[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-167363-en":3,"doc-seo-167363-105":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},167363,1099514068365,"Aurelia","https://ap-avatar.wpscdn.com/avatar/10000253d8d9f28188e?_k=1776742907772140068",4,"Exam","Triangles - Advanced Pages","Advanced study notes on triangles, covering key triangle types (scalene, isosceles, equilateral, and isosceles right-angled) with defining properties and angle-side relationships. Fundamental triangle properties summarize inequality relations among sides, angle-sum rules for interior angles and exterior angles, and classifications of acute, right, and obtuse triangles using the c² comparison to a²+b². The document further explains congruence criteria (SSS, SAS, ASA, AAS, and RHS) and similarity theorems for similar triangles (AAA, SSS, and SAS).","10.TRIANGLE  \n# ■   Triangles\n\n\n| Types of triangles   | Definition/Property   | Diagram   |\n| --- | --- | --- |\n| Scalene triangle   | (i)A triangle in which none of the two sides are  \u003Cbr>equal is called a scalene triangle  \u003Cbr>(ii)All the three angles are also different   | B  \u003Cbr>a≠b≠c   |\n| Isosceles triangle   | (i)A triangles in which at least two sides are equal is  \u003Cbr>called an isosceles triangle.  \u003Cbr>(ii)In this triangle,the angles opposite to the  \u003Cbr>congruent sides are also equal  \u003Cbr>(iii)2 medians,2 altitudes equal.  \u003Cbr>(iv)Internal bisectors of 2 angles are equal.  \u003Cbr>(v)Bisector of vertical angle bisects the base and  \u003Cbr>perpendicular to the base.  \u003Cbr>(vi)May be acute,obtuse or right angled triangle..   | C  \u003Cbr>B  \u003Cbr>AB=AC  \u003Cbr>∠B=∠C   |\n| Equilateral triangle   | (i)A triangle in which all the three sides are equal is  \u003Cbr>called an equilateral triangle.  \u003Cbr>(ii)In this triangle each angle is congruent and equal  \u003Cbr>to 60°  \u003Cbr>iii)Always acute angled.  \u003Cbr>(iv)Incentre,circumcentre,orthocentre and centroid  \u003Cbr>coincide.  \u003Cbr>(v)Point of intersection of altitude,medians and  \u003Cbr>angular bisectors is same.   | C  \u003Cbr>AB=BC=AC  \u003Cbr>∠A=∠B=∠C=60°   |\n| Isosceles right  \u003Cbr>angled triangle  \u003Cbr>AB=BC   | (i)2 sides are equal  \u003Cbr>(ii)Angle included by the equal sides is 90°.  \u003Cbr>(iii)Side opposite to 90°is hypotenuse and is the  \u003Cbr>greatest side.  \u003Cbr>(iv)Median to the hypotenuse is half of the hypotenuse  \u003Cbr>(v)Of the two acute angles,if one is 30°.The smallest  \u003Cbr>side is half of the greatest side or the side opposite  \u003Cbr>to 30°is half of hypotenuse.   | C   |\n\n# ■ Fundamental properties of triangles\n\n● Sum of any two sides is always greater than the third side.  \n● The difference of any two sides is always less than the third side.  \n● Greater angle has a greater side opposite to it and smaller angle has a smaller side opposite toit i.e.,if two sides of triangle are not congruent then the angle opposite to the greater side is greater.  \n● Let a,b and cbe the three sides of a △ABC and c is the largest side,then  \n▶ if c²\u003Ca²+b²,the triangle is acute angle triangle  \n▶ if c²=a²+b²,the triangle is right angled triangle  \n▶if c²>a²+b²,the triangle is obtuse angle triangle  \n   \n   \n● The sum of all the three interior angles is always 180°  \ni.e ∠CAB+∠ABC+∠BCA=180°  \nF  \n● The sum of three(ordered)exterior angles of a angles is 360°  \nF  \nFig.(i)  \nFig.(ii)  \nIn fig(i):∠FAC+∠ECB+∠DBA=360°  \nIn fig(ii):∠FAB+∠DBC+∠ECA=360°  \n● A triangle must have at least two acute angles  \n● In atriangle,the measure of an exterior angle equals the sum of the measures of the interior opposite angles.  \n● The measure of an exterior angle of a triangle is greater than the measure of each of the opposite interior angles.  \n(u十36=100  \n# ■ Congruence of triangles\n\n\n| Test   | Property   | Diagram   |\n| --- | --- | --- |\n| S-S-S   | (Side-Side-Side)  \u003Cbr>If the three sides of one triangle are equal to the  \u003Cbr>corresponding three sides of the other triangle,then the two  \u003Cbr>triangles are congruent  \u003Cbr>AB≌PQ,AC≌PR,BC=QR  \u003Cbr>△ABC≌△PQR  \u003Cbr>∴   |  |\n| S-A-S   | (Side-Angle-Side)  \u003Cbr>If two sides and the included angle between them be congruent to  \u003Cbr>the corresponding sides and the angle included between them,  \u003Cbr>of the other triangle then the two triangles are congruent.  \u003Cbr>AB≥PQ,∠ABC≌∠PQR,BC=QR  \u003Cbr>△ABC≌△PQR   |  |\n| A-S-A   | (Angle-Side-Angle)  \u003Cbr>If two angles and the included side of a triangle are congruent  \u003Cbr>to the corresponding angles and the included  \u003Cbr>side of the other triangle,then the two triangles are congruent.  \u003Cbr>∠ABC≥∠PQR,BC≥QR,∠ACB≥∠PRQ  \u003Cbr>△ABC≥△PQR   |  |\n| A-A-S   | (Angle-Angle-Side)  \u003Cbr>If two angles and a side other than the included side of a one  \u003Cbr>triangle are congruent to the corresponding angles and a  \u003Cbr>corresponding side other than the included side of the other  \u003Cbr>triangle,then the two triangles are congruent.  \u003Cbr>∠ABC≌∠PQR,∠ACB≥∠PRQ  \u003Cbr>(or AB≥PQ)  \u003Cbr>AC≥PR  \u003Cbr>an","cbCairy5N3uJO7PO","https://ap.wps.com/l/cbCairy5N3uJO7PO","pdf",13255199,1,17,"English","en",105,"# Triangles\n## Types of triangles\n## Fundamental properties of triangles\n## Congruence of triangles\n## Theorems related to similar triangles\n## Important Definition","[{\"question\":\"What are the main types of triangles and their defining conditions?\",\"answer\":\"The notes distinguish scalene, isosceles, equilateral, and isosceles right-angled triangles, using which sides are equal and how the angles relate (e.g., equilateral angles are 60°).\"},{\"question\":\"How are acute, right, and obtuse triangles identified using side lengths?\",\"answer\":\"Let c be the largest side. If c² \\u003c a²+b² the triangle is acute; if c² = a²+b² it is right-angled; if c² \\u003e a²+b² it is obtuse.\"},{\"question\":\"What congruence tests are listed for proving two triangles congruent?\",\"answer\":\"The document lists SSS, SAS, ASA (A-S-A), AAS (Angle-Angle-Side), and RHS (Right angle-Hypotenuse-Side) as congruence criteria.\"}]","Triangles - Advanced Pages | PDF",1788212117,43,{"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},"triangles-advanced-pages","",{"@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/triangles-advanced-pages/167363/",{"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","2026-09-11","2026-08-31",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 are the main types of triangles and their defining conditions?","Question",{"text":75,"@type":76},"The notes distinguish scalene, isosceles, equilateral, and isosceles right-angled triangles, using which sides are equal and how the angles relate (e.g., equilateral angles are 60°).","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How are acute, right, and obtuse triangles identified using side lengths?",{"text":80,"@type":76},"Let c be the largest side. If c² \u003C a²+b² the triangle is acute; if c² = a²+b² it is right-angled; if c² > a²+b² it is obtuse.",{"name":82,"@type":73,"acceptedAnswer":83},"What congruence tests are listed for proving two triangles congruent?",{"text":84,"@type":76},"The document lists SSS, SAS, ASA (A-S-A), AAS (Angle-Angle-Side), and RHS (Right angle-Hypotenuse-Side) as congruence criteria.","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},[93,97,101,104,109,114,119,124,129,132,136],{"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":125,"doc_module":4,"doc_module_name":46,"category_name":126,"show_sort_weight":127,"slug":128},9,"Religion & Spirituality",20,"religion-spirituality",{"id":127,"doc_module":4,"doc_module_name":46,"category_name":130,"show_sort_weight":127,"slug":131},"World Cup","world-cup",{"id":133,"doc_module":4,"doc_module_name":46,"category_name":134,"show_sort_weight":133,"slug":135},10,"Lifestyle","lifestyle",{"id":137,"doc_module":4,"doc_module_name":46,"category_name":138,"show_sort_weight":105,"slug":139},19,"General","general"]