[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-207769-en":3,"doc-seo-207769-105":29,"detail-sidebar-cat-0-en-105":89},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":4,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":11,"language":21,"language_code":22,"site_id":23,"html_lang":22,"table_of_contents":24,"faqs":25,"seo_title":26,"seo_description":14,"update_tm":27,"read_time":28},207769,687207412472,"Angel","https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d",4,"Exam","CIRCLE THEOREMS - Teacher Notes - Oxford GCSE Maths","Teacher notes for an Oxford GCSE Maths activity on circle theorems delivered using TI-Nspire. Students interactively explore key properties by dragging points and lines on the TI-Nspire screen, building understanding through observation rather than formal proof. The notes guide six sections covering: central versus circumference angles, equal angles from the same arc and segment, right angles in semicircles, opposite angles in cyclic quadrilaterals summing to 180°, and the perpendicular relationship between tangents and radii. A final angle puzzle uses two tangents meeting at a point.","Oxford GCSE Maths Barrie Galpin and Jay Timotheus  \nCIRCLE THEOREMS  \nTeacher Notes  \nReferences  \nFoundations   \nFoundations Plus   \nHigher G2 .5  \nHigher Plus G1 .1 and G1 .2  \nIntroduction  \nStudents are able to explore the following circle theorems by moving lines and points around the TI-Nspire screen:  \n• the angle at the centre is twice the angle at the circumference;  \n• angles from the same arc in the same segment are equal;  \n• the angle in a semicircle is a right angle;  \n• opposite angles in a cyclic quadrilateral add up to 180°;  \n• the tangent to a circle at a point is perpendicular to the radius at that point.  \nThese activities do not prove the circle theorems. Instead they give students the opportunity to get a sense of how the circle theorems work, as they move points and lines around and see what happens.  \nResources  \nThe TI-Nspire document CircleTheorems.tns is needed for this activity.  \nA 3-page student handout guides students in the use of the TI-Nspire document TI-Nspire skills students will need  \nTransferring a document to the handheld  \nOpening a document on the handheld  \nMoving between pages of a document  \nMoving from one part of a split screen to another  \nThe activity  \nThe activity is designed for use by students working individually on TI-Nspire handhelds. It can also be demonstrated on a screen using the TI-Nspire Navigator System, which also makes it easy to compare and discuss students’different results.  \nThe student notes give guidance and instructions for using the TI-Nspire document. They are divided into six sections and there are notes on each section below  \nThroughout the activity all points marked with an empty circle may be dragged.  \nTeacher notes Page 1  \nOxford GCSE Maths Barrie Galpin and Jay Timotheus  \n1. The angle at the centre of a circle is twice the angle at the circumference  \nStudents explore the diagram, moving the points A, B, C, or D around the circle. They aim to make the angle at the centre twice the angle at the circumference and find that this is only possible when the two angles are defined by the same arc.  \nPage 1.2 gives this summary of the theorem. Points may be moved around the circle to further illustrate its veracity.  \n2. Angles from the same arc in the same segment are equal  \nStudents choose points and drag them around the circle. They notice which angles change and which stay the same.  \nThey discover that the angle they drag will not change. This is because by dragging they are generating many angles from the same arc in the same segment.  \nStudents verify their understanding of the theorem by trying to measure the angle at Q (page 2 .2) . They can drag point B over point Q to show that the angles at B and Q are equal.  \nThis is the summary of the theorem on page 2.3.  \nMoving points M and N provides a very vivid illustration of the theorem.  \nTeacher notes Page 2  \nOxford GCSE Maths Barrie Galpin and Jay Timotheus  \n3. The angle in a semicircle is a right angle  \nIn this section, by moving the vertices of a triangle around acircle, students try to make each of the three angles in turn equal to 90 ° . Through the activity they recognise that, for an angle to be 90°, one of the three sides of the triangle must become a diameter, splitting the circle into two semicircles. It follows that the angle in a semicircle must always be a right angle.  \nThis statement of the theorem appears on page 3.2.  \n4. Opposite angles of a cyclic quadrilateral add to 180°  \nStudents move point Q around the circle. They find that when the angle remains in the same segment it does not change (since angles in the same segment are equal) . However, when the point Q moves to the opposite segment, the angle at Q instantly changes.  \nStudents discover that the sum of the two angles is 180 ° and then apply this to a cyclic quadrilateral.  \nOn page 4.2 the fact that angles in opposite segments add to 180 ° allows missing angles in the quadrilateral to be found.  \nOn page 4.3 the","cbCaiuSZgqLicRKw","https://ap.wps.com/l/cbCaiuSZgqLicRKw","pdf",662683,1,"English","en",105,"# Introduction\n# Resources\n## The TI-Nspire activity\n## Student handout and skills\n# The activity\n## Section 1: Central angle twice circumference angle\n## Section 2: Equal angles from the same arc in the same segment\n## Section 3: Angle in a semicircle is a right angle\n## Section 4: Opposite angles in a cyclic quadrilateral add to 180°\n## Section 5: Tangents are perpendicular to radii\n## Section 6: An angle puzzle with two tangents","[{\"question\":\"What is the purpose of the TI-Nspire circle theorems activities?\",\"answer\":\"Students explore circle theorems by moving lines and points to see how the relationships behave. The activities develop intuition through investigation rather than providing formal proofs.\"},{\"question\":\"How do students discover the relationship between the central angle and the circumference angle?\",\"answer\":\"They move points to try to make the central angle twice the circumference angle and observe that this is possible only when both angles are defined by the same arc.\"},{\"question\":\"How does the activity use tangents to teach the radius relationship?\",\"answer\":\"Students move a line until two intersection points coincide and a message indicates the angle between the line and the radius is 90°, showing tangents are perpendicular to radii at the point of contact.\"}]","CIRCLE THEOREMS - Teacher Notes - Oxford GCSE Maths | PDF",1788601090,10,{"code":4,"msg":30,"data":31},"ok",{"site_id":23,"language":22,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":84,"head_meta":86,"extra_data":88,"updated_unix":27},"circle-theorems-teacher-notes-oxford-gcse-maths","",{"@graph":35,"@context":83},[36,52,66],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/exam/",3,{"item":51,"name":13,"@type":42,"position":11},"https://docshare.wps.com/document/circle-theorems-teacher-notes-oxford-gcse-maths/207769/",{"url":51,"name":13,"@type":53,"author":54,"headline":13,"publisher":56,"fileFormat":59,"inLanguage":22,"description":14,"dateModified":60,"datePublished":60,"encodingFormat":59,"isAccessibleForFree":61,"interactionStatistic":62},"DigitalDocument",{"name":9,"@type":55},"Person",{"url":40,"name":57,"@type":58},"DocShare","Organization","application/pdf","2026-09-05",true,{"@type":63,"interactionType":64,"userInteractionCount":4},"InteractionCounter",{"@type":65},"ViewAction",{"@type":67,"mainEntity":68},"FAQPage",[69,75,79],{"name":70,"@type":71,"acceptedAnswer":72},"What is the purpose of the TI-Nspire circle theorems activities?","Question",{"text":73,"@type":74},"Students explore circle theorems by moving lines and points to see how the relationships behave. The activities develop intuition through investigation rather than providing formal proofs.","Answer",{"name":76,"@type":71,"acceptedAnswer":77},"How do students discover the relationship between the central angle and the circumference angle?",{"text":78,"@type":74},"They move points to try to make the central angle twice the circumference angle and observe that this is possible only when both angles are defined by the same arc.",{"name":80,"@type":71,"acceptedAnswer":81},"How does the activity use tangents to teach the radius relationship?",{"text":82,"@type":74},"Students move a line until two intersection points coincide and a message indicates the angle between the line and the radius is 90°, showing tangents are perpendicular to radii at the point of contact.","https://schema.org",{"og:url":51,"og:type":85,"og:title":13,"og:site_name":57,"og:description":14},"article",{"robots":87,"canonical":51},"index,follow",{"doc_id":7,"site_id":23},{"code":4,"msg":5,"data":90},[91,95,99,102,107,112,117,122,127,130,133],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":92,"show_sort_weight":93,"slug":94},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":96,"show_sort_weight":97,"slug":98},"Literature",80,"literature",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":100,"slug":101},70,"exam",{"id":103,"doc_module":4,"doc_module_name":45,"category_name":104,"show_sort_weight":105,"slug":106},5,"Comic",60,"comic",{"id":108,"doc_module":4,"doc_module_name":45,"category_name":109,"show_sort_weight":110,"slug":111},6,"Technology",50,"technology",{"id":113,"doc_module":4,"doc_module_name":45,"category_name":114,"show_sort_weight":115,"slug":116},7,"Healthcare",40,"healthcare",{"id":118,"doc_module":4,"doc_module_name":45,"category_name":119,"show_sort_weight":120,"slug":121},8,"Research & Report",30,"research-report",{"id":123,"doc_module":4,"doc_module_name":45,"category_name":124,"show_sort_weight":125,"slug":126},9,"Religion & Spirituality",20,"religion-spirituality",{"id":125,"doc_module":4,"doc_module_name":45,"category_name":128,"show_sort_weight":125,"slug":129},"World Cup","world-cup",{"id":28,"doc_module":4,"doc_module_name":45,"category_name":131,"show_sort_weight":28,"slug":132},"Lifestyle","lifestyle",{"id":134,"doc_module":4,"doc_module_name":45,"category_name":135,"show_sort_weight":103,"slug":136},19,"General","general"]