[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-207767-en":3,"doc-seo-207767-105":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},207767,5909887256941,"Mason","https://ap-avatar.wpscdn.com/davatar_9964176cb1d06d4a9deccf72a44ae3dc",4,"Exam","Circle theorems - A Level Links - 2b. Circles","Circle theorems for A Level geometry, focused on circles and related angle and line relationships. Covers key concepts including chords and tangents, tangent-radius right angles, equal tangents from an external point, semicircle right angles, and central-to-circumference angle doubling. Includes the same-arc equal angles in segments, cyclic quadrilateral opposite angles summing to 180°, and the alternate segment theorem linking tangent–chord angles to angles in the alternate segment. Provides worked examples and practice questions to justify results using theorems.","Circle theorems  \nA LEVEL LINKS  \nScheme of work: 2b. Circles – equation of a circle, geometric problems on a grid  \nKey points  \n• A chord is a straight line joining two points on the circumference of a circle.  \nSo AB is a chord.  \n• A tangent is a straight line that touches the circumference of a circle at only one point. The angle between a tangent and the radius is 90° .  \n• Two tangents on a circle that meet at a point outside the circle are equal in length. So AC = BC.  \n• The angle in a semicircle is a right angle. So angle ABC = 90° .  \n• When two angles are subtended by the same arc, the angle at the centre of a circle is twice the angle at the circumference.  \nSo angle AOB = 2 × angle ACB.  \n• Angles subtended by the same arc at the circumference are equal. This means that angles in the same segment are equal.  \nSo angle ACB = angle ADB and  \nangle CAD = angle CBD.  \n• A cyclic quadrilateral is a quadrilateral with all four vertices on the circumference of a circle. Opposite angles in a cyclic quadrilateral total 180° . So x + y = 180° and p + q = 180° .  \n• The angle between a tangent and chord is equal to the angle in the alternate segment, this is known as the alternate segment theorem. So angle BAT = angle ACB.  \nExamples  \nExample 1 Work out the size of each angle  \nmarked with a letter.  \nGive reasons for your answers.  \n| Angle a = 360° − 92°\u003Cbr>= 268°\u003Cbr>as the angles in a full turn total 360° .\u003Cbr>Angle b = 268° ÷ 2\u003Cbr>= 134°\u003Cbr>as when two angles are subtended by the same arc, the angle at the centre of acircle is twice the angle at the circumference. | 1 The angles in a full turn total 360° .\u003Cbr>2 Angles a and b are subtended by the same arc, so angle b is half of angle a. |\n| --- | --- |\n\nExample 2 Work out the size of the angles in the triangle.  \nGive reasons for your answers.  \n| Angles are 90°, 2c and c.\u003Cbr>90° + 2c + c = 180° 90° + 3c = 180°\u003Cbr>3c = 90°\u003Cbr>c = 30°\u003Cbr>2c = 60°\u003Cbr>The angles are 30°, 60° and 90° as the angle in a semi-circle is a right angle and the angles in a triangle total 180° . | 1 The angle in a semicircle is a right angle.\u003Cbr>2 Angles in a triangle total 180° .\u003Cbr>3 Simplify and solve the equation. |\n| --- | --- |\n\nExample 3 Work out the size of each angle marked with a letter.  \nGive reasons for your answers.  \n| Angle d = 55° as angles subtended by the same arc are equal.\u003Cbr>Angle e = 28° as angles subtended by the same arc are equal. | 1 Angles subtended by the same arc are equal so angle 55° and angle dare equal.\u003Cbr>2 Angles subtended by the same arc are equal so angle 28° and angle e are equal. |\n| --- | --- |\n\nExample 4 Work out the size of each angle marked with a letter.  \nGive reasons for your answers.  \n| Angle f = 180° − 94°\u003Cbr>= 86°\u003Cbr>as opposite angles in a cyclic quadrilateral total 180° . | 1 Opposite angles in a cyclic quadrilateral total 180° so angle 94° and angle f total 180° .\u003Cbr>(continued on next page) |\n| --- | --- |\n\nExample 5  \n\n| Angle g = 180° − 86°= 84°\u003Cbr>as angles on a straight line total 180° .\u003Cbr>Angle h = angle f = 86° as angles subtended by the same arc are equal. | 2 Angles on a straight line total 180° so angle f and angle g total 180° .\u003Cbr>3 Angles subtended by the same arc are equal so angle f and angle h are equal. |\n| --- | --- |\n\nWork out the size of each angle marked with a letter. Give reasons for your answers.  \n| Angle i = 53° because of the alternate segment theorem.\u003Cbr>Angle j = 53° because it is the alternate angle to 53° .\u003Cbr>Angle k = 180° − 53° − 53°= 74°\u003Cbr>as angles in a triangle total 180° . | 1 The angle between a tangent and chord is equal to the angle in the alternate segment.\u003Cbr>2 As there are two parallel lines, angle 53° is equal to angle j because they are alternate angles.\u003Cbr>3 The angles in a triangle total 180°, so i +j + k = 180° . |\n| --- | --- |\n\nExample 6 XZ and YZ are two tangents to a circle with centre O.  \nProve that triangles XZO and YZO are congruent.  \n| Angle OXZ = 90° and angle OYZ = 90° as the angles","cbCaitYEupkylcq2","https://ap.wps.com/l/cbCaitYEupkylcq2","pdf",877860,1,9,"English","en",105,"# Scheme of work: 2b. Circles\n## Key points\n## Examples\n## Practice\n## Extend","[{\"question\":\"What is the relationship between a tangent and the radius of a circle at the point of contact?\",\"answer\":\"The angle between a tangent and the radius is 90°.\"},{\"question\":\"How do angles subtended by the same arc relate on a circle?\",\"answer\":\"Angles subtended by the same arc at the circumference are equal, and the angle at the centre is twice the angle at the circumference.\"},{\"question\":\"What is special about a cyclic quadrilateral’s opposite angles?\",\"answer\":\"Opposite angles in a cyclic quadrilateral total 180°, so x + y = 180° and p + q = 180°.\"}]","Circle theorems - A Level Links - 2b. Circles | PDF",1788601085,23,{"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-a-level-links-2b-circles","",{"@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-a-level-links-2b-circles/207767/",{"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 is the relationship between a tangent and the radius of a circle at the point of contact?","Question",{"text":75,"@type":76},"The angle between a tangent and the radius is 90°.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How do angles subtended by the same arc relate on a circle?",{"text":80,"@type":76},"Angles subtended by the same arc at the circumference are equal, and the angle at the centre is twice the angle at the circumference.",{"name":82,"@type":73,"acceptedAnswer":83},"What is special about a cyclic quadrilateral’s opposite angles?",{"text":84,"@type":76},"Opposite angles in a cyclic quadrilateral total 180°, so x + y = 180° and p + q = 180°.","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},[93,97,101,104,109,114,119,124,128,131,135],{"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":21,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":105,"slug":138},19,"General","general"]