[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-386864-105":59,"doc-detail-386864-en":130},{"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":123,"head_meta":125,"extra_data":127,"updated_unix":129},105,"en","ml-geometry-4-3-sss-and-sas","ML% Geometry 4-3 - SSS and SAS","","This lesson explains how to prove triangles are congruent using the SSS and SAS Congruence Postulates. It starts by contrasting the full set of corresponding side and angle congruence requirements with the reduced information needed for SSS and SAS. Worked examples show formal statement-reason proof structures, including using diagonal bracing to justify rigidity and using coordinate geometry with the Distance Formula to confirm congruent triangles. Practice directions include guided vocabulary, concept, and skill checks.",{"@graph":69,"@context":122},[70,84,105],{"@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/ml-geometry-4-3-sss-and-sas/386864/",{"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/ml-geometry-4-3-sss-and-sas/386864.png","ImageObject",300,407,{"name":92,"@type":93},"Well Done","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-30","2026-09-24",true,{"@type":102,"interactionType":103,"userInteractionCount":56},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"How does the SSS Congruence Postulate prove two triangles are congruent?","Question",{"text":112,"@type":113},"If three sides of one triangle are congruent to three sides of a second triangle, then the triangles are congruent. The lesson uses this to conclude triangle congruence based on side matches only.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"What information is required for the SAS Congruence Postulate?",{"text":117,"@type":113},"Two sides and the included angle between them must be congruent in both triangles. With that shared included angle, the lesson concludes the triangles are congruent.",{"name":119,"@type":110,"acceptedAnswer":120},"When solving, how do you choose between SSS and SAS?",{"text":121,"@type":113},"Check the diagram to see whether you have three corresponding side pairs (SSS) or two side pairs plus the included angle (SAS). The lesson’s example shows that matching corresponding sides can justify using SSS when enough information is present.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},386864,1790377831,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":19,"category_name":20,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":56,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":39,"language":139,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":140,"faqs":141,"seo_title":142,"seo_description":67,"update_tm":143,"read_time":46,"preview_page_count":4},5909892210082,"https://ap-avatar.wpscdn.com/avatar/56001069da27c9f41e8?x-image-process=image/resize,m_fixed,w_180,h_180&k=1789614185077863061","Page 1 of 8  \nWhat you should learn  \n GOAL 1 Prove that triangles are congruent using the SSS and SAS Congruence Postulates.  \nUse congruence postulates in real-life problems, such as bracing a structure in Example 5.  \nWhy you should learn it  \n􀀃 Congruence postulates help you see why triangles make things stable, such asthe seaplane’s wing below and the objects in  \nExs. 30 and 31. L  \nProving Triangles are Congruent: SSS and SAS  \nHow much do you need to know about two triangles to prove that they are congruent? In Lesson 4.2, you learned that if all six pairs of corresponding parts (sides and angles) are congruent, then the triangles are congruent.  \nB E  \nF  \nA  \nC D  \nIf Sides are and Angles arethen Triangles are  \ncongruent congruent congruent  \n1. ÆAB £ ÆDE 4. ™A £ ™D ¤ABC £ ¤DEF  \n2. ÆBC £ ÆEF 5. ™B £ ™E  \n3. ÆAC £ ÆDF 6. ™C £ ™F  \nIn this lesson and the next, you will learn that you do not need all six of the pieces of information above to prove that the triangles are congruent. For example, if all three pairs of corresponding sides are congruent, then the SSS Congruence Postulate guarantees that the triangles are congruent.  \nPOSTULATE  \nPOSTULATE 19 Side-Side-Side (SSS) Congruence Postulate  \nIf three sides of one triangle are congruent to three sides of a second  \ntriangle, then the two triangles are congruent. If Side ÆMN £ ÆQR,  \nSide ÆNP £ ÆRS, and  \nSide ÆPM £ ÆSQ, then ¤MNP £ ¤QRS.  \nM  \nPN  \nq  \nR  \nS  \nEXAMPLE1 Using the SSS Congruence Postulate  \nProve that ¤PQW £ ¤TSW.  \nParagraph Proof The marks on the diagram show that ÆPQ £ÆTS, ÆPW £ ÆTW, and ÆQW £ ÆSW.  \nP  \nq W S  \nT  \n􀀂 So, by the SSS Congruence Postulate, you know that ¤PQW £ ¤TSW.  \n212 Chapter 4 Congruent Triangles  \nACTIVITY  \nConstruction  Copying a Triangle  \nFollow the steps below to construct a triangle that is congruent to a given ¤ABC.  \nC  \nA B  \n1  \nConstruct ÆDE so that it is congruent to ÆAB .(See page 104 for the construction.)  \nOpen your compass to the length AC. Use this length to draw an arc with the compass point at D.  \n3  \nDraw an arc with radius BC and center E that intersects the arc from Step 2. Label the intersection point F.  \nD E  \n4 Draw ¤DEF. By the  \nSSS Congruence Postulate,¤ABC £ ¤DEF.  \nThe SSS Congruence Postulate is a shortcut for proving two triangles are congruent without using all six pairs of corresponding parts. The postulate below is a shortcut that uses two sides and the angle that is included between the sides.  \nPOSTULATE  \nPOSTULATE 20 Side-Angle-Side (SAS) Congruence Postulate  \nIf two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the two triangles  \nare congruent.  \nIf Side ÆPQ £ ÆWX, Angle ™Q £ ™X, and Side ÆQS £ ÆXY,  \nthen ¤PQS £ ¤WXY.  \nXS  \nq  \nP  \nW Y  \nEXAMPLE2 Using the SAS Congruence Postulate  \nProve that ¤AEB £ ¤DEC.  \nB C  \nA D  \nStatements Reasons  \n1. ÆAE £ ÆDE, ÆBE £ ÆCE  \n2. ™1 £ ™2  \n3. ¤AEB £ ¤DEC  \n1. Given  \n2. Vertical Angles Theorem  \n3. SAS Congruence Postulate  \n4.3 Proving Triangles are Congruent: SSS and SAS 213  \nLogical Reasoning  \nEXAMPLE3 Choosing Which Congruence Postulate to Use  \nDecide whether enough information is given in the diagram to prove that  \n¤PQR £ ¤PSR. If there is enough information, state the congruence postulate  \nyou would use.  \nSOLUTION  \nq  \nP  \nS  \nR  \nParagraph Proof The marks on the diagram show that ÆPQ £ ÆPS and  \nÆQR £ ÆSR . By the Reflexive Property of Congruence, ÆRP £ ÆRP . Because the sides of ¤PQR are congruent to the corresponding sides of ¤PSR, you can use the SSS Congruence Postulate to prove that the triangles are congruent.  \nProving Triangles Congruent  \nEXAMPLE 4  \nProof  \nARCHITECTURE You are designing the window shown  \nin the photo. You want to make ¤DRA congruent to ¤DRG. You design the window so that ÆDR fi ÆAG and ÆRA £ ÆRG . Can you conclude that ¤DRA £ ¤DRG?  \nSOLUTION  \nTo begin, copy the diagram and label it using the given information. Then write t","cbCaiv8h9nfSQnCo","https://ap.wps.com/l/cbCaiv8h9nfSQnCo","pdf",1013548,"English","# What you should learn\n## GOAL 1\n# Why you should learn it\n# Proving Triangles are Congruent: SSS and SAS\n## Side-Side-Side (SSS) Congruence Postulate\n## Example 1 Using the SSS Congruence Postulate\n## Activity: Construction Copying a Triangle\n## Side-Angle-Side (SAS) Congruence Postulate\n## Example 2 Using the SAS Congruence Postulate\n## Example 3 Choosing Which Congruence Postulate to Use\n## Example 4 Architecture window congruence\n## Example 5 Triangular Frameworks are Rigid\n## Example 6 Congruent Triangles in a Coordinate Plane\n# Guided Practice","[{\"question\":\"How does the SSS Congruence Postulate prove two triangles are congruent?\",\"answer\":\"If three sides of one triangle are congruent to three sides of a second triangle, then the triangles are congruent. The lesson uses this to conclude triangle congruence based on side matches only.\"},{\"question\":\"What information is required for the SAS Congruence Postulate?\",\"answer\":\"Two sides and the included angle between them must be congruent in both triangles. With that shared included angle, the lesson concludes the triangles are congruent.\"},{\"question\":\"When solving, how do you choose between SSS and SAS?\",\"answer\":\"Check the diagram to see whether you have three corresponding side pairs (SSS) or two side pairs plus the included angle (SAS). The lesson’s example shows that matching corresponding sides can justify using SSS when enough information is present.\"}]","ML% Geometry 4-3 - SSS and SAS | PDF",1790277580]