Transum Software

Proof of Circle Theorems

Arrange the stages of the proofs for the standard circle theorems in the correct order.

Circle Theorems Help Video More on Circles More on Angles

Drag the statements proving the theorem into the correct order.

	Similarly ∠AOC = 180° – 2 x ∠OCA OB = OC (radii of circle) ∠BOA = 2∠BCA Q.E.D. Construct radius OC ∠COB = 180° – 2 x ∠BCO (Angle sum of triangle OBC) To prove: ∠BOA = 2∠BCA ∠BCO = ∠OBC (equal angles in isosceles triangle) ∠BOA = 2(∠BCO + ∠OCA) ∠BOA = 360° – (180° – 2 x ∠BCO + 180° – 2 x ∠OCA) ∴ OBC is an isosceles triangle
	∴ 2 x ∠ABD = 2 x ∠ACD ∠AOD = 2 x ∠ABD (angle at centre twice angle at circumference) ∠AOD = 2 x ∠ACD (angle at centre twice angle at circumference) ∠ABD = ∠ACD Q.E.D. Construct radii from A and D To prove: ∠ABD = ∠ACD
	Similarly in triangle BCO ∠OCB = ∠OBC To prove: ∠ABC = 90° ∴ ∠OAB = ∠OBA (equal angles in isosceles triangle ABO) ∠OAB + ∠OBA + ∠OCB + ∠OBC = 180° (Angle sum of triangle ABC) OA = OB (radii) ∴ 2(∠OBA + ∠OBC) = 180° ∴ ABO is an isosceles triangle (two equal sides) ∠ABC = 90° Q.E.D. Construct the radius OB ∴ ∠OBA + ∠OBC = 90°
	The obtuse and reflex angles at O add up to 360° (angles at a point) Similarly the obtuse angle AOC = 2 x ∠CDA To prove ∠ABC + ∠CDA = 180° ∴ 2 x ∠ABC + 2 x ∠CDA = 360° Reflex ∠AOC = 2 x ∠ABC (angle at centre twice angle at circumference) ∠ABC + ∠CDA = 180° Q.E.D. Construct the radii OA and OC
	2 x ∠CAB = 2 x ∠CBD (from [1] above) ∠OBC + ∠CBD = 90° (angle between radius and tangent) [2] ∠CAB = ∠CBD Q.E.D. 2 x ∠OBC + ∠COB = 180° (angle sum of triangle) [3] Obtuse ∠COB = 2 x ∠CAB (angle at centre twice angle at circumference) [1] ∠COB = 2 x ∠CBD To prove ∠CAB = ∠CBD ∠OBC = ∠OCB (equal angles in isosceles triangle OBC) Construct the radii OB and OC 2 x ∠OBC + ∠COB = 2(∠OBC + ∠CBD) (from [2] and [3] above)

Instructions Try your best to answer the questions above. Type your answers into the boxes provided leaving no spaces. As you work through the exercise regularly click the "check" button. If you have any wrong answers, do your best to do corrections but if there is anything you don't understand, please ask your teacher for help. When you have got all of the questions correct you may want to print out this page and paste it into your exercise book. If you keep your work in an ePortfolio you could take a screen shot of your answers and paste that into your Maths file.

	You are learning this because future ideas are built on previous ideas and the more ideas you learn the more power you have to learn anything, ever! The world's best football player engages in rigorous training, practicing skills repeatedly to prepare for the big game. While the specific actions rehearsed during these exercises are rarely needed in the match, the very process of performing these drills enhances their overall prowess as a top player. more...	More Activities: Maths Map Go Maths Games Puzzles Times Tables TablesMaster iPad Maths Investigations Exam Questions
Mathematicians are not the people who find Maths easy; they are the people who enjoy how mystifying, puzzling and hard it is. Are you a mathematician? Comment recorded on the 16 March 'Starter of the Day' page by Mrs A Milton, Ysgol Ardudwy: "I have used your starters for 3 years now and would not have a lesson without one! Fantastic way to engage the pupils at the start of a lesson." Comment recorded on the 1 February 'Starter of the Day' page by M Chant, Chase Lane School Harwich: "My year five children look forward to their daily challenge and enjoy the problems as much as I do. A great resource - thanks a million."		Each month a newsletter is published containing details of the new additions to the Transum website and a new puzzle of the month. The newsletter is then duplicated as a podcast which is available on the major delivery networks. You can listen to the podcast while you are commuting, exercising or relaxing. Transum breaking news is available on Twitter @Transum and if that's not enough there is also a Transum Facebook page.
Featured Activity Strategy Games We have collected together lots of strategy games for you to enjoy. Many of them have mathematical principles involved in their design so would make a good activity for a Mathematics lesson.	Answers There are answers to this exercise but they are available in this space to teachers, tutors and parents who have logged in to their Transum subscription on this computer. A Transum subscription unlocks the answers to the online exercises, quizzes and puzzles. It also provides the teacher with access to quality external links on each of the Transum Topic pages and the facility to add to the collection themselves. Subscribers can manage class lists, lesson plans and assessment data in the Class Admin application and have access to reports of the Transum Trophies earned by class members. If you would like to enjoy ad-free access to the thousands of Transum resources, receive our monthly newsletter, unlock the printable worksheets and see our Maths Lesson Finishers then sign up for a subscription now: Subscribe
Go Maths Learning and understanding Mathematics, at every level, requires learner engagement. Mathematics is not a spectator sport. Sometimes traditional teaching fails to actively involve students. One way to address the problem is through the use of interactive activities and this web site provides many of those. The Go Maths page is an alphabetical list of free activities designed for students in Secondary/High school. Maths Map Are you looking for something specific? An exercise to supplement the topic you are studying at school at the moment perhaps. Navigate using our Maths Map to find exercises, puzzles and Maths lesson starters grouped by topic.
Teachers
If you found this activity useful don't forget to record it in your scheme of work or learning management system. The short URL, ready to be copied and pasted, is as follows:	Alternatively, if you use Google Classroom, all you have to do is click on the green icon below in order to add this activity to one of your classes.	It may be worth remembering that if Transum.org should go offline for whatever reason, there is a mirror site at Transum.info that contains most of the resources that are available here on Transum.org. When planning to use technology in your lesson always have a plan B!
Do you have any comments? It is always useful to receive feedback and helps make this free resource even more useful for those learning Mathematics anywhere in the world. Click here to enter your comments.

For Students:

For Teachers:

For All:

Transum.org is a proud supporter of the kidSAFE Seal Program

©1997-2024 WWW.TRANSUM.ORG

Proof of Circle Theorems

This is a video playlist.

Don't wait until you have finished the exercise before you click on the 'Check' button. Click it often as you work through the questions to see if you are answering them correctly.

Answers to this exercise are available lower down this page when you are logged in to your Transum account. If you don’t yet have a Transum subscription one can be very quickly set up if you are a teacher, tutor or parent.

Curriculum Reference

See the National Curriculum page for links to related online activities and resources.

Close