[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-207771-en":3,"doc-seo-207771-105":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},207771,3985741905716,"Kyle","https://ap-avatar.wpscdn.com/davatar_994ba38a5ba835b3df7d355c54d3ed8d",4,"Exam","Circle Definitions and Theorems - Key Geometry Concepts","Circle Definitions and Theorems compiles essential vocabulary and major results from circle geometry. It defines key parts such as center, radius, diameter, chord, arc, tangent, and secant, along with their relationships using central angles, inscribed angles, and intercepted arcs. It then summarizes core theorem set-ups including chord and arc congruence, perpendicularity properties, angle theorems (inscribed, semicircle, cyclic quadrilateral), and several tangent- and secant-based exterior angle formulas, supporting standardized problem solving and review.","CIRCLE DEFINITIONS AND THEOREMS  \nDEFINITIONS  \n\n| Circle- The set of points in a plane equidistant from 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 a circle are congruent, then they determine two central angles that are\u003Cbr>congruent. | |  |\n| --- | --- | --- |\n| Chord Arcs Theorem If two chords in acircle are congruent, then their intercepted arcs\u003Cbr>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\u003Cbr>Two congruent chords in a circle are\u003Cbr>equidistant\u003Cbr>from the center of the circle. | | |\n| Perpendicular Bisector of a Chord Theorem The 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 a circle from a point outside the circle are congruent. | |  |\n\nTHEOREMS  \n\n| Inscribed Angle Theorem The measure of an angle inscribed in a circle is one-half the measure of th","cbCainZlJ7OnoZca","https://ap.wps.com/l/cbCainZlJ7OnoZca","pdf",638499,1,8,"English","en",105,"# Definitions\n## Core circle terms\n## Angles and arcs\n# Theorems\n## Inscribed-angle relationships\n## Chord and arc theorems\n## Tangent and secant angle theorems\n## Cyclic quadrilaterals and parallel-line arc results\n# Additional definitions","[{\"question\":\"What does “radius,” “diameter,” and “circumference” mean in these notes?\",\"answer\":\"The notes define radius as a segment from the center to a point on the circle, diameter as a segment through the center with endpoints on the circle, and circumference as the distance around the circle’s edge.\"},{\"question\":\"How do chord congruence results relate to central angles and arcs?\",\"answer\":\"If two chords are congruent, the corresponding central angles are congruent (Chord Central Angles Theorem). It also implies their intercepted arcs are congruent (Chord Arcs Theorem).\"},{\"question\":\"What angle formulas are used for tangent- and secant-based angles outside a circle?\",\"answer\":\"For tangent-chord angles, the angle equals half the intercepted arc measure. For two secants outside the circle, the angle equals half the difference of the intercepted arcs. Similar one-half-difference logic is used for tangent-secant and tangent-tangent exterior angles.\"}]","Circle Definitions and Theorems - Key Geometry Concepts | PDF",1788601094,20,{"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":85,"head_meta":87,"extra_data":89,"updated_unix":28},"circle-definitions-and-theorems-key-geometry-concepts","",{"@graph":36,"@context":84},[37,53,67],{"@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-definitions-and-theorems-key-geometry-concepts/207771/",{"url":52,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":61,"encodingFormat":60,"isAccessibleForFree":62,"interactionStatistic":63},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-09-05",true,{"@type":64,"interactionType":65,"userInteractionCount":4},"InteractionCounter",{"@type":66},"ViewAction",{"@type":68,"mainEntity":69},"FAQPage",[70,76,80],{"name":71,"@type":72,"acceptedAnswer":73},"What does “radius,” “diameter,” and “circumference” mean in these notes?","Question",{"text":74,"@type":75},"The notes define radius as a segment from the center to a point on the circle, diameter as a segment through the center with endpoints on the circle, and circumference as the distance around the circle’s edge.","Answer",{"name":77,"@type":72,"acceptedAnswer":78},"How do chord congruence results relate to central angles and arcs?",{"text":79,"@type":75},"If two chords are congruent, the corresponding central angles are congruent (Chord Central Angles Theorem). It also implies their intercepted arcs are congruent (Chord Arcs Theorem).",{"name":81,"@type":72,"acceptedAnswer":82},"What angle formulas are used for tangent- and secant-based angles outside a circle?",{"text":83,"@type":75},"For tangent-chord angles, the angle equals half the intercepted arc measure. For two secants outside the circle, the angle equals half the difference of the intercepted arcs. Similar one-half-difference logic is used for tangent-secant and tangent-tangent exterior angles.","https://schema.org",{"og:url":52,"og:type":86,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":88,"canonical":52},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":91},[92,96,100,103,108,113,118,122,126,129,133],{"id":20,"doc_module":4,"doc_module_name":46,"category_name":93,"show_sort_weight":94,"slug":95},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":97,"show_sort_weight":98,"slug":99},"Literature",80,"literature",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":101,"slug":102},70,"exam",{"id":104,"doc_module":4,"doc_module_name":46,"category_name":105,"show_sort_weight":106,"slug":107},5,"Comic",60,"comic",{"id":109,"doc_module":4,"doc_module_name":46,"category_name":110,"show_sort_weight":111,"slug":112},6,"Technology",50,"technology",{"id":114,"doc_module":4,"doc_module_name":46,"category_name":115,"show_sort_weight":116,"slug":117},7,"Healthcare",40,"healthcare",{"id":21,"doc_module":4,"doc_module_name":46,"category_name":119,"show_sort_weight":120,"slug":121},"Research & Report",30,"research-report",{"id":123,"doc_module":4,"doc_module_name":46,"category_name":124,"show_sort_weight":29,"slug":125},9,"Religion & Spirituality","religion-spirituality",{"id":29,"doc_module":4,"doc_module_name":46,"category_name":127,"show_sort_weight":29,"slug":128},"World Cup","world-cup",{"id":130,"doc_module":4,"doc_module_name":46,"category_name":131,"show_sort_weight":130,"slug":132},10,"Lifestyle","lifestyle",{"id":134,"doc_module":4,"doc_module_name":46,"category_name":135,"show_sort_weight":104,"slug":136},19,"General","general"]