[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-207764-en":3,"doc-seo-207764-105":30,"detail-sidebar-cat-0-en-105":91},{"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":20,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":27,"seo_description":14,"update_tm":28,"read_time":29},207764,962084931830,"Jacob","https://ap-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd",4,"Exam","Circle Theorems (Higher Only) - Worksheet - Geometry and Measures","Worksheet notes for GCSE Maths focusing on circle theorems at Higher tier. It defines a tangent and explains how angles involving a radius and tangent form a right angle, then develops the rule that an angle at the centre is twice the angle at the circumference. The material includes worked examples with labelled diagrams and step-by-step calculations, followed by formal proof outlines using properties of perpendicular lines, isosceles triangles, angle sums, and straight-line angles.","GCSE Maths – Geometry and Measures  \nCircle Theorems (Higher Only)  \nNotes  \nWORKSHEET  \nCircle Theorems  \nA theorem is a statement which can be proven to be true. Circle theorems involve properties of circles.  \nYou will need to be able to identify, use and prove seven circle theorems. We will go through each one of them in detail. The order of the following theorems does not matter.  \nTheorem 1  \nWhen a radius meets a tangent, the angle will always be 90 °  \n•  \n•  \n•  \n•  \nA tangent is a straight line which touches the circle once at a single point on the circumference of the circle. The position of the tangent does not matter, it can be anywhere around the circle.  \nPoint 􀢕 in the diagram is the centre of the circle. The line connecting the centre of the circle and the tangent is the radius of the circle.  \nWhere the radius and the tangent meet, they make a right-angle.  \n| Example: In the following diagram, find the value of 􀝔 .\u003Cbr>Diagram not drawn to scale.\u003Cbr>|  |\n| --- | --- |\n| 1. Label the unlabelled points.\u003Cbr>􀜱􀜤: radius of the circle\u003Cbr>􀜣􀜤: tangent to the circle\u003Cbr>Using theorem 1: Angle 􀜱􀜤􀜣 = 90°\u003Cbr>since the radius meets the tangent at that point.\u003Cbr>2. Use the property of angles in a triangle to find angle 􀝔 . Sum ofall angles inside a triangle sum to 180°: | |\n| 42° + 90° + 􀝔 = 180°􀝔 = 180° − 90° − 42° = 􀫝􀫡 ° |  |\n\nProof of Theorem 1  \nThere is no proof that you need to remember for this theorem because it comes directly from the definition of a tangent. The definition of the tangent is that it is perpendicular to the radius. Therefore, if the line the radius meets is specified as a tangent , the angle between them will be 90° because they are perpendicular to each other.  \nHowever, below is a proof that explains why the angle is 90° .  \nWe will use the following diagram:  \nSTEP 1: Make the necessary assumptions which are opposite to what you are trying to prove.  \n􀜱􀜣􀜤 is a right-angled triangle.  \nSo, if 􀜱􀜣 is not perpendicular to 􀜣􀜤 , then we will assume 􀜱􀜤 is perpendicular to 􀜣􀜤 .  \nTherefore, angle 􀡻􀡮􀡭 = 􀫢􀫙 °  \nSTEP 2: Progress with the assumptions and prove that they are wrong and impossible to be true.  \nIf angle 􀜱􀜤􀜣 = 90° :  \n􀜱􀜣 should be the longest side because it is opposite the biggest angle.  \nThis makes 􀜱􀜣 the hypotenuse.  \nTherefore, 􀡻􀡭 > 􀡻􀡮 .  \nHowever, 􀡻􀡭 = 􀡻􀡯 because both are radii of the circle, therefore, 􀡻􀡯 > 􀡻􀡮 as well. But,  \n􀡻􀡮 = 􀡻􀡯 + 􀡯􀡮  \nThis means 􀜱􀜥 cannot be greater than 􀜱􀜤 and as 􀜱􀜣 = 􀜱􀜥, 􀜱􀜣 cannot be greater than 􀜱􀜤 .  \nTherefore, 􀜱􀜤 is the hypotenuse which means 􀡻􀡭􀡮 = 􀫢􀫙 ° .  \nThis proves our assumption that angle 􀡻􀡮􀡭 = 􀫢􀫙 ° was wrong.  \nHence, for 􀜱􀜣􀜤 to be a right angled triangle we must have angle 􀡻􀡭􀡮 = 􀫢􀫙 ° .  \nTheorem 2  \nAngle at the centre of the circle is twice the angle at the circumference  \n• Point 􀢕 in the diagram is the centre of the circle.  \n• It is important that angle 􀢞 is at the circumference of the circle.  \n• An isosceles triangle can be made from the two radii of the circle.  \n• The representation on the right is not the only representation possible. The following are other ways the theorem can be applied.  \n| Example: In the following diagram, find the value of 􀝔 .\u003Cbr>Diagram not drawn to scale.\u003Cbr>|\n| --- |\n| Using theorem 2, the angle at the centre is twice the angle at the circumference, so:\u003Cbr>220°\u003Cbr>􀝔 = = 􀫚􀫚􀫙 °\u003Cbr>2 |\n\n\n| Example: In the following diagram, find the value of 􀝔 if 􀝕 = 35.\u003Cbr>Diagram not drawn to scale.\u003Cbr>|\n| --- |\n| This is a more challenging question. If you spot the isosceles triangle, then you can easily work your way to the answer.\u003Cbr>1. Label all the unlabelled points.\u003Cbr>To find angle 􀝔, we will first need to find\u003Cbr>angle 􀜣􀜱􀜤 at the centre.\u003Cbr>2. Identify the isosceles triangle within the circle.\u003Cbr>􀜱􀜣 = radius of the circle\u003Cbr>􀜱􀜤 = radius of the circle\u003Cbr>Therefore, 􀜱􀜣 = 􀜱􀜤 and hence triangle 􀜱􀜣􀜤 is isosceles.\u003Cbr>3. Use the property of isosceles triangles to find the angle at the centre.\u003Cbr>Property of isosceles triangle is that","cbCaigCiKmGErwpE","https://ap.wps.com/l/cbCaigCiKmGErwpE","pdf",1125729,1,17,"English","en",105,"# Circle Theorems (Higher Only) - Notes\n## Theorem 1: Radius meets tangent at 90°\n## Proof of Theorem 1\n## Theorem 2: Centre angle is twice circumference angle\n## Proof of Theorem 2\n## Theorem 3: Two tangents from a point are equal","[{\"question\":\"What does Theorem 1 say about a radius and a tangent?\",\"answer\":\"When a radius meets a tangent, the angle is always 90°. This uses the fact that a tangent is perpendicular to the radius at the point of contact.\"},{\"question\":\"How does Theorem 2 relate angles at the centre and at the circumference?\",\"answer\":\"An angle at the centre of the circle is twice the angle at the circumference. The proof uses isosceles triangles formed by radii and angle properties like sums to 180°.\"},{\"question\":\"What is the key idea behind Theorem 3?\",\"answer\":\"Two tangents drawn from a single point to the circle are equal in length from that point to the tangency points. The notes explain this by forming an isosceles triangle from the two tangents.\"}]","Circle Theorems (Higher Only) - Worksheet - Geometry and Measures | PDF",1788601080,43,{"code":4,"msg":31,"data":32},"ok",{"site_id":24,"language":23,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"circle-theorems-higher-only-worksheet-geometry-and-measures","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/exam/",3,{"item":52,"name":13,"@type":43,"position":11},"https://docshare.wps.com/document/circle-theorems-higher-only-worksheet-geometry-and-measures/207764/",{"url":52,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-09-08","2026-09-05",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What does Theorem 1 say about a radius and a tangent?","Question",{"text":75,"@type":76},"When a radius meets a tangent, the angle is always 90°. This uses the fact that a tangent is perpendicular to the radius at the point of contact.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does Theorem 2 relate angles at the centre and at the circumference?",{"text":80,"@type":76},"An angle at the centre of the circle is twice the angle at the circumference. The proof uses isosceles triangles formed by radii and angle properties like sums to 180°.",{"name":82,"@type":73,"acceptedAnswer":83},"What is the key idea behind Theorem 3?",{"text":84,"@type":76},"Two tangents drawn from a single point to the circle are equal in length from that point to the tangency points. The notes explain this by forming an isosceles triangle from the two tangents.","https://schema.org",{"og:url":52,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":52},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,104,109,114,119,124,129,132,136],{"id":20,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":102,"slug":103},70,"exam",{"id":105,"doc_module":4,"doc_module_name":46,"category_name":106,"show_sort_weight":107,"slug":108},5,"Comic",60,"comic",{"id":110,"doc_module":4,"doc_module_name":46,"category_name":111,"show_sort_weight":112,"slug":113},6,"Technology",50,"technology",{"id":115,"doc_module":4,"doc_module_name":46,"category_name":116,"show_sort_weight":117,"slug":118},7,"Healthcare",40,"healthcare",{"id":120,"doc_module":4,"doc_module_name":46,"category_name":121,"show_sort_weight":122,"slug":123},8,"Research & Report",30,"research-report",{"id":125,"doc_module":4,"doc_module_name":46,"category_name":126,"show_sort_weight":127,"slug":128},9,"Religion & Spirituality",20,"religion-spirituality",{"id":127,"doc_module":4,"doc_module_name":46,"category_name":130,"show_sort_weight":127,"slug":131},"World Cup","world-cup",{"id":133,"doc_module":4,"doc_module_name":46,"category_name":134,"show_sort_weight":133,"slug":135},10,"Lifestyle","lifestyle",{"id":137,"doc_module":4,"doc_module_name":46,"category_name":138,"show_sort_weight":105,"slug":139},19,"General","general"]