[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-207776-en":3,"doc-seo-207776-105":30,"detail-sidebar-cat-0-en-105":90},{"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":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},207776,549768702563,"Fahsai","https://ap-avatar.wpscdn.com/avatar/8000c4aa63b76e948b?x-image-process=image/resize,m_fixed,w_180,h_180&k=1786536092046926083",4,"Exam","GCSE Mathematics Grade 8/9 Proof of Circle Theorems Solutions","This document contains solutions for proving circle theorems for GCSE Mathematics, targeting Grade 8/9. It includes proofs for the angle subtended by an arc at the centre, the angle subtended by a semicircle, angles in the same segment, opposite angles of a cyclic quadrilateral, and the alternate segment theorem. Each proof is presented with a diagram and step-by-step logical reasoning. The materials are designed for exam preparation, requiring students to have specific stationery and a scientific calculator. Instructions emphasize answering all questions, showing working, and using provided spaces. The document is a valuable resource for students aiming for high grades in geometry.","# GCSEEdexcel\n\nMathematicsGrade 8/9  \nMaterials  \nFor this paper you must have:  \n● Ruler  \n·Pencil,Rubber,Protractor and Compass  \n·Scientific calculator,which you are expected to use when appropriate  \n# Instructions\n\n● Answerall questions  \n·Answer questions in the space provided  \n·All working must be shown  \n·Do all rough work in this book.Cross out any rough work you don't want to be marked  \n# Information\n\n·The marks for the questions are shown in brackets  \nName:  \nDate:  \n# PROOF OF CIRCLE THEOREMS\n\nMark Score(%)  \nLeave  \nblank  \nProve that the angle subtended by an arc at the centre of a circle is twice the angle subtended atany point on the circumference.  \nAngle BOC =K  \nAngle BOA=4  \n∴Angle.AOC =360-x-4  \n2 angles in isosceles triangle are the same  \nAngle  \n360--4=2(180     )  \nLeave  \nblank  \nC  \nA  \nProve the angle subtended at the circumference by a semicircle is a right angle.  \nAngie AOB=x  \nAngle COB=4  \n∴x+y=180°   argles ina straight uine odd up to 180°  \n2 angles in an isosceles triangle are the same  \nIf x+y=180°  ,  \n=180-90  \nzq0°  \nwww.examqa.com  \nLeave  \nblank  \nD  \nProve that angles in the same segment are equal.  \nAngle COD =x  \n→angles at circumference are half of theangle a the centre  \nwww.examqa.com  \nB  \nA  \nProve that opposite angles of a cyclic quadrilateral sum to 180°  \niet angle AOC(top)=Klet angie AOC (botom)=4  \nAngle飞(+9=360   →angle ata point is 360  \nand→angies at a circumference is half ofthe angle at the centre  \nif x+y=360°,  then  \nLeave  \nblank  \nwww.examqa.com  \nLeave  \nblank  \nE  \nProve the alternate segment theorem.  \nlet Angle BAE=K  \n∴Angle BAC=90-x →when tangent meets radius is qo·  \nAngle 9CB=x→angies ina triangle add up to 180“∴(80-90-90-x)=90~90+x =飞  \nAngle ADB=x→angles in same segment are equal  \nwww.examqa.com","cbCaigLdb0OPs9hV","https://ap.wps.com/l/cbCaigLdb0OPs9hV","pdf",893606,1,6,"English","en",105,"# Proof of Circle Theorems\n## Prove that the angle subtended by an arc at the centre of a circle is twice the angle subtended at any point on the circumference.\n## Prove the angle subtended at the circumference by a semicircle is a right angle.\n## Prove that angles in the same segment are equal.\n## Prove that opposite angles of a cyclic quadrilateral sum to 180°\n## Prove the alternate segment theorem.","[{\"question\":\"What mathematical tools are required for this exam paper?\",\"answer\":\"For this paper, students are required to have a ruler, pencil, rubber, protractor, compass, and a scientific calculator.\"},{\"question\":\"What are the key circle theorems proven in this document?\",\"answer\":\"This document proves the theorem that the angle at the centre is twice the angle at the circumference, the angle in a semicircle is a right angle, angles in the same segment are equal, opposite angles of a cyclic quadrilateral sum to 180°, and the alternate segment theorem.\"},{\"question\":\"What is the purpose of this document?\",\"answer\":\"This document serves as a solution guide for proving essential circle theorems in GCSE Mathematics, specifically for students aiming for Grade 8/9, acting as exam preparation material.\"}]","GCSE Mathematics Grade 8/9 Proof of Circle Theorems Solutions | 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