[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-386027-105":59,"doc-detail-386027-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","geometry-module-1-topic-3-congruence-through-transformations-lesson-2-asa-sas-and-sss-proving-triangle-congruence-theorems","Geometry - Module 1 - Topic 3 - Congruence Through Transformations - Lesson 2 - ASA, SAS, and SSS: Proving Triangle Congruence Theorems","","Geometry Module 1, Topic 3, Lesson 2 explains proving triangle congruence theorems using rigid motion transformations. Students apply isometries to preserve distances and angle measures, use reflections to map points across a line, and use the perpendicular bisector relationship to guide reasoning. Activities develop ASA, SAS, and SSS proofs, include measurement-based investigations to determine when fewer than six part measures are sufficient, and discuss non-examples such as AAA and the risk of misapplying SSS to AAA.",{"@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/geometry-module-1-topic-3-congruence-through-transformations-lesson-2-asa-sas-and-sss-proving-triangle-congruence-theorems/386027/",{"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/geometry-module-1-topic-3-congruence-through-transformations-lesson-2-asa-sas-and-sss-proving-triangle-congruence-theorems/386027.png","ImageObject",300,407,{"name":92,"@type":93},"Riley","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-11-01","2026-09-24",true,{"@type":102,"interactionType":103,"userInteractionCount":52},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"What key properties do students use to prove triangle congruence theorems in this lesson?","Question",{"text":112,"@type":113},"Students use rigid motions where isometries preserve distances and angle measures, reflections map points across a line, and a point is equidistant from two points exactly when it lies on their perpendicular bisector.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"Why might measuring all six corresponding parts of two triangles be unnecessary?",{"text":117,"@type":113},"Students conclude that congruence can be established with fewer than six corresponding parts, because certain angle relationships and side relationships imply the remaining corresponding parts must also be congruent.",{"name":119,"@type":110,"acceptedAnswer":120},"What misconception does the lesson warn about regarding AAA and SSS?",{"text":121,"@type":113},"Students may incorrectly extend SSS thinking to include AAA. A counterexample is given: two different-sized isosceles right triangles can match under AAA without being congruent.","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},386027,1790913119,{"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":52,"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":41,"preview_page_count":4},1374391975076,"https://ap-avatar.wpscdn.com/avatar/14000253ca4ec9f6853?x-image-process=image/resize,m_fixed,w_180,h_180&k=1783305029341752051","Geometry | Module 1 | Topic 3 | Congruence Through Transformations  \nLesson 2  \nASA, SAS, and SSS:  \nProving Triangle Congruence Theorems  \nLesson Overview  \nStudents use what they have learned in the previous topic: (1) isometries preserve distancesand angle measures, (2) any point in the plane can be reflected across a line to map to another point in the plane, and (3) a point is equidistant from two other points if and only if it lies on their perpendicular bisector. They use these facts to create and verify proofs of the SSS, SAS, and ASA Congruence Theorems using rigid motion transformations. Students then explore some non-examples of congruence theorems (AAA and SSA).  \nLesson Video(s): The aligned lesson overview video(s) provide additional instruction for students on the key concepts in this lesson and can be found alongside the digital interactive student lesson.  \nTEKS: G.6B, G.6C  \nLesson Structure and Pacing: 3 Days  \nDay 1  \nEngage  \nGetting Started: Necessary Conditions  \nDevelop  \nActivity 2.1: Congruent Line Segments by Reflection  \nDay 2  \nActivity 2.2: Side-Side-Side Congruence  \nActivity 2.3: Side-Angle-Side Congruence  \nDay 3  \nActivity 2.4: Angle-Side-Angle Congruence  \nActivity 2.5: Non-Examples of Congruence Theorems  \nDemonstrate  \nTalk the Talk: The Right Combination  \nAdapted Virtual Facilitation Notes Lesson 2: ASA, SAS, and SSS • 1  \nGetting Started: Necessary Conditions  \nAsynchronous Facilitation Notes  \nIn this activity, students use measuring tools to determine whether two triangles are congruent in free response questions. They conclude that measuring all six corresponding parts of both triangles may not be necessary to demonstrate two triangles are congruent. Note: students will need a ruler and protractor or will use the ruler and protractor on the toolbar for this activity.  \nSynchronous Facilitation Notes  \nIn this activity, students use measuring tools to determine whether two triangles are congruent. They conclude that measuring all six corresponding parts of both triangles may not be necessary to demonstrate two triangles are congruent.  \nHave the students work with a partner or in a group to complete Questions 1 and 2. Share responses as a class.  \nAs students work, look for  \n• Comments regarding experimenting with triangles in middle school, when they determined whether a unique triangle could be constructed from a combination of three measures of angles and/or sides.  \n• The realization that if two pairs of corresponding angles in a triangle are congruent, then the third pair of corresponding angles are also congruent.  \nMisconception  \nStudents may incorrectly extend what they remember about SSS to include AAA. If this is the case, ask students to provide a counterexample. One such counterexample is two different-sized isosceles right triangles.  \nQuestions to ask  \n• What unit of measure did you use to measure the sides of each triangle?  \n• What side of nRNL is congruent to side ? Side ¯AC? Side ¯BC?  \n• What angle of nRNL is congruent to /A? /B? /C?  \n• Do you think you need to measure all six angles and sides of two triangles to determine whether the triangles are congruent? Why or why not?  \n• Do you think measuring only 5 pairs of corresponding angles and sides would be sufficient to prove the triangles are congruent? What about 4 pairs of corresponding sides and angles? What about 3 pairs of corresponding sides and angles?  \n• Is there a point in your measuring that you knew you could stop because you knew the remainder of the sides and angles would have to be congruent as well? If so, when?  \nSummary  \nIf two triangles are congruent, three pairs of corresponding sides are equal in length and three pairs of corresponding angles are equal in measure. It is possible to conclude two triangles are congruent using fewer than six pairs of corresponding parts.  \nAdapted Virtual Facilitation Notes Lesson 2: ASA, SAS, and SSS • 2  \nActivity 2.1:  \nCongruent Line Segments by Reflection","cbCaifqnXJV3io5R","https://ap.wps.com/l/cbCaifqnXJV3io5R","pdf",673128,12,"English","# Lesson Overview\n## Lesson Structure and Pacing\n## Engage: Getting Started - Necessary Conditions\n## Develop: Activity 2.1 - Congruent Line Segments by Reflection\n## Develop: Activity 2.2 - Side-Side-Side Congruence\n## Develop: Activity 2.3 - Side-Angle-Side Congruence\n## Develop: Activity 2.4 - Angle-Side-Angle Congruence\n## Develop: Activity 2.5 - Non-Examples of Congruence Theorems\n## Demonstrate: Talk the Talk - The Right Combination\n## Getting Started: Asynchronous and Synchronous Facilitation Notes","[{\"question\":\"What key properties do students use to prove triangle congruence theorems in this lesson?\",\"answer\":\"Students use rigid motions where isometries preserve distances and angle measures, reflections map points across a line, and a point is equidistant from two points exactly when it lies on their perpendicular bisector.\"},{\"question\":\"Why might measuring all six corresponding parts of two triangles be unnecessary?\",\"answer\":\"Students conclude that congruence can be established with fewer than six corresponding parts, because certain angle relationships and side relationships imply the remaining corresponding parts must also be congruent.\"},{\"question\":\"What misconception does the lesson warn about regarding AAA and SSS?\",\"answer\":\"Students may incorrectly extend SSS thinking to include AAA. A counterexample is given: two different-sized isosceles right triangles can match under AAA without being congruent.\"}]","Geometry - Module 1 - Topic 3 - Congruence Through Transformations - Lesson 2 - ASA, SAS, and SSS: Proving Triangle Congruence Theorems | PDF",1790271820]