[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-491085-105":59,"doc-detail-491085-en":131},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":124,"head_meta":126,"extra_data":128,"updated_unix":130},105,"en","circle-definitions-and-theorems-definitions","Circle Definitions and Theorems - Definitions","","Circle definitions and theorems covering core geometry vocabulary and key relationships. The material defines circle, radius, circumference, congruent circles, diameter, chord, tangent, secant, inscribed angles, central angles, and arcs, including minor, major, and semicircle arcs. It also explains arc naming conventions and the role of endpoints. Theorems provided include connections between chords and central angles, chord and arc congruence, chord-perpendicular properties, and tangent-to-radius perpendicularity, plus inscribed angle and cyclic quadrilateral results.",{"@graph":69,"@context":123},[70,84,106],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":20,"@type":76,"position":81},"https://docshare.wps.com/document/exam/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/circle-definitions-and-theorems-definitions/491085/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/circle-definitions-and-theorems-definitions/491085.png","ImageObject",300,407,{"name":92,"@type":93},"Patrick","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-11-06","2026-10-01",true,{"@type":102,"interactionType":103,"userInteractionCount":105},"InteractionCounter",{"@type":104},"ViewAction",17,{"@type":107,"mainEntity":108},"FAQPage",[109,115,119],{"name":110,"@type":111,"acceptedAnswer":112},"What is the difference between a minor arc and a major arc?","Question",{"text":113,"@type":114},"A minor arc has a measure less than 180°, while a major arc has a measure greater than 180°. A semicircle arc has a measure equal to 180°.","Answer",{"name":116,"@type":111,"acceptedAnswer":117},"How are arcs named, and how do you indicate the major arc?",{"text":118,"@type":114},"Arcs are named by their endpoints, such as arc AB. Unless stated otherwise, it refers to the minor arc (the shorter one). To indicate the major arc, use three letters and include an extra point, like arc ACB.",{"name":120,"@type":111,"acceptedAnswer":121},"What does the Inscribed Angle Theorem say?",{"text":122,"@type":114},"The measure of an angle inscribed in a circle is one-half the measure of the central angle that intercepts the same arc.","https://schema.org",{"og:url":83,"og:type":125,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":127,"canonical":83},"index,follow",{"doc_id":129,"site_id":62},491085,1791690893,{"code":4,"msg":5,"data":132},{"doc_id":129,"user_id":133,"nickname":92,"user_avatar":134,"doc_module":4,"category_id":19,"category_name":20,"doc_title":65,"doc_description":67,"doc_content":135,"file_id":136,"file_url":137,"file_type":138,"file_size":139,"view_count":105,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":19,"language":140,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":67,"update_tm":144,"read_time":52,"preview_page_count":4},549758146520,"https://ap-avatar.wpscdn.com/avatar/80002397d8c0411e94?_k=1775819394049821470","CIRCLE DEFINITIONS AND THEOREMS  \nDEFINITIONS  \n\n| Circle- The set of points in a plane equidistant form a given point(the center of the circle) .\u003Cbr>Radius-A segment from the center of the circle to a point on the circle(the distance from the center to a point on the circle.)\u003Cbr>Circumference – distance around the edge of the circle\u003Cbr>Congruent Circles-two circles with the same radius. | |\n| --- | --- |\n| Diameter – A segment that goes through the center of the circle, with both endpoints on the edge of the circle.\u003Cbr>Chord-A line segment that goes from one point to another on the circle's circumference. | \u003Cbr>\u003Cbr>DI is the diameter.\u003Cbr>\u003Cbr>CH isa chord.\u003Cbr>DO and OI are both radii. Diameter 􀀠 2* radius. Radius 􀀠 12 * diameter |\n| Tangent – a line that intersects a circle at only one point. The radius at the point of tangency is perpendicular to the tangent line.\u003Cbr>\u003Cbr>TG isa tangent line.\u003Cbr>The point of tangency isthe point T.\u003Cbr>OTis a radius and TG 􀁁 OT\u003Cbr>Secant – a line that intersects a circle at two points.\u003Cbr>SE isa secant line. | |\n| Inscribed Angle-an angle made from points sitting on the circle's edge. | \u003Cbr>A and C are \"end points\"B is the \"apex point\" |\n\n\n| Central angle-an angle with vertex at the center of the circle\u003Cbr>Arc – part of the circumference (edge) of the circle. The measure of an arc is equal to the measure of\u003Cbr>the central angle formed by its endponts. | | AB is an arc\u003Cbr>􀂑AOB isa central angle. m􀂑AOB 􀀠 mAB\u003Cbr>|\n| --- | --- | --- |\n| Arcs are named by their endpoints. The dashed arc to the right would be called \"arc AB\" . or \"arc BA\", the order of the endpoints does not matter. As a shorthand this can be written as the letters AB with a curving line above them Example:  which is read \"arc AB\" .\u003Cbr>Notice that this naming can be ambiguous. For example it may mean the major arc AB, where you go the long way around the bottom of the circle. Unless stated otherwise, it always means the minor arc-the shortest of the two.\u003Cbr>If you want to indicate the major arc, add an extra point and use three letters in the name. For example in the\u003Cbr>diagram on the right the major arc is indicated by ACB which is the long arc from A to B going around the bottom via C.\u003Cbr>There are two measures ofan arc\u003Cbr>1. The length of the arc\u003Cbr>2. The angle of the arc\u003Cbr>Here is a semicircle arc with a central angle of 180° it covers exactly half of the circumference.The endpoints A and B lie on the diameter of the circle.When naming this arc, we use an extra point C, now we have arc ACB. The third point tells us which half of the circumference the semicircle arc covers. If there are just two points wepresume that the named arc is the smallest one on the circumference, the minor arc (as long as the arc is not a semicircle where both arcs are the same size) .\u003Cbr>| \u003Cbr>Minor arc – arc whose measure is less that 180 degrees.\u003Cbr>Major arc – arc whose measure is greater than 180 degrees.\u003Cbr>Semicircle arc-arc whose measure = 180 degrees.\u003Cbr>|  |\n\n\n| Chord Central Angles Theorem If two chords in acircle are congruent, then they\u003Cbr>determine two central angles that are congruent. | |  |\n| --- | --- | --- |\n| Chord Arcs Theorem If two chords in a circle are congruent, then their intercepted arcs are congruent. | |  |\n| Perpendicular to a Chord Theorem The perpendicular from the center of a circle to a chord is the bisector of the chord. | |  |\n| Chord Distance to Center Theorem Two congruent chords in a circle are equidistant\u003Cbr>from the center of the circle. | | |\n| Perpendicular Bisector of a Chord Theorem The\u003Cbr>perpendicular bisector of a chord\u003Cbr>passes through the center of the circle. |  |  |\n| Tangent Theorem A tangent to a circle is perpendicular to the radius drawn to the point of tangency. | |  |\n| Tangent Segments Theorem Tangent segments to acircle from a point outside the\u003Cbr>circle are congruent. | |  |\n\nTHEOREMS  \n\n| Inscribed Angle Theorem The measure of an angle inscribed in a circle is one-half\u003Cbr>the measure of the","cbCaihtX122Toea6","https://ap.wps.com/l/cbCaihtX122Toea6","pdf",401353,"English","# Definitions\n## Basic circle parts\n## Diameter, chord, tangent, secant\n## Inscribed and central angles\n## Arcs and their naming\n# Theorems\n## Chords and arcs theorems\n## Perpendicular and bisector theorems\n## Tangent theorems\n## Inscribed angle and cyclic quadrilateral theorems","[{\"question\":\"What is the difference between a minor arc and a major arc?\",\"answer\":\"A minor arc has a measure less than 180°, while a major arc has a measure greater than 180°. A semicircle arc has a measure equal to 180°.\"},{\"question\":\"How are arcs named, and how do you indicate the major arc?\",\"answer\":\"Arcs are named by their endpoints, such as arc AB. Unless stated otherwise, it refers to the minor arc (the shorter one). To indicate the major arc, use three letters and include an extra point, like arc ACB.\"},{\"question\":\"What does the Inscribed Angle Theorem say?\",\"answer\":\"The measure of an angle inscribed in a circle is one-half the measure of the central angle that intercepts the same arc.\"}]","Circle Definitions and Theorems - Definitions | PDF",1790845370]