[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-386023-105":59,"doc-detail-386023-en":131},{"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":124,"head_meta":126,"extra_data":128,"updated_unix":130},105,"en","chapter-7-congruence-postulates-theorems","Chapter 7 - Congruence Postulates & Theorems","","Chapter 7 explains congruence in geometry by defining congruent objects as having the same size and shape, and uses polygons and quadrilaterals to illustrate how congruence requires matching sides and angles in corresponding positions. The chapter then focuses on triangles, describing the idea of fitting and coinciding, and introduces congruence postulates based on side-side-side (SSS), side-angle-side (SAS), and angle-side-angle (ASA). It also discusses triangle vocabulary such as opposite and included, and shows how to structure congruence proofs with given statements and reasons.",{"@graph":69,"@context":123},[70,84,106],{"@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-7-congruence-postulates-theorems/386023/",{"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-7-congruence-postulates-theorems/386023.png","ImageObject",300,407,{"name":92,"@type":93},"Theodore","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-11-04","2026-09-24",true,{"@type":102,"interactionType":103,"userInteractionCount":105},"InteractionCounter",{"@type":104},"ViewAction",15,{"@type":107,"mainEntity":108},"FAQPage",[109,115,119],{"name":110,"@type":111,"acceptedAnswer":112},"What does it mean for two shapes or polygons to be congruent?","Question",{"text":113,"@type":114},"Congruent objects have the same size and shape. For polygons, all corresponding sides and all corresponding angles must match in the correct order.","Answer",{"name":116,"@type":111,"acceptedAnswer":117},"How do you determine whether two triangles are congruent?",{"text":118,"@type":114},"Two triangles are congruent when corresponding sides and angles match. The chapter emphasizes using congruence postulates and matching the corresponding parts correctly after naming the triangles.",{"name":120,"@type":111,"acceptedAnswer":121},"What are the three main congruence postulates covered in this chapter?",{"text":122,"@type":114},"The chapter presents SSS (three sides), SAS (two sides and the included angle), and ASA (two angles and the included side). Each postulate states that the triangles are congruent when the specified corresponding parts are congruent.","https://schema.org",{"og:url":83,"og:type":125,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":127,"canonical":83},"index,follow",{"doc_id":129,"site_id":62},386023,1790377818,{"code":4,"msg":5,"data":132},{"doc_id":129,"user_id":133,"nickname":92,"user_avatar":134,"doc_module":4,"category_id":19,"category_name":20,"doc_title":65,"doc_description":67,"doc_content":135,"file_id":136,"file_url":137,"file_type":138,"file_size":139,"view_count":105,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":105,"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},7971461740886,"https://ap-avatar.wpscdn.com/davatar_3d24733baf745e90a7e4bdd5f77d97b2","Chapter 7  \nCongruence Postulates &Theorems- Δ ’s  \nIn math, the word congruent is used to describe objects that have the same size and shape. When you traced things when you were a little kid, you were using congruence. Neat, don’t you think?  \nStop signs would be examples of congruent shapes. Since a stop sign has 8 sides, they would be congruent octagons.  \nCongruent polygons – polygons are congruent when all the sides and all  \nthe angles of one polygon are congruent to all the angles and all the sides of another polygon respectively.  \nB  \nA  \nD  \nC  \nM  \nP  \nN  \nO  \n∠A≅∠M, ∠B≅∠N, ∠C≅∠O, and ∠D≅∠P Quad ABCD ≅ Quad MNOP  \nTo show the two quadrilaterals are congruent, I need to show eight congruences – 4 angles and 4 sides respectively.  \nHow I name my congruent polygons as congruent is very important. The respective angles and sides must appear in the same order. So, when we write  \nQuad ABCD ≅ Quad MNOP  \nThat means position is important. ∠C corresponds to ∠O, notice they are both in the third position.  \nCopyright © 2003 Hanlonmath 1  \nWe are going to look specifically at triangles. To determine if two triangles are congruent, they must have the same size and shape. They must fit on top of each other, they must coincide.  \nMathematically, we say all the sides and angles of one triangle must be congruent to the corresponding sides and angles of another triangle.  \nA  \nB C  \nE F  \nIn other words, we would have to show angles A, B, and C were congruent ( ≅ ) to angles D,  E and F respectively, then show ΑΒ , BC , and AC were ≅ to DE , EF , and DF respectively.  \nWe would have to show those six relationships.  \nAh, but there is good news. If I gave everyone reading this three sticks of length 10”, 8”, and 7”, then asked them to glue the ends together to make triangles, something interesting happens. When I collect the triangles, they all fit very nicely on top of each other, they have the same size and shape, they coincide – they are congruent!  \nWhy’s that good news? Because rather than showing all the angles and all the sides of one triangle are congruent to all the sides and all the angles of another triangle (6 relationships), I was able to determine congruence just using the 3 sides-a shortcut.  \nThat observation leads us to the side, side, side congruence postulate.  \nSSS Postulate If three sides of one triangle are congruent,  \nrespectively, to three sides of another triangle, then the triangles are congruent.  \nCopyright © 2003 Hanlonmath 2  \nA  \nB C  \nD  \nE F  \nI could write that as ∆BCA ≅ ∆EFD. Notice how the ≅ angles and sides have the same position. Someone else could have written that relationship as ∆ACB≅∆DFE. Again notice how you name the first triangle effects how you arrange the letters in the second triangle.  \nThere are 6 ways I could have written that congruence statement. What you need to remember is the corresponding parts must line up. Using that notation allows me to determine the congruent parts without looking at a diagram.  \nHere are the 6 ways, notice how the angles and sides are aligned.  \n∆ABC ≅ ∆DEF  \n∆BCA ≅ ∆EFD  \n∆CAB ≅ ∆FDE  \n∆BAC ≅ ∆EDF  \n∆ACB ≅ ∆DFE  \n∆CBA ≅ ∆FED  \nSo, if I chose ∆CAB ≅ ∆FDE, notice how angles C and F correspond, angles A and D, and angles B and E. Now look at the order of the sides; side CA corresponds to side FD, side AB corresponds to DE, and CB corresponds to FE. Without looking at the picture, I can name these relationships.  \nLearning the vocabulary and notation is extremely important for success in any field. To be clear, let’s examine the words “opposite” and “included” in the context of triangles.  \nLooking at the following triangle and the phrases in the box, these are important for you to know.  \nCopyright © 2003 Hanlonmath 3  \nC  \nA  B  \n∠B lies opposite AC   ∠B is included between ABand BC  \nBC lies opposite ∠A  \nBC is included between ∠Band ∠C  \nExample: 1 Given: XZ ≅ XW and YZ ≅ YW  \nProve: ∆XYZ ≅ ∆XYW  \nMarking my congruent sides.  \nIn order to prove triang","cbCaind8hZIoO9U1","https://ap.wps.com/l/cbCaind8hZIoO9U1","pdf",598215,"English","# Chapter 7 Congruence Postulates & Theorems\n## Congruence in polygons and quadrilaterals\n## Triangle congruence and corresponding parts\n## SSS Congruence Postulate\n## Naming triangles and correspondence order\n## Triangle vocabulary: opposite and included\n## Example proof using SSS\n## SAS and ASA Congruence Postulates\n## Studying and applying postulates for proofs","[{\"question\":\"What does it mean for two shapes or polygons to be congruent?\",\"answer\":\"Congruent objects have the same size and shape. For polygons, all corresponding sides and all corresponding angles must match in the correct order.\"},{\"question\":\"How do you determine whether two triangles are congruent?\",\"answer\":\"Two triangles are congruent when corresponding sides and angles match. The chapter emphasizes using congruence postulates and matching the corresponding parts correctly after naming the triangles.\"},{\"question\":\"What are the three main congruence postulates covered in this chapter?\",\"answer\":\"The chapter presents SSS (three sides), SAS (two sides and the included angle), and ASA (two angles and the included side). Each postulate states that the triangles are congruent when the specified corresponding parts are congruent.\"}]","Chapter 7 - Congruence Postulates & Theorems | PDF",1790271813,38]