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class="vector-toc-text"> <span class="vector-toc-numb">2.1.1</span> <span>Side-side-angle</span> </div> </a> <ul id="toc-Side-side-angle-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Angle-angle-angle" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Angle-angle-angle"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1.2</span> <span>Angle-angle-angle</span> </div> </a> <ul id="toc-Angle-angle-angle-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-CPCTC" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#CPCTC"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>CPCTC</span> </div> </a> <ul id="toc-CPCTC-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Definition_of_congruence_in_analytic_geometry" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Definition_of_congruence_in_analytic_geometry"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Definition of congruence in analytic geometry</span> </div> </a> <ul id="toc-Definition_of_congruence_in_analytic_geometry-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Congruent_conic_sections" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Congruent_conic_sections"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Congruent conic sections</span> </div> </a> <ul id="toc-Congruent_conic_sections-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Congruent_polyhedra" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Congruent_polyhedra"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Congruent polyhedra</span> </div> </a> <ul id="toc-Congruent_polyhedra-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Congruent_triangles_on_a_sphere" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Congruent_triangles_on_a_sphere"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Congruent triangles on a sphere</span> </div> </a> <ul id="toc-Congruent_triangles_on_a_sphere-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notation" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notation"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Notation</span> </div> </a> <ul id="toc-Notation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Congruence (geometry)</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 56 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-56" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">56 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AA%D8%B7%D8%A7%D8%A8%D9%82_(%D9%87%D9%86%D8%AF%D8%B3%D8%A9)" title="تطابق (هندسة) – Arabic" lang="ar" hreflang="ar" data-title="تطابق (هندسة)" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/%C3%81ngulos_congruentes" title="Ángulos congruentes – Asturian" lang="ast" hreflang="ast" data-title="Ángulos congruentes" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%B8%E0%A6%B0%E0%A7%8D%E0%A6%AC%E0%A6%B8%E0%A6%AE%E0%A6%A4%E0%A6%BE_(%E0%A6%9C%E0%A7%8D%E0%A6%AF%E0%A6%BE%E0%A6%AE%E0%A6%BF%E0%A6%A4%E0%A6%BF)" title="সর্বসমতা (জ্যামিতি) – Bangla" lang="bn" hreflang="bn" data-title="সর্বসমতা (জ্যামিতি)" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BD%D0%B3%D1%80%D1%83%D1%8D%D0%BD%D1%82%D0%BB%D1%8B%D2%A1_(%D0%B3%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%8F)" title="Конгруэнтлыҡ (геометрия) – Bashkir" lang="ba" hreflang="ba" data-title="Конгруэнтлыҡ (геометрия)" data-language-autonym="Башҡортса" data-language-local-name="Bashkir" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%95%D0%B4%D0%BD%D0%B0%D0%BA%D0%B2%D0%BE%D1%81%D1%82" title="Еднаквост – Bulgarian" lang="bg" hreflang="bg" data-title="Еднаквост" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Podudarnost_(geometrija)" title="Podudarnost (geometrija) – Bosnian" lang="bs" hreflang="bs" data-title="Podudarnost (geometrija)" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Congru%C3%A8ncia_(geometria)" title="Congruència (geometria) – Catalan" lang="ca" hreflang="ca" data-title="Congruència (geometria)" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BD%D0%B3%D1%80%D1%83%D1%8D%D0%BD%D1%82%D0%BB%C4%83%D1%85_(%D0%B3%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8)" title="Конгруэнтлăх (геометри) – Chuvash" lang="cv" hreflang="cv" data-title="Конгруэнтлăх (геометри)" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-sn mw-list-item"><a href="https://sn.wikipedia.org/wiki/Chiwirano" title="Chiwirano – Shona" lang="sn" hreflang="sn" data-title="Chiwirano" data-language-autonym="ChiShona" data-language-local-name="Shona" class="interlanguage-link-target"><span>ChiShona</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Cyfathiant" title="Cyfathiant – Welsh" lang="cy" hreflang="cy" data-title="Cyfathiant" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Kongruens" title="Kongruens – Danish" lang="da" hreflang="da" data-title="Kongruens" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Kongruenz_(Geometrie)" title="Kongruenz (Geometrie) – German" lang="de" hreflang="de" data-title="Kongruenz (Geometrie)" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Kongruentsus" title="Kongruentsus – Estonian" lang="et" hreflang="et" data-title="Kongruentsus" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Congruencia_(geometr%C3%ADa)" title="Congruencia (geometría) – Spanish" lang="es" hreflang="es" data-title="Congruencia (geometría)" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Kongruentzia_(geometria)" title="Kongruentzia (geometria) – Basque" lang="eu" hreflang="eu" data-title="Kongruentzia (geometria)" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%87%D9%85%E2%80%8C%D9%86%D9%87%D8%B4%D8%AA%DB%8C_(%D9%87%D9%86%D8%AF%D8%B3%D9%87)" title="هم‌نهشتی (هندسه) – Persian" lang="fa" hreflang="fa" data-title="هم‌نهشتی (هندسه)" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Congruence_(g%C3%A9om%C3%A9trie)" title="Congruence (géométrie) – French" lang="fr" hreflang="fr" data-title="Congruence (géométrie)" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Congruencia_(xeometr%C3%ADa)" title="Congruencia (xeometría) – Galician" lang="gl" hreflang="gl" data-title="Congruencia (xeometría)" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%95%A9%EB%8F%99_(%EA%B8%B0%ED%95%98%ED%95%99)" title="합동 (기하학) – Korean" lang="ko" hreflang="ko" data-title="합동 (기하학)" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%B8%E0%A4%B0%E0%A5%8D%E0%A4%B5%E0%A4%BE%E0%A4%82%E0%A4%97%E0%A4%B8%E0%A4%AE%E0%A4%A4%E0%A4%BE" title="सर्वांगसमता – Hindi" lang="hi" hreflang="hi" data-title="सर्वांगसमता" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Sukladnost_(geometrija)" title="Sukladnost (geometrija) – Croatian" lang="hr" hreflang="hr" data-title="Sukladnost (geometrija)" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Kongruo_(geometrio)" title="Kongruo (geometrio) – Ido" lang="io" hreflang="io" data-title="Kongruo (geometrio)" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Kongruen" title="Kongruen – Indonesian" lang="id" hreflang="id" data-title="Kongruen" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Congruenza_(geometria)" title="Congruenza (geometria) – Italian" lang="it" hreflang="it" data-title="Congruenza (geometria)" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%97%D7%A4%D7%99%D7%A4%D7%94" title="חפיפה – Hebrew" lang="he" hreflang="he" data-title="חפיפה" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BD%D0%B3%D1%80%D1%83%D1%8D%D0%BD%D1%82%D1%82%D1%96%D0%BB%D1%96%D0%BA" title="Конгруэнттілік – Kazakh" lang="kk" hreflang="kk" data-title="Конгруэнттілік" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Congruentia_(geometria)" title="Congruentia (geometria) – Latin" lang="la" hreflang="la" data-title="Congruentia (geometria)" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Egybev%C3%A1g%C3%B3s%C3%A1g" title="Egybevágóság – Hungarian" lang="hu" hreflang="hu" data-title="Egybevágóság" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%A1%D0%BA%D0%BB%D0%B0%D0%B4%D0%BD%D0%BE%D1%81%D1%82" title="Складност – Macedonian" lang="mk" hreflang="mk" data-title="Складност" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Kongruen_(geometri)" title="Kongruen (geometri) – Malay" lang="ms" hreflang="ms" data-title="Kongruen (geometri)" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%91%E1%80%95%E1%80%BA%E1%80%90%E1%80%B0%E1%80%8A%E1%80%AE%E1%80%81%E1%80%BC%E1%80%84%E1%80%BA%E1%80%B8" title="ထပ်တူညီခြင်း – Burmese" lang="my" hreflang="my" data-title="ထပ်တူညီခြင်း" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="Burmese" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Congruentie_(meetkunde)" title="Congruentie (meetkunde) – Dutch" lang="nl" hreflang="nl" data-title="Congruentie (meetkunde)" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%9B%B3%E5%BD%A2%E3%81%AE%E5%90%88%E5%90%8C" title="図形の合同 – Japanese" lang="ja" hreflang="ja" data-title="図形の合同" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Kongruens_(geometri)" title="Kongruens (geometri) – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Kongruens (geometri)" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Kongruentlik" title="Kongruentlik – Uzbek" lang="uz" hreflang="uz" data-title="Kongruentlik" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Przystawanie_(geometria)" title="Przystawanie (geometria) – Polish" lang="pl" hreflang="pl" data-title="Przystawanie (geometria)" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Congru%C3%AAncia_(geometria)" title="Congruência (geometria) – Portuguese" lang="pt" hreflang="pt" data-title="Congruência (geometria)" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ksh mw-list-item"><a href="https://ksh.wikipedia.org/wiki/Kongruenz_(Jeometri)" title="Kongruenz (Jeometri) – Colognian" lang="ksh" hreflang="ksh" data-title="Kongruenz (Jeometri)" data-language-autonym="Ripoarisch" data-language-local-name="Colognian" class="interlanguage-link-target"><span>Ripoarisch</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Congruen%C8%9B%C4%83_(geometrie)" title="Congruență (geometrie) – Romanian" lang="ro" hreflang="ro" data-title="Congruență (geometrie)" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BD%D0%B3%D1%80%D1%83%D1%8D%D0%BD%D1%82%D0%BD%D0%BE%D1%81%D1%82%D1%8C_(%D0%B3%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%8F)" title="Конгруэнтность (геометрия) – Russian" lang="ru" hreflang="ru" data-title="Конгруэнтность (геометрия)" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Kongruenca_(gjeometri)" title="Kongruenca (gjeometri) – Albanian" lang="sq" hreflang="sq" data-title="Kongruenca (gjeometri)" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Congruence" title="Congruence – Simple English" lang="en-simple" hreflang="en-simple" data-title="Congruence" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Skladnost_(geometrija)" title="Skladnost (geometrija) – Slovenian" lang="sl" hreflang="sl" data-title="Skladnost (geometrija)" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D9%BE%DB%8E%DA%A9%DA%A9%DB%95%D9%88%D8%AA%D9%88%D9%88%DB%8C%DB%8C_(%D8%A6%DB%95%D9%86%D8%AF%D8%A7%D8%B2%DB%95)" title="پێککەوتوویی (ئەندازە) – Central Kurdish" lang="ckb" hreflang="ckb" data-title="پێککەوتوویی (ئەندازە)" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/Podudarnost_(geometrija)" title="Podudarnost (geometrija) – Serbian" lang="sr" hreflang="sr" data-title="Podudarnost (geometrija)" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Podudarnost_(geometrija)" title="Podudarnost (geometrija) – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Podudarnost (geometrija)" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li 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src="//upload.wikimedia.org/wikipedia/en/thumb/1/1b/Semi-protection-shackle.svg/20px-Semi-protection-shackle.svg.png" decoding="async" width="20" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/1/1b/Semi-protection-shackle.svg/30px-Semi-protection-shackle.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/1/1b/Semi-protection-shackle.svg/40px-Semi-protection-shackle.svg.png 2x" data-file-width="512" data-file-height="512" /></a></span></div></div> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Relationship between two figures of the same shape and size, or mirroring each other</div> <p class="mw-empty-elt"> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Congruent_non-congruent_triangles.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/12/Congruent_non-congruent_triangles.svg/260px-Congruent_non-congruent_triangles.svg.png" decoding="async" width="260" height="104" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/12/Congruent_non-congruent_triangles.svg/390px-Congruent_non-congruent_triangles.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/12/Congruent_non-congruent_triangles.svg/520px-Congruent_non-congruent_triangles.svg.png 2x" data-file-width="568" data-file-height="227" /></a><figcaption>The two triangles on the left are congruent. The third is <a href="/wiki/Similarity_(geometry)" title="Similarity (geometry)">similar</a> to them. The last triangle is neither congruent nor similar to any of the others. Congruence permits alteration of some properties, such as location and orientation, but leaves others unchanged, like <a href="/wiki/Distance" title="Distance">distances</a> and <a href="/wiki/Angle" title="Angle">angles</a>. The unchanged properties are called <a href="/wiki/Invariant_(mathematics)" title="Invariant (mathematics)">invariants</a>.</figcaption></figure> <p>In <a href="/wiki/Geometry" title="Geometry">geometry</a>, two figures or objects are <b>congruent</b> if they have the same <a href="/wiki/Shape" title="Shape">shape</a> and <a href="/wiki/Size" title="Size">size</a>, or if one has the same shape and size as the <a href="/wiki/Mirror_image" title="Mirror image">mirror image</a> of the other.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>More formally, two sets of <a href="/wiki/Point_(geometry)" title="Point (geometry)">points</a> are called <b>congruent</b> if, and only if, one can be transformed into the other by an <a href="/wiki/Isometry" title="Isometry">isometry</a>, i.e., a combination of <a href="/wiki/Rigid_motion" class="mw-redirect" title="Rigid motion">rigid motions</a>, namely a <a href="/wiki/Translation_(geometry)" title="Translation (geometry)">translation</a>, a <a href="/wiki/Rotation" title="Rotation">rotation</a>, and a <a href="/wiki/Reflection_(mathematics)" title="Reflection (mathematics)">reflection</a>. This means that either object can be repositioned and reflected (but not resized) so as to coincide precisely with the other object. Therefore, two distinct plane figures on a piece of paper are congruent if they can be cut out and then matched up completely. Turning the paper over is permitted. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Angle-angle-side_triangle_congruence.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/Angle-angle-side_triangle_congruence.svg/220px-Angle-angle-side_triangle_congruence.svg.png" decoding="async" width="220" height="147" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/Angle-angle-side_triangle_congruence.svg/330px-Angle-angle-side_triangle_congruence.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/60/Angle-angle-side_triangle_congruence.svg/440px-Angle-angle-side_triangle_congruence.svg.png 2x" data-file-width="135" data-file-height="90" /></a><figcaption>This diagram illustrates the geometric principle of angle-angle-side triangle congruence: given triangle ABC and triangle A'B'C', triangle ABC is congruent with triangle A'B'C' if and only if: angle CAB is congruent with angle C'A'B', and angle ABC is congruent with angle A'B'C', and BC is congruent with B'C'. Note <a href="/wiki/Hatch_mark#Congruency_notation" title="Hatch mark">hatch marks</a> are used here to show angle and side equalities.</figcaption></figure> <p>In elementary geometry the word <i>congruent</i> is often used as follows.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> The word <i>equal</i> is often used in place of <i>congruent</i> for these objects. </p> <ul><li>Two <a href="/wiki/Line_segment" title="Line segment">line segments</a> are congruent if they have the same length.</li> <li>Two <a href="/wiki/Angle" title="Angle">angles</a> are congruent if they have the same measure.</li> <li>Two <a href="/wiki/Circle" title="Circle">circles</a> are congruent if they have the same diameter.</li></ul> <p>In this sense, <i>two plane figures are congruent</i> implies that their corresponding characteristics are "congruent" or "equal" including not just their corresponding sides and angles, but also their corresponding diagonals, perimeters, and areas. </p><p>The related concept of <a href="/wiki/Similarity_(geometry)" title="Similarity (geometry)">similarity</a> applies if the objects have the same shape but do not necessarily have the same size. (Most definitions consider congruence to be a form of similarity, although a minority require that the objects have different sizes in order to qualify as similar.) </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Determining_congruence_of_polygons">Determining congruence of polygons</h2></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Quadrilateral_congruence.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cf/Quadrilateral_congruence.png/333px-Quadrilateral_congruence.png" decoding="async" width="333" height="214" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/c/cf/Quadrilateral_congruence.png 1.5x" data-file-width="379" data-file-height="244" /></a><figcaption>The orange and green quadrilaterals are congruent; the blue is not congruent to them. All three have the same <a href="/wiki/Perimeter" title="Perimeter">perimeter</a> and <a href="/wiki/Area" title="Area">area</a>. (The ordering of the sides of the blue quadrilateral is "mixed" which results in two of the interior angles and one of the diagonals not being congruent.)</figcaption></figure> <p>For two polygons to be congruent, they must have an equal number of sides (and hence an equal number—the same number—of vertices). Two polygons with <i>n</i> sides are congruent if and only if they each have numerically identical sequences (even if clockwise for one polygon and counterclockwise for the other) side-angle-side-angle-... for <i>n</i> sides and <i>n</i> angles. </p><p>Congruence of polygons can be established graphically as follows: </p> <ul><li>First, match and label the corresponding vertices of the two figures.</li> <li>Second, draw a vector from one of the vertices of one of the figures to the corresponding vertex of the other figure. <i>Translate</i> the first figure by this vector so that these two vertices match.</li> <li>Third, <i>rotate</i> the translated figure about the matched vertex until one pair of <a href="/wiki/Corresponding_sides" class="mw-redirect" title="Corresponding sides">corresponding sides</a> matches.</li> <li>Fourth, <i>reflect</i> the rotated figure about this matched side until the figures match.</li></ul> <p>If at any time the step cannot be completed, the polygons are not congruent. </p> <div class="mw-heading mw-heading2"><h2 id="Congruence_of_triangles">Congruence of triangles</h2></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Solution_of_triangles" title="Solution of triangles">Solution of triangles</a></div> <p>Two <a href="/wiki/Triangle" title="Triangle">triangles</a> are congruent if their corresponding <a href="/wiki/Edge_(geometry)" title="Edge (geometry)">sides</a> are equal in length, and their corresponding <a href="/wiki/Angle" title="Angle">angles</a> are equal in measure. </p><p>Symbolically, we write the congruency and incongruency of two triangles <span class="texhtml">△<i>ABC</i></span> and <span class="texhtml">△<i>A′B′C′</i></span> as follows: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ABC\cong A'B'C'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>B</mi> <mi>C</mi> <mo>≅</mo> <msup> <mi>A</mi> <mo>′</mo> </msup> <msup> <mi>B</mi> <mo>′</mo> </msup> <msup> <mi>C</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ABC\cong A'B'C'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/831487b988352b643b14f52b10017985f068407f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.731ex; height:2.509ex;" alt="{\displaystyle ABC\cong A'B'C'}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ABC\ncong A'B'C'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>B</mi> <mi>C</mi> <mo>≆</mo> <msup> <mi>A</mi> <mo>′</mo> </msup> <msup> <mi>B</mi> <mo>′</mo> </msup> <msup> <mi>C</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ABC\ncong A'B'C'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/862e408fb0446733b6f758ccfd2a6dd87a2e2082" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.731ex; height:2.843ex;" alt="{\displaystyle ABC\ncong A'B'C'}"></span></dd></dl> <p>In many cases it is sufficient to establish the equality of three corresponding parts and use one of the following results to deduce the congruence of the two triangles. </p> <div class="mw-heading mw-heading3"><h3 id="Determining_congruence">Determining congruence</h3></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Congruent_triangles.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/72/Congruent_triangles.svg/220px-Congruent_triangles.svg.png" decoding="async" width="220" height="275" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/72/Congruent_triangles.svg/330px-Congruent_triangles.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/72/Congruent_triangles.svg/440px-Congruent_triangles.svg.png 2x" data-file-width="300" data-file-height="375" /></a><figcaption>The shape of a triangle is determined up to congruence by specifying two sides and the angle between them (SAS), two angles and the side between them (ASA) or two angles and a corresponding adjacent side (AAS). Specifying two sides and an adjacent angle (SSA), however, can yield two distinct possible triangles.</figcaption></figure> <p>Sufficient evidence for congruence between two triangles in <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a> can be shown through the following comparisons: </p> <ul><li><b>SAS</b> (side-angle-side): If two pairs of sides of two triangles are equal in length, and the included angles are equal in measurement, then the triangles are congruent.</li> <li><b>SSS</b> (side-side-side): If three pairs of sides of two triangles are equal in length, then the triangles are congruent.</li> <li><b>ASA</b> (angle-side-angle): If two pairs of angles of two triangles are equal in measurement, and the included sides are equal in length, then the triangles are congruent.</li></ul> <p>The ASA postulate is attributed to <a href="/wiki/Thales_of_Miletus" title="Thales of Miletus">Thales of Miletus</a>. In most systems of axioms, the three criteria – SAS, SSS and ASA – are established as <a href="/wiki/Theorem" title="Theorem">theorems</a>. In the <a href="/wiki/School_Mathematics_Study_Group" title="School Mathematics Study Group">School Mathematics Study Group</a> system <b>SAS</b> is taken as one (#15) of 22 postulates. </p> <ul><li><b>AAS</b> (angle-angle-side): If two pairs of angles of two triangles are equal in measurement, and a pair of corresponding non-included sides are equal in length, then the triangles are congruent. AAS is equivalent to an ASA condition, by the fact that if any two angles are given, so is the third angle, since their sum should be 180°. ASA and AAS are sometimes combined into a single condition, <b>AAcorrS</b> – any two angles and a corresponding side.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup></li> <li><b>RHS</b> (right-angle-hypotenuse-side), also known as <b>HL</b> (hypotenuse-leg): If two right-angled triangles have their hypotenuses equal in length, and a pair of other sides are equal in length, then the triangles are congruent.</li></ul> <div class="mw-heading mw-heading4"><h4 id="Side-side-angle">Side-side-angle</h4></div> <p>The SSA condition (side-side-angle) which specifies two sides and a non-included angle (also known as ASS, or angle-side-side) does not by itself prove congruence. In order to show congruence, additional information is required such as the measure of the corresponding angles and in some cases the lengths of the two pairs of corresponding sides. There are a few possible cases: </p><p>If two triangles satisfy the SSA condition and the length of the side opposite the angle is greater than or equal to the length of the adjacent side (SSA, or long side-short side-angle), then the two triangles are congruent. The opposite side is sometimes longer when the corresponding angles are acute, but it is <i>always</i> longer when the corresponding angles are right or obtuse. Where the angle is a right angle, also known as the hypotenuse-leg (HL) postulate or the right-angle-hypotenuse-side (RHS) condition, the third side can be calculated using the <a href="/wiki/Pythagorean_theorem" title="Pythagorean theorem">Pythagorean theorem</a> thus allowing the SSS postulate to be applied. </p><p>If two triangles satisfy the SSA condition and the corresponding angles are acute and the length of the side opposite the angle is equal to the length of the adjacent side multiplied by the sine of the angle, then the two triangles are congruent. </p><p>If two triangles satisfy the SSA condition and the corresponding angles are acute and the length of the side opposite the angle is greater than the length of the adjacent side multiplied by the sine of the angle (but less than the length of the adjacent side), then the two triangles cannot be shown to be congruent. This is the <a href="/wiki/Ambiguous_case" class="mw-redirect" title="Ambiguous case">ambiguous case</a> and two different triangles can be formed from the given information, but further information distinguishing them can lead to a proof of congruence. </p> <div class="mw-heading mw-heading4"><h4 id="Angle-angle-angle">Angle-angle-angle</h4></div> <p>In Euclidean geometry, AAA (angle-angle-angle) (or just AA, since in Euclidean geometry the angles of a triangle add up to 180°) does not provide information regarding the size of the two triangles and hence proves only <a href="/wiki/Similarity_(geometry)" title="Similarity (geometry)">similarity</a> and not congruence in Euclidean space. </p><p>However, in <a href="/wiki/Spherical_geometry" title="Spherical geometry">spherical geometry</a> and <a href="/wiki/Hyperbolic_geometry" title="Hyperbolic geometry">hyperbolic geometry</a> (where the sum of the angles of a triangle varies with size) AAA is sufficient for congruence on a given curvature of surface.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="CPCTC"><span class="anchor" id="CPCTC"></span> CPCTC</h3></div> <p>This <a href="/wiki/Acronym" title="Acronym">acronym</a> stands for <i>Corresponding Parts of Congruent Triangles are Congruent</i>, which is an abbreviated version of the definition of congruent triangles.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p>In more detail, it is a succinct way to say that if triangles <span class="texhtml mvar" style="font-style:italic;">ABC</span> and <span class="texhtml mvar" style="font-style:italic;">DEF</span> are congruent, that is, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \triangle ABC\cong \triangle DEF,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">△</mi> <mi>A</mi> <mi>B</mi> <mi>C</mi> <mo>≅</mo> <mi mathvariant="normal">△</mi> <mi>D</mi> <mi>E</mi> <mi>F</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \triangle ABC\cong \triangle DEF,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6eae89d6a70f8813eb27b471d343b4dad2b4d3de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.591ex; height:2.509ex;" alt="{\displaystyle \triangle ABC\cong \triangle DEF,}"></span></dd></dl> <p>with corresponding pairs of angles at vertices <span class="texhtml mvar" style="font-style:italic;">A</span> and <span class="texhtml mvar" style="font-style:italic;">D</span>; <span class="texhtml mvar" style="font-style:italic;">B</span> and <span class="texhtml mvar" style="font-style:italic;">E</span>; and <span class="texhtml mvar" style="font-style:italic;">C</span> and <span class="texhtml mvar" style="font-style:italic;">F</span>, and with corresponding pairs of sides <span class="texhtml mvar" style="font-style:italic;">AB</span> and <span class="texhtml mvar" style="font-style:italic;">DE</span>; <span class="texhtml mvar" style="font-style:italic;">BC</span> and <span class="texhtml mvar" style="font-style:italic;">EF</span>; and <span class="texhtml mvar" style="font-style:italic;">CA</span> and <span class="texhtml mvar" style="font-style:italic;">FD</span>, then the following statements are true: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {AB}}\cong {\overline {DE}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>B</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>≅</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>D</mi> <mi>E</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {AB}}\cong {\overline {DE}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6aa4eb177e18cf62014ae0ace26755a3f1575f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.575ex; height:3.009ex;" alt="{\displaystyle {\overline {AB}}\cong {\overline {DE}}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {BC}}\cong {\overline {EF}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>B</mi> <mi>C</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>≅</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>E</mi> <mi>F</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {BC}}\cong {\overline {EF}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/994739a1d9b87fb2edabf451a8292d824c7321d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.603ex; height:3.009ex;" alt="{\displaystyle {\overline {BC}}\cong {\overline {EF}}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {AC}}\cong {\overline {DF}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>C</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>≅</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>D</mi> <mi>F</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {AC}}\cong {\overline {DF}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e44cf69bc8cae29c4a6e557645fd8b74296a13de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.731ex; height:3.009ex;" alt="{\displaystyle {\overline {AC}}\cong {\overline {DF}}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \angle BAC\cong \angle EDF}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∠</mi> <mi>B</mi> <mi>A</mi> <mi>C</mi> <mo>≅</mo> <mi mathvariant="normal">∠</mi> <mi>E</mi> <mi>D</mi> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \angle BAC\cong \angle EDF}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e639c7ff9f3fd907dfadb745817b96dcfa2d2f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:17.169ex; height:2.176ex;" alt="{\displaystyle \angle BAC\cong \angle EDF}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \angle ABC\cong \angle DEF}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∠</mi> <mi>A</mi> <mi>B</mi> <mi>C</mi> <mo>≅</mo> <mi mathvariant="normal">∠</mi> <mi>D</mi> <mi>E</mi> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \angle ABC\cong \angle DEF}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d032d33bfbc146f5d9cd03c7650cf1eed29b60e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:17.169ex; height:2.176ex;" alt="{\displaystyle \angle ABC\cong \angle DEF}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \angle BCA\cong \angle EFD.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∠</mi> <mi>B</mi> <mi>C</mi> <mi>A</mi> <mo>≅</mo> <mi mathvariant="normal">∠</mi> <mi>E</mi> <mi>F</mi> <mi>D</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \angle BCA\cong \angle EFD.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7105a1268007df1c292a69c9ecaf01ba8d7ce56c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:17.816ex; height:2.176ex;" alt="{\displaystyle \angle BCA\cong \angle EFD.}"></span></dd></dl> <p>The statement is often used as a justification in elementary geometry proofs when a conclusion of the congruence of parts of two triangles is needed after the congruence of the triangles has been established. For example, if two triangles have been shown to be congruent by the <i>SSS</i> criteria and a statement that corresponding angles are congruent is needed in a proof, then CPCTC may be used as a justification of this statement. </p><p>A related theorem is <b>CPCFC</b>, in which "triangles" is replaced with "figures" so that the theorem applies to any pair of <a href="/wiki/Polygon" title="Polygon">polygons</a> or <a href="/wiki/Polyhedron" title="Polyhedron">polyhedrons</a> that are congruent. </p> <div class="mw-heading mw-heading2"><h2 id="Definition_of_congruence_in_analytic_geometry">Definition of congruence in analytic geometry</h2></div> <p>In a <a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean system</a>, congruence is fundamental; it is the counterpart of equality for numbers. In <a href="/wiki/Analytic_geometry" title="Analytic geometry">analytic geometry</a>, congruence may be defined intuitively thus: two mappings of figures onto one Cartesian coordinate system are congruent if and only if, for <i>any</i> two points in the first mapping, the <a href="/wiki/Euclidean_distance" title="Euclidean distance">Euclidean distance</a> between them is equal to the Euclidean distance between the corresponding points in the second mapping. </p><p>A more formal definition states that two <a href="/wiki/Subset" title="Subset">subsets</a> <i>A</i> and <i>B</i> of <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a> <b>R</b><sup><i>n</i></sup> are called congruent if there exists an <a href="/wiki/Isometry" title="Isometry">isometry</a> <i>f</i> : <b>R</b><sup><i>n</i></sup> → <b>R</b><sup><i>n</i></sup> (an element of the <a href="/wiki/Euclidean_group" title="Euclidean group">Euclidean group</a> <i>E</i>(<i>n</i>)) with <i>f</i>(<i>A</i>) = <i>B</i>. Congruence is an <a href="/wiki/Equivalence_relation" title="Equivalence relation">equivalence relation</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Congruent_conic_sections">Congruent conic sections</h2></div> <p>Two conic sections are congruent if their <a href="/wiki/Eccentricity_(mathematics)" title="Eccentricity (mathematics)">eccentricities</a> and one other distinct parameter characterizing them are equal. Their eccentricities establish their shapes, equality of which is sufficient to establish similarity, and the second parameter then establishes size. Since two <a href="/wiki/Circle" title="Circle">circles</a>, <a href="/wiki/Parabola" title="Parabola">parabolas</a>, or <a href="/wiki/Rectangular_hyperbola" class="mw-redirect" title="Rectangular hyperbola">rectangular hyperbolas</a> always have the same eccentricity (specifically 0 in the case of circles, 1 in the case of parabolas, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4afc1e27d418021bf10898eb44a7f5f315735ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}}"></span> in the case of rectangular hyperbolas), two circles, parabolas, or rectangular hyperbolas need to have only one other common parameter value, establishing their size, for them to be congruent. </p> <div class="mw-heading mw-heading2"><h2 id="Congruent_polyhedra">Congruent polyhedra</h2></div> <p>For two <a href="/wiki/Polyhedra" class="mw-redirect" title="Polyhedra">polyhedra</a> with the same combinatorial type (that is, the same number <i>E</i> of edges, the same number of <a href="/wiki/Face_(geometry)" title="Face (geometry)">faces</a>, and the same number of sides on corresponding faces), there exists a set of <i>E</i> measurements that can establish whether or not the polyhedra are congruent.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> The number is tight, meaning that less than <i>E</i> measurements are not enough if the polyhedra are generic among their combinatorial type. But less measurements can work for special cases. For example, <a href="/wiki/Cube" title="Cube">cubes</a> have 12 edges, but 9 measurements are enough to decide if a polyhedron of that combinatorial type is congruent to a given regular cube. </p> <div class="mw-heading mw-heading2"><h2 id="Congruent_triangles_on_a_sphere">Congruent triangles on a sphere</h2></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Solving_triangles#Solving_spherical_triangles" class="mw-redirect" title="Solving triangles">Solving triangles § Solving spherical triangles</a>, and <a href="/wiki/Spherical_trigonometry#Solution_of_triangles" title="Spherical trigonometry">Spherical trigonometry § Solution of triangles</a></div> <p>As with plane triangles, on a sphere two triangles sharing the same sequence of angle-side-angle (ASA) are necessarily congruent (that is, they have three identical sides and three identical angles).<sup id="cite_ref-Bolin_9-0" class="reference"><a href="#cite_note-Bolin-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> This can be seen as follows: One can situate one of the vertices with a given angle at the south pole and run the side with given length up the prime meridian. Knowing both angles at either end of the segment of fixed length ensures that the other two sides emanate with a uniquely determined trajectory, and thus will meet each other at a uniquely determined point; thus ASA is valid. </p><p>The congruence theorems side-angle-side (SAS) and side-side-side (SSS) also hold on a sphere; in addition, if two spherical triangles have an identical angle-angle-angle (AAA) sequence, they are congruent (unlike for plane triangles).<sup id="cite_ref-Bolin_9-1" class="reference"><a href="#cite_note-Bolin-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p><p>The plane-triangle congruence theorem angle-angle-side (AAS) does not hold for spherical triangles.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> As in plane geometry, side-side-angle (SSA) does not imply congruence. </p> <div class="mw-heading mw-heading2"><h2 id="Notation">Notation</h2></div> <p>A symbol commonly used for congruence is an equals symbol with a <a href="/wiki/Tilde" title="Tilde">tilde</a> above it, <b><span class="texhtml">≅</span></b>, corresponding to the <a href="/wiki/Unicode" title="Unicode">Unicode</a> character 'approximately equal to' (U+2245). In the UK, the three-bar equal sign <i><span class="texhtml">≡</span></i> (U+2261) is sometimes used. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div> <ul><li><a href="/wiki/Euclidean_plane_isometry" title="Euclidean plane isometry">Euclidean plane isometry</a></li> <li><a href="/wiki/Isometry" title="Isometry">Isometry</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFClaphamNicholson2009" class="citation web cs1">Clapham, C.; Nicholson, J. (2009). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20131029203826/http://web.cortland.edu/matresearch/OxfordDictionaryMathematics.pdf">"Oxford Concise Dictionary of Mathematics, Congruent Figures"</a> <span class="cs1-format">(PDF)</span>. Addison-Wesley. p. 167. Archived from the original on 29 October 2013<span class="reference-accessdate">. Retrieved <span class="nowrap">2 June</span> 2017</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Oxford+Concise+Dictionary+of+Mathematics%2C+Congruent+Figures&rft.pages=167&rft.pub=Addison-Wesley&rft.date=2009&rft.aulast=Clapham&rft.aufirst=C.&rft.au=Nicholson%2C+J.&rft_id=http%3A%2F%2Fweb.cortland.edu%2Fmatresearch%2FOxfordDictionaryMathematics.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACongruence+%28geometry%29" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_web" title="Template:Cite web">cite web</a>}}</code>: CS1 maint: bot: original URL status unknown (<a href="/wiki/Category:CS1_maint:_bot:_original_URL_status_unknown" title="Category:CS1 maint: bot: original URL status unknown">link</a>)</span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://mathopenref.com/congruent.html">"Congruence"</a>. Math Open Reference. 2009<span class="reference-accessdate">. Retrieved <span class="nowrap">2 June</span> 2017</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Congruence&rft.pub=Math+Open+Reference&rft.date=2009&rft_id=http%3A%2F%2Fmathopenref.com%2Fcongruent.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACongruence+%28geometry%29" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFParr1970" class="citation book cs1">Parr, H. E. (1970). <i>Revision Course in School mathematics</i>. Mathematics Textbooks Second Edition. G Bell and Sons Ltd. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-7135-1717-4" title="Special:BookSources/0-7135-1717-4"><bdi>0-7135-1717-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Revision+Course+in+School+mathematics&rft.series=Mathematics+Textbooks+Second+Edition&rft.pub=G+Bell+and+Sons+Ltd.&rft.date=1970&rft.isbn=0-7135-1717-4&rft.aulast=Parr&rft.aufirst=H.+E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACongruence+%28geometry%29" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCornel2002" class="citation book cs1"><a href="/wiki/Antonio_Coronel" class="mw-redirect" title="Antonio Coronel">Cornel, Antonio</a> (2002). <i>Geometry for Secondary Schools</i>. Mathematics Textbooks Second Edition. Bookmark Inc. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/971-569-441-1" title="Special:BookSources/971-569-441-1"><bdi>971-569-441-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Geometry+for+Secondary+Schools&rft.series=Mathematics+Textbooks+Second+Edition&rft.pub=Bookmark+Inc.&rft.date=2002&rft.isbn=971-569-441-1&rft.aulast=Cornel&rft.aufirst=Antonio&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACongruence+%28geometry%29" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJacobs1974" class="citation cs2">Jacobs, Harold R. (1974), <a rel="nofollow" class="external text" href="https://archive.org/details/geometry0000jaco/page/160/mode/2up"><i>Geometry</i></a>, W.H. Freeman, p. 160, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-7167-0456-0" title="Special:BookSources/0-7167-0456-0"><bdi>0-7167-0456-0</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Geometry&rft.pages=160&rft.pub=W.H.+Freeman&rft.date=1974&rft.isbn=0-7167-0456-0&rft.aulast=Jacobs&rft.aufirst=Harold+R.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fgeometry0000jaco%2Fpage%2F160%2Fmode%2F2up&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACongruence+%28geometry%29" class="Z3988"></span> Jacobs uses a slight variation of the phrase</span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.cliffsnotes.com/study-guides/geometry/triangles/congruent-triangles">"Congruent Triangles"</a>. Cliff's Notes<span class="reference-accessdate">. Retrieved <span class="nowrap">2014-02-04</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Congruent+Triangles&rft.pub=Cliff%27s+Notes&rft_id=https%3A%2F%2Fwww.cliffsnotes.com%2Fstudy-guides%2Fgeometry%2Ftriangles%2Fcongruent-triangles&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACongruence+%28geometry%29" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBorisovDickinsonHastings2010" class="citation journal cs1">Borisov, Alexander; Dickinson, Mark; Hastings, Stuart (March 2010). "A Congruence Problem for Polyhedra". <i><a href="/wiki/American_Mathematical_Monthly" class="mw-redirect" title="American Mathematical Monthly">American Mathematical Monthly</a></i>. <b>117</b> (3): 232–249. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/0811.4197">0811.4197</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.4169%2F000298910X480081">10.4169/000298910X480081</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:8166476">8166476</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Mathematical+Monthly&rft.atitle=A+Congruence+Problem+for+Polyhedra&rft.volume=117&rft.issue=3&rft.pages=232-249&rft.date=2010-03&rft_id=info%3Aarxiv%2F0811.4197&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A8166476%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.4169%2F000298910X480081&rft.aulast=Borisov&rft.aufirst=Alexander&rft.au=Dickinson%2C+Mark&rft.au=Hastings%2C+Stuart&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACongruence+%28geometry%29" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCreech" class="citation web cs1">Creech, Alexa. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20131111162553/http://146.163.152.131/teaching/projects/creech_final.pdf">"A Congruence Problem"</a> <span class="cs1-format">(PDF)</span>. Archived from <a rel="nofollow" class="external text" href="http://146.163.152.131/teaching/projects/creech_final.pdf">the original</a> <span class="cs1-format">(PDF)</span> on November 11, 2013.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=A+Congruence+Problem&rft.aulast=Creech&rft.aufirst=Alexa&rft_id=http%3A%2F%2F146.163.152.131%2Fteaching%2Fprojects%2Fcreech_final.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACongruence+%28geometry%29" class="Z3988"></span></span> </li> <li id="cite_note-Bolin-9"><span class="mw-cite-backlink">^ <a href="#cite_ref-Bolin_9-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Bolin_9-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBolin2003" class="citation web cs1">Bolin, Michael (September 9, 2003). <a rel="nofollow" class="external text" href="http://math.iit.edu/~mccomic/420/notes/Bolin_spherical.pdf#page=6">"Exploration of Spherical Geometry"</a> <span class="cs1-format">(PDF)</span>. pp. 6–7. <a rel="nofollow" class="external text" href="https://ghostarchive.org/archive/20221009/http://math.iit.edu/~mccomic/420/notes/Bolin_spherical.pdf#page=6">Archived</a> <span class="cs1-format">(PDF)</span> from the original on 2022-10-09.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Exploration+of+Spherical+Geometry&rft.pages=6-7&rft.date=2003-09-09&rft.aulast=Bolin&rft.aufirst=Michael&rft_id=http%3A%2F%2Fmath.iit.edu%2F~mccomic%2F420%2Fnotes%2FBolin_spherical.pdf%23page%3D6&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACongruence+%28geometry%29" class="Z3988"></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" 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