[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-386795-105":59,"doc-detail-386795-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-5-congruent-triangles-using-congruent-triangles","Chapter 5 - Congruent Triangles - Using Congruent Triangles","","This document section focuses on congruent triangles and related geometry skills, including angles of triangles, congruent polygons, and proof strategies. It explains how to prove triangle congruence using SAS, SSS, ASA, and AAS, and includes practical work such as using congruent triangles and coordinate proofs. Coordinate geometry examples demonstrate applying the midpoint and distance formulas, and solving linear equations with variables on both sides. Additional content covers defining congruence with “if and only if” statements and classifying statements as definitions, postulates, or theorems.",{"@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-5-congruent-triangles-using-congruent-triangles/386795/",{"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-congruent-triangles-using-congruent-triangles/386795.png","ImageObject",300,407,{"name":92,"@type":93},"Ezra","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",12,{"@type":107,"mainEntity":108},"FAQPage",[109,115,119],{"name":110,"@type":111,"acceptedAnswer":112},"What are the main methods to prove triangle congruence covered in this section?","Question",{"text":113,"@type":114},"The section covers proving triangle congruence by SAS, SSS, and by ASA and AAS.","Answer",{"name":116,"@type":111,"acceptedAnswer":117},"How are the midpoint and distance formulas used here?",{"text":118,"@type":114},"It shows how to find the midpoint coordinates of a segment and how to compute the distance between two points using the distance formula, including a worked example and approximate evaluation.",{"name":120,"@type":111,"acceptedAnswer":121},"What does it mean that a definition is an “if and only if” statement?",{"text":122,"@type":114},"The document explains that a definition uses biconditional logic: the forward and reverse conditional statements both hold, meaning each condition implies the other.","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},386795,1790377450,{"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":140,"language":141,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":142,"faqs":143,"seo_title":144,"seo_description":67,"update_tm":145,"read_time":146,"preview_page_count":4},1099514068035,"https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c","5 Congruent Triangles  \n5.1 Angles of Triangles  \n5.2 Congruent Polygons  \n5.3 Proving Triangle Congruence by SAS  \n5.4 Equilateral and Isosceles Triangles  \n5.5 Proving Triangle Congruence by SSS  \n5.6 Proving Triangle Congruence by ASA and AAS  \n5.7 Using Congruent Triangles  \n5.8 Coordinate Proofs  \nSEE the Big Idea  \nHang Glider (p. 252)  \nBarn (p. 224)  \nPainting (p. 211)  \nf  \nfeguard  \nTower  \nPainti  \nng  \n(  \n(p 21  \n1)  \nHome Decor  \n(p. 231)  \n(p. 217)  \nMaintaining Mathematical Proficiency  \nUsing the Midpoint and Distance Formulas  \nExample 1 The endpoints ofA—B are A(−2, 3) and B(4, 7) . Find the coordinates of the midpoint M.  \nUse the Midpoint Formula.  \nM ( , ) = M ( , )  \n= M(1, 5)  \n The coordinates of the midpoint M are (1, 5) . Example 2 Find the distance between C (0, −5) and D(3, 2) .  \nCD = √(x2 − x1)2 + (y2 − y 1)2 Distance Formula  \n——  \n= √(3 − 0)2 + [2 − (—5)]2 Substitute.  \n= √32 + 72 Subtract.  \n= √9 + 49 Evaluate powers.  \n= √—58 Add.  \n≈ 7.6 Use a calculator.  \n—  \n The distance between C(0, −5) and D(3, 2) is about 7.6.  \nFind the coordinates of the midpoint M of the segment with the given endpoints. Then find the distance between the two points.  \n1. P(−4, 1) and Q(0, 7) 2. G(3, 6) and H(9, −2) 3. U(−1, −2) and V(8, 0)  \nSolving Equations with Variables on Both Sides  \nExample 3 Solve 2 − 5x = −3x.  \n2 − 5x = −3x Write the equation.  \n+5x  +5x Add 5x to each side.  \n2 = 2x Simplify.  \n =  Divide each side by 2.  \n1 = x Simplify.  \n The solution is x = 1.  \nSolve the equation.  \n4. 7x + 12 = 3x 5. 14 − 6t = t 6. 5p + 10 = 8p + 1  \n7. w + 13 = 11w − 7 8. 4x + 1 = 3 − 2x 9. z − 2 = 4 + 9z  \n10. ABSTRACT REASONING Is it possible to find the length of a segment in a coordinate plane without using the Distance Formula? Explain your reasoning.  \nDynamic Solutions available [at BigIdeasMath.com](at BigIdeasMath.com)  \n205  \nMathematical Processes  \nMathematically proficient students understand and use given definitions.  \nDefinitions, Postulates, and Theorems  \nCore Concept  \nDefinitions and Biconditional Statements  \nA definition is always an “if and only if” statement. Here is an example.  \nDefinition: Two geometric figures are congruent figures if and only if there is a rigid motion or a composition of rigid motions that maps one of the figures onto the other.  \nBecause this is a definition, it is a biconditional statement. It implies the following two conditional statements.  \n1. If two geometric figures are congruent figures, then there is a rigid motion or a composition of rigid motions that maps one of the figures onto the other.  \n2. If there is a rigid motion or a composition of rigid motions that maps one geometric figure onto another, then the two geometric figures are congruent figures.  \nDefinitions, postulates, and theorems are the building blocks of geometry. In two-column proofs, the statements in the reason column are almost always definitions, postulates, or theorems.  \nIdentifying Definitions, Postulates, and Theorems  \nClassify each statement as a definition, a postulate, or a theorem.  \na. If two lines are cut by a transversal so that alternate interior angles are congruent, then the lines are parallel.  \nb. If two coplanar lines have no point of intersection, then the lines are parallel.  \nc. If there is a line and a point not on the line, then there is exactly one line through the point parallel to the given line.  \nSOLUTION  \na. This is a theorem. It is the Alternate Interior Angles Converse Theorem (Theorem 3.6) studied in Section 3.3.  \nb. This is the definition of parallel lines.  \nc. This is a postulate. It is the Parallel Postulate (Postulate 3.1) studied in Section 3.1. In Euclidean geometry, it is assumed, not proved, to be true.  \nMonitoring Progress  \nClassify each statement as a definition, a postulate, or a theorem. Explain your reasoning.  \n1. In a coordinate plane, two nonvertical lines are perpendicular if and only if the product of their slopes is −1.  \n2. If two","cbCaingjeDdsQhaN","https://ap.wps.com/l/cbCaingjeDdsQhaN","pdf",8813308,64,"English","# 5 Congruent Triangles\n## 5.1 Angles of Triangles\n## 5.2 Congruent Polygons\n## 5.3 Proving Triangle Congruence by SAS\n## 5.4 Equilateral and Isosceles Triangles\n## 5.5 Proving Triangle Congruence by SSS\n## 5.6 Proving Triangle Congruence by ASA and AAS\n## 5.7 Using Congruent Triangles\n## 5.8 Coordinate Proofs\n## Definitions, Postulates, and Theorems\n## Monitoring Progress","[{\"question\":\"What are the main methods to prove triangle congruence covered in this section?\",\"answer\":\"The section covers proving triangle congruence by SAS, SSS, and by ASA and AAS.\"},{\"question\":\"How are the midpoint and distance formulas used here?\",\"answer\":\"It shows how to find the midpoint coordinates of a segment and how to compute the distance between two points using the distance formula, including a worked example and approximate evaluation.\"},{\"question\":\"What does it mean that a definition is an “if and only if” statement?\",\"answer\":\"The document explains that a definition uses biconditional logic: the forward and reverse conditional statements both hold, meaning each condition implies the other.\"}]","Chapter 5 - Congruent Triangles - Using Congruent Triangles | PDF",1790276161,161]