journey.
Where can I download the Knowunity app?
You can download the app from Google Play Store and Apple App Store.
Is Knowunity really free of charge?
That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.
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4.9/5
App Store
4.8/5
Google Play
The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.
Stefan S
iOS user
This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.
Samantha Klich
Android user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
Anna
iOS user
Best app on earth! no words because it’s too good
Thomas R
iOS user
Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.
Basil
Android user
This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.
Rohan U
Android user
I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.
Xander S
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now
Paul T
iOS user
The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.
Stefan S
iOS user
This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.
Samantha Klich
Android user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
Anna
iOS user
Best app on earth! no words because it’s too good
Thomas R
iOS user
Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.
Basil
Android user
This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.
Rohan U
Android user
I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.
Xander S
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now
Paul T
iOS user