[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-1-en-105":3,"doc-seo-573391-105":53,"doc-detail-573391-en":126},{"code":4,"msg":5,"data":6},0,"success",[7,14,19,24,29,34,39,44,49],{"id":8,"doc_module":9,"doc_module_name":10,"category_name":11,"show_sort_weight":12,"slug":13},11,1,"Template","Presentations",90,"presentations",{"id":15,"doc_module":9,"doc_module_name":10,"category_name":16,"show_sort_weight":17,"slug":18},12,"Resumes",80,"resumes",{"id":20,"doc_module":9,"doc_module_name":10,"category_name":21,"show_sort_weight":22,"slug":23},14,"Invoices",70,"invoices",{"id":25,"doc_module":9,"doc_module_name":10,"category_name":26,"show_sort_weight":27,"slug":28},15,"Posters",60,"posters",{"id":30,"doc_module":9,"doc_module_name":10,"category_name":31,"show_sort_weight":32,"slug":33},16,"Social Media",50,"social-media",{"id":35,"doc_module":9,"doc_module_name":10,"category_name":36,"show_sort_weight":37,"slug":38},17,"Forms",40,"forms",{"id":40,"doc_module":9,"doc_module_name":10,"category_name":41,"show_sort_weight":42,"slug":43},18,"Letters",30,"letters",{"id":45,"doc_module":9,"doc_module_name":10,"category_name":46,"show_sort_weight":47,"slug":48},21,"Paper Templates",5,"papers-templates",{"id":50,"doc_module":9,"doc_module_name":10,"category_name":51,"show_sort_weight":4,"slug":52},158,"General","general-158",{"code":4,"msg":54,"data":55},"ok",{"site_id":56,"language":57,"slug":58,"title":59,"keywords":60,"description":61,"schema_data":62,"social_meta":119,"head_meta":121,"extra_data":123,"updated_unix":125},105,"en","mathematics-webinar-circle-geometry-part-1","Mathematics Webinar - Circle Geometry - Part 1","","This educational material provides a comprehensive exploration of circle geometry theorems and their rigorous mathematical proofs. The document outlines fundamental properties, including the relationship between the angle subtended by a chord at the center of a circle and its corresponding angle at the circumference, as well as characteristics of cyclic quadrilaterals where opposite angles sum to 180 degrees. Detailed proof procedures, geometric diagrams, and structured worked examples are presented to guide students through the application of these theorems in solving complex geometric problems. The content emphasizes foundational principles such as the bisecting properties of radii relative to chords and perpendicularity conditions within circles, serving as a valuable resource for students aiming to master high-school level geometry and analytical reasoning.",{"@graph":63,"@context":118},[64,80,101],{"@type":65,"itemListElement":66},"BreadcrumbList",[67,71,74,77],{"item":68,"name":69,"@type":70,"position":9},"https://docshare.wps.com","Home","ListItem",{"item":72,"name":10,"@type":70,"position":73},"https://docshare.wps.com/template/",2,{"item":75,"name":51,"@type":70,"position":76},"https://docshare.wps.com/template/general/",3,{"item":78,"name":59,"@type":70,"position":79},"https://docshare.wps.com/template/mathematics-webinar-circle-geometry-part-1/573391/",4,{"url":78,"name":59,"@type":81,"image":82,"author":87,"headline":59,"publisher":90,"fileFormat":93,"inLanguage":57,"description":61,"dateModified":94,"datePublished":95,"encodingFormat":93,"isAccessibleForFree":96,"interactionStatistic":97},"DigitalDocument",{"url":83,"@type":84,"width":85,"height":86},"https://docshare.wps.com/thumbnails/mathematics-webinar-circle-geometry-part-1/573391.png","ImageObject",442,249,{"name":88,"@type":89},"Rhys","Person",{"url":68,"name":91,"@type":92},"DocShare","Organization","application/pdf","2026-10-17","2026-10-14",true,{"@type":98,"interactionType":99,"userInteractionCount":79},"InteractionCounter",{"@type":100},"ViewAction",{"@type":102,"mainEntity":103},"FAQPage",[104,110,114],{"name":105,"@type":106,"acceptedAnswer":107},"What is the relationship between the angle subtended by a chord at the center and at the circumference?","Question",{"text":108,"@type":109},"According to the circle theorem, the angle subtended by a chord at the center of a circle is twice the size of the angle that it subtends at the circumference.","Answer",{"name":111,"@type":106,"acceptedAnswer":112},"What is the property of opposite angles in a cyclic quadrilateral?",{"text":113,"@type":109},"In a cyclic quadrilateral, the sum of the opposite angles is always 180 degrees.",{"name":115,"@type":106,"acceptedAnswer":116},"How does a radius affect a chord in circle geometry?",{"text":117,"@type":109},"A radius that is perpendicular to a chord bisects the chord, and conversely, a radius that bisects a chord is perpendicular to it.","https://schema.org",{"og:url":78,"og:type":120,"og:title":59,"og:site_name":91,"og:description":61},"article",{"robots":122,"canonical":78},"index,follow",{"doc_id":124,"site_id":56},573391,1792012975,{"code":4,"msg":5,"data":127},{"doc_id":124,"user_id":128,"nickname":88,"user_avatar":129,"doc_module":9,"category_id":50,"category_name":51,"doc_title":59,"doc_description":61,"doc_content":130,"file_id":131,"file_url":132,"file_type":133,"file_size":134,"view_count":79,"is_deleted":4,"is_public":9,"is_downloadable":9,"audit_status":9,"page_count":135,"language":136,"language_code":57,"site_id":56,"html_lang":57,"table_of_contents":137,"faqs":138,"seo_title":139,"seo_description":61,"update_tm":140,"read_time":141},687207024643,"https://ap-avatar.wpscdn.com/davatar_3d24733baf745e90a7e4bdd5f77d97b2","Proofs Of Theorems  \n6 Marks out of 150  \nTheorem Proof  \nPROOF REQUIRED  \nTheorem Proof  \nPROOF REQUIRED  \nTheorem Proof  \nPROOF REQUIRED  \nTheorem Proof  \nPROOF REQUIRED  \nTheorems and Worked Examples  \n෡􀜰1 = 90°; radius bisects chord Theorem 1 AB=BC ; radius ꓕ chord  \n෠􀜲1 = 90°; radius bisects chord  \nAP=PB ; radius ꓕ chord  \nWORKED EXAMPLE 1  \n( I DO)  \nWORKED EXAMPLE 1 SOLUTION  \nWORKED EXAMPLE 2 (I DO)","cbCaigLAjwxum5hf","https://ap.wps.com/l/cbCaigLAjwxum5hf","pdf",2404508,72,"English","# Circle Geometry Theorems\n## Theorem Proofs\n## Worked Examples","[{\"question\":\"What is the relationship between the angle subtended by a chord at the center and at the circumference?\",\"answer\":\"According to the circle theorem, the angle subtended by a chord at the center of a circle is twice the size of the angle that it subtends at the circumference.\"},{\"question\":\"What is the property of opposite angles in a cyclic quadrilateral?\",\"answer\":\"In a cyclic quadrilateral, the sum of the opposite angles is always 180 degrees.\"},{\"question\":\"How does a radius affect a chord in circle geometry?\",\"answer\":\"A radius that is perpendicular to a chord bisects the chord, and conversely, a radius that bisects a chord is perpendicular to it.\"}]","Mathematics Webinar - Circle Geometry - Part 1 | PDF",1791938834,25]