[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-386024-105":59,"doc-detail-386024-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","chapter-5-congruence-triangles-methods-for-showing-triangles-to-be-congruent","Chapter 5 - Congruence & Triangles - Methods for Showing Triangles to be Congruent","","Chapter 5 focuses on identifying and proving triangle congruence using multiple standard criteria. It explains how to match congruent figures through corresponding parts and introduces CPCTC. The lesson compares common congruence methods, including SSS, SAS, ASA, and AAS, and highlights the roles of included and non-included sides or angles. Worked examples guide determining corresponding angles and sides, writing congruence statements, and constructing two-column proofs with shared and vertical angles.",{"@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/chapter-5-congruence-triangles-methods-for-showing-triangles-to-be-congruent/386024/",{"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/chapter-5-congruence-triangles-methods-for-showing-triangles-to-be-congruent/386024.png","ImageObject",300,407,{"name":92,"@type":93},"Levi","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-11-01","2026-09-24",true,{"@type":102,"interactionType":103,"userInteractionCount":44},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"How do you show two triangles are congruent using corresponding parts?","Question",{"text":112,"@type":113},"Identify the matching corresponding vertices, then use corresponding angles and corresponding sides to form a congruence statement. If triangles are congruent, CPCTC confirms corresponding parts are congruent.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"When using SSS, what information is needed?",{"text":117,"@type":113},"You need three sides of one triangle congruent to three sides of a second triangle. If all three side pairs match, the triangles are congruent by the SSS Congruence Postulate.",{"name":119,"@type":110,"acceptedAnswer":120},"What is the difference between the included and non-included side in ASA and AAS?",{"text":121,"@type":113},"For ASA, the side between the two angles is the included side. For AAS, the side between the two angles is not included; instead, a non-included side is used along with two congruent angles and their corresponding side condition.","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},386024,1790507157,{"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":44,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":139,"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":145,"preview_page_count":4},7971461740909,"https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d","Chapter 5: Congruence & Triangles  \nMethods for Showing Triangles to be Congruent  \nSSS side-side-side  \nright  \nSAS side-angle-side  \nAAS angle-angle-side  \nASA angle-side-angle  \nHL angle-hypotenuse-leg  \n5.1 CONGRUENCE & TRIANGLES  \nGoal: Identify congruent triangles and corresponding parts  \n∂ Congruence  \n􀂾 Corresponding parts—  \nKey Vocabulary:  \n􀁸 Corresponding parts  \n􀁸 Congruent figures  \n􀂾 Figures are congruent if   \n􀂃  CPCTC—  \n∂ Congruent Parts of Congruent Triangles  \n􀂃 In the diagram below, triangle XYZ is congruent to triangle MNP.  \n􀂃 Written as: 􀁕XYZ 􀀣 􀁕MNP. Congruencstatemente  \n􀁸 The notation shows the corresponding vertices and thus the corresponding angles & sides.  \nCorresponding Angles  Corresponding Sides  \nExamples: Determine whether the angles or sides are corresponding angles, corresponding sides, or neither.  \nGiven that 􀁕XYZ 􀀣 􀁕EFD, determine the congruent side or angle that corresponds to the side or angle.  \nExample 9: Given that 􀁕STU 􀀣 􀁕YXZ, list all corresponding congruent parts.  \nAngles  \nExample 10: Given 􀁕ABC 􀀣 􀁕DEF, find the length of and m􀂑B.  \nExample 11: Which congruence statement is correct? Why?  \nSides  \n∂ Determine Whether  Triangles are  Congruent  \n􀂾 In the figure, || and ||  \n􀂃 Determine whether two triangles are congruent. If they are, write a congruence statement.  \n􀁸 Remember that when parallel lines are cut by a transversal, the alternate interior angles are congruent.  \n􀂃 Start by listing the side(s) &/or angles you know to be congruent.  \n􀂃 Then list any info you can deduce from the figure.  \nBe on the lookout for vertical angles and shared sides of both triangles.  \nExamples: Determining Congruence  \nDetermine whether the triangles are congruent. If they are, write a congruence statement.  \n12. 13.  \nExample 14: Given that 􀁕FGH􀀣 􀁕TSR, determine which congruence statement does not describe the triangles.  \na. 􀁕HGF 􀀣 􀁕RST b. 􀁕FHG 􀀣 􀁕TRS c. 􀁕GFH􀀣 􀁕SRT  \nExamples: Geo-Gebra  \n15. If 􀁕RST 􀀣 􀁕XYZ, find the value of n.  \nTo do this problem you must identify the angle congruent to 􀂑Y.  \n16. If 􀁕RQP 􀀣 􀁕ONM, find the value of m.   To do this problem you must identify the side congruent to  \n17. Refer to Jillian’s work below and answer the questions that follow.  \na. How does Jillian know that 􀂑BAC 􀀣 􀂑DAE?  \nb. Is 􀁕ABC 􀀣 􀁕ADE? Explain your reasoning.  \n5.2 PROVING TRIANGLES ARE CONGRUENT: SSS & SAS  \nGoal: Show triangles are congruent using SSS & SAS.  \nAccessing Prior Knowledge: Pre-Proof Warm-Ups [http://www.regentsprep.org/Regents/math/geometry/GP3/preproof.htm](http://www.regentsprep.org/Regents/math/geometry/GP3/preproof.htm)  \n∂ Side – Side – Side Congruence Postulate (SSS)  \n􀂾 If three sides of one triangle are congruent to three sides of a second triangle, then the two triangles are congruent.  \n􀂃  Sides:  \nExamples: Using the SSS Congruence Postulate  \nDoes the diagram give enough information to use the SSS Congruence Postulate? Explain your reasoning.  \n1.  \n2.  \nList those angles &/or sides you know to be congruent:  \nList those angles &/or sides you know to be congruent:  \n∂ The “Included” Angle  \n􀂾 In 􀁕ABC,   is the included angle between sides  \n.  \nExample 3: Identifying the Included Angle  \nUse the diagram to name the included angle between the two given sides.  \na.  \nb.  \nc.  \nd.  \ne.  \n∂ Side – Angle – Side Congruence Postulate (SAS)  \n􀂾 If 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 are congruent.  \n􀂃 Side: Angle: Side:   \nExamples: Using the SAS Congruence Postulate  \nDoes the diagram give enough information to use the SAS Congruence Postulate? Explain your reasoning.  \n4.  \n5.  \nList those angles &/or sides you know to be congruent:  \nList those angles &/or sides you know to be congruent:  \n6. What additional information is needed to prove the two triangles are congruent by the SAS Congruence Postulate?  \n∂ Writing Proofs  \n􀂾 Proof–  \n􀂃 The proof begins with the   ","cbCaiv1HtlJZxjsY","https://ap.wps.com/l/cbCaiv1HtlJZxjsY","pdf",2248976,21,"English","# 5.1 Congruence & Triangles\n## Methods for Showing Triangles to be Congruent\n## Congruent Parts of Congruent Triangles\n## Determine Whether Triangles are Congruent\n# 5.2 Proving Triangles Are Congruent: SSS & SAS\n## Side-Side-Side Congruence Postulate (SSS)\n## The “Included” Angle\n## Side-Angle-Side Congruence Postulate (SAS)\n## Writing Proofs\n# 5.3 Proving Triangles Are Congruent: ASA & AAS\n## The “Included” Side\n## Angle-Side-Angle Congruence Postulate\n## The “Non-Included” Side","[{\"question\":\"How do you show two triangles are congruent using corresponding parts?\",\"answer\":\"Identify the matching corresponding vertices, then use corresponding angles and corresponding sides to form a congruence statement. If triangles are congruent, CPCTC confirms corresponding parts are congruent.\"},{\"question\":\"When using SSS, what information is needed?\",\"answer\":\"You need three sides of one triangle congruent to three sides of a second triangle. If all three side pairs match, the triangles are congruent by the SSS Congruence Postulate.\"},{\"question\":\"What is the difference between the included and non-included side in ASA and AAS?\",\"answer\":\"For ASA, the side between the two angles is the included side. For AAS, the side between the two angles is not included; instead, a non-included side is used along with two congruent angles and their corresponding side condition.\"}]","Chapter 5 - Congruence & Triangles - Methods for Showing Triangles to be Congruent | PDF",1790271815,53]