{"id":116,"date":"2022-04-13T08:23:08","date_gmt":"2022-04-13T08:23:08","guid":{"rendered":"http:\/\/9thclass.deltapublications.in\/?page_id=116"},"modified":"2025-10-31T09:35:00","modified_gmt":"2025-10-31T09:35:00","slug":"g-8-sss-sas-and-asa-theorems","status":"publish","type":"page","link":"https:\/\/9thclass.deltapublications.in\/index.php\/g-8-sss-sas-and-asa-theorems\/","title":{"rendered":"G.8 SSS, SAS and ASA Theorems"},"content":{"rendered":"\n<h2 class=\"wp-block-heading has-text-align-center has-text-color\" style=\"color:#00056d;text-transform:uppercase\"><strong>SSS, SAS and ASA Theorems<\/strong><\/h2>\n\n\n\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-9d6595d7 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\" style=\"flex-basis:100%\">\n<div style=\"position: relative; width: 100%; height: 0; padding-top: 56.2500%;\n padding-bottom: 0; box-shadow: 0 2px 8px 0 rgba(63,69,81,0.16); margin-top: 1.6em; margin-bottom: 0.9em; overflow: hidden;\n border-radius: 8px; will-change: transform;\">\n  <iframe loading=\"lazy\" style=\"position: absolute; width: 100%; height: 100%; top: 0; left: 0; border: none; padding: 0;margin: 0;\"\n    src=\"https:\/\/www.canva.com\/design\/DAG3WX1ha-s\/ChPw2qFiSCGYskTTh7jiVg\/watch?embed\" allowfullscreen=\"allowfullscreen\" allow=\"fullscreen\">\n  <\/iframe>\n<\/div>\n<a href=\"https:&#x2F;&#x2F;www.canva.com&#x2F;design&#x2F;DAG3WX1ha-s&#x2F;ChPw2qFiSCGYskTTh7jiVg&#x2F;watch?utm_content=DAG3WX1ha-s&amp;utm_campaign=designshare&amp;utm_medium=embeds&amp;utm_source=link\" target=\"_blank\" rel=\"noopener\">Design<\/a> by Delta publications\n<\/div>\n<\/div>\n\n\n\n<p class=\"has-text-color has-huge-font-size\" style=\"color:#74008b;text-transform:capitalize\"><strong>key notes :<\/strong><\/p>\n\n\n\n<h3 class=\"wp-block-heading has-text-color has-link-color has-large-font-size wp-elements-a107070e09e6c966f60906806804d4bd\" style=\"color:#000060\">\ud83d\udd3a <strong>SSS (Side-Side-Side) Congruence Theorem<\/strong><\/h3>\n\n\n\n<p>\ud83d\udca1 <strong>Statement:<\/strong><br>If <strong>three sides<\/strong> of one triangle are <strong>equal<\/strong> to the <strong>three sides<\/strong> of another triangle, then the triangles are <strong>congruent<\/strong>.<\/p>\n\n\n\n<p>\ud83d\udccf <strong>Symbolically:<\/strong><br>If AB = PQ, BC = QR, and CA = RP \ud83d\udc49 then <strong>\u25b3ABC \u2245 \u25b3PQR<\/strong><\/p>\n\n\n\n<p>\ud83c\udfaf <strong>Meaning:<\/strong><br>All sides are the same length \u2014 so the triangles are exactly the same shape and size!<\/p>\n\n\n\n<p>\ud83e\udde0 <strong>Tip:<\/strong><br>No need to check angles \u2014 sides are enough! \u2705<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading has-text-color has-link-color has-large-font-size wp-elements-bcc02596b2825e80feedc71dfa7ba382\" style=\"color:#000060\">\ud83d\udcd0 <strong>SAS (Side-Angle-Side) Congruence Theorem<\/strong><\/h3>\n\n\n\n<p>\ud83d\udca1 <strong>Statement:<\/strong><br>If <strong>two sides and the included angle<\/strong> (the angle between those sides) of one triangle are <strong>equal<\/strong> to <strong>two sides and the included angle<\/strong> of another triangle, then the triangles are <strong>congruent<\/strong>.<\/p>\n\n\n\n<p>\ud83d\udccf <strong>Symbolically:<\/strong><br>If AB = PQ, \u2220B = \u2220Q, and BC = QR \ud83d\udc49 then <strong>\u25b3ABC \u2245 \u25b3PQR<\/strong><\/p>\n\n\n\n<p>\ud83c\udfaf <strong>Meaning:<\/strong><br>Two equal sides with the angle between them equal ensure both triangles are identical.<\/p>\n\n\n\n<p>\ud83e\udde0 <strong>Remember:<\/strong><br>The <strong>included angle<\/strong> must be the one <strong>between the two sides<\/strong> \u2014 not outside them! \ud83d\udd3a<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading has-text-color has-link-color has-large-font-size wp-elements-76697e89b2a74d106b1cb80589102425\" style=\"color:#000060\">\ud83d\udccf <strong>ASA (Angle-Side-Angle) Congruence Theorem<\/strong><\/h3>\n\n\n\n<p>\ud83d\udca1 <strong>Statement:<\/strong><br>If <strong>two angles and the included side<\/strong> (the side between the angles) of one triangle are <strong>equal<\/strong> to the corresponding parts of another triangle, then the triangles are <strong>congruent<\/strong>.<\/p>\n\n\n\n<p>\ud83d\udccf <strong>Symbolically:<\/strong><br>If \u2220A = \u2220P, AB = PQ, and \u2220B = \u2220Q \ud83d\udc49 then <strong>\u25b3ABC \u2245 \u25b3PQR<\/strong><\/p>\n\n\n\n<p>\ud83c\udfaf <strong>Meaning:<\/strong><br>Two angles and the side between them determine the triangle completely.<\/p>\n\n\n\n<p>\ud83e\udde0 <strong>Tip:<\/strong><br>Angles + the included side = perfect match! \ud83d\udcaf<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading has-text-color has-link-color has-large-font-size wp-elements-dee21bbca8e9b0dd76e016a7d403edc8\" style=\"color:#000060\">\ud83d\udcab <strong>Why Congruence Matters<\/strong><\/h3>\n\n\n\n<p>\u2705 Congruent triangles have <strong>equal sides and equal angles<\/strong>.<br>\u2705 Used to <strong>prove geometric results<\/strong>, <strong>construct figures<\/strong>, and <strong>solve real-world problems<\/strong>.<br>\u2705 Congruence helps us understand <strong>symmetry<\/strong> and <strong>similarity<\/strong> in shapes! \ud83c\udf0d<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading has-text-color has-link-color has-large-font-size wp-elements-ae88624871d6e387b97b56992e1281ef\" style=\"color:#000060\">\ud83c\udf08 <strong>Quick Recap Table:<\/strong><\/h3>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-fixed-layout\"><thead><tr><th>\ud83d\udd22 Theorem<\/th><th>\ud83e\udde9 Given Equal Parts<\/th><th>\ud83c\udfaf Condition<\/th><th>\ud83c\udfc6 Result<\/th><\/tr><\/thead><tbody><tr><td><strong>SSS<\/strong><\/td><td>3 sides<\/td><td>All sides equal<\/td><td>Triangles congruent<\/td><\/tr><tr><td><strong>SAS<\/strong><\/td><td>2 sides + included angle<\/td><td>Angle between sides equal<\/td><td>Triangles congruent<\/td><\/tr><tr><td><strong>ASA<\/strong><\/td><td>2 angles + included side<\/td><td>Side between angles equal<\/td><td>Triangles congruent<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading has-text-color has-link-color has-large-font-size wp-elements-d1b3a5236d5b3a2d343e46ffb8f844b2\" style=\"color:#000060\">\ud83e\udded <strong>Bonus Tip:<\/strong><\/h3>\n\n\n\n<p>When triangles are congruent (\u2245), their <strong>corresponding parts are equal<\/strong> \u2014 this is called <strong>CPCTC<\/strong> (Corresponding Parts of Congruent Triangles are Congruent)! \ud83c\udfc5<\/p>\n\n\n\n<p class=\"has-text-align-center has-text-color has-large-font-size\" style=\"color:#105000\"><strong>Learn with an example<\/strong><\/p>\n\n\n\n<div class=\"wp-block-group has-background has-large-font-size\" style=\"background-color:#f6bdbd\"><div class=\"wp-block-group__inner-container is-layout-constrained wp-block-group-is-layout-constrained\">\n<div class=\"wp-block-group has-background-background-color has-background\"><div class=\"wp-block-group__inner-container is-layout-constrained wp-block-group-is-layout-constrained\">\n<p class=\"has-text-color\" style=\"color:#b00012\"><strong>Which&nbsp;rule explains why these triangles are&nbsp;congruent?<\/strong><\/p>\n\n\n<div class=\"wp-block-image is-resized\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"500\" height=\"500\" src=\"https:\/\/9thclass.deltapublications.in\/wp-content\/uploads\/2023\/06\/Untitled-design-1-2.png\" alt=\"\" class=\"wp-image-8507\" style=\"width:300px;height:auto\" srcset=\"https:\/\/9thclass.deltapublications.in\/wp-content\/uploads\/2023\/06\/Untitled-design-1-2.png 500w, https:\/\/9thclass.deltapublications.in\/wp-content\/uploads\/2023\/06\/Untitled-design-1-2-300x300.png 300w, https:\/\/9thclass.deltapublications.in\/wp-content\/uploads\/2023\/06\/Untitled-design-1-2-150x150.png 150w\" sizes=\"auto, (max-width: 500px) 100vw, 500px\" \/><\/figure><\/div>\n\n\n<ul class=\"wp-block-list\">\n<li>ASA <\/li>\n\n\n\n<li>SSS<\/li>\n\n\n\n<li>SAS<\/li>\n\n\n\n<li>These triangles cannot be proven congruent.<\/li>\n<\/ul>\n<\/div><\/div>\n\n\n\n<p>First,&nbsp;look for congruent sides and&nbsp;angles.<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"500\" height=\"500\" src=\"https:\/\/9thclass.deltapublications.in\/wp-content\/uploads\/2023\/06\/Untitled_design__1_-removebg-preview-5.png\" alt=\"\" class=\"wp-image-8508\" style=\"aspect-ratio:1;width:387px;height:auto\" srcset=\"https:\/\/9thclass.deltapublications.in\/wp-content\/uploads\/2023\/06\/Untitled_design__1_-removebg-preview-5.png 500w, https:\/\/9thclass.deltapublications.in\/wp-content\/uploads\/2023\/06\/Untitled_design__1_-removebg-preview-5-300x300.png 300w, https:\/\/9thclass.deltapublications.in\/wp-content\/uploads\/2023\/06\/Untitled_design__1_-removebg-preview-5-150x150.png 150w\" sizes=\"auto, (max-width: 500px) 100vw, 500px\" \/><\/figure><\/div>\n\n\n<p>CF \u2245 GH and BF \u2245 BG .<\/p>\n\n\n\n<p>Next, notice that \u2220CBF and \u2220GBH are vertical angles. Since vertical angles are congruent, \u2220CBF\u2245\u2220GBH.<\/p>\n\n\n\n<p>Finally, put the three congruency statements in order. BF is between CF and \u2220CBF, and BG is between GH and \u2220GBH in the diagram.<\/p>\n\n\n\n<p>CF \u2245 GH                          Side<\/p>\n\n\n\n<p>BF \u2245 BG                       Side<\/p>\n\n\n\n<p>\u2220CBF \u2245 \u2220GBH          Angle<\/p>\n\n\n\n<p>In order, the congruent sides and angles form <strong>SSA<\/strong>. This is not one of the three ways to show that triangles are congruent. There is not enough information to prove that the triangles are congruent.<\/p>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-group has-background has-large-font-size\" style=\"background-color:#fbd1ef\"><div class=\"wp-block-group__inner-container is-layout-constrained wp-block-group-is-layout-constrained\">\n<div class=\"wp-block-group has-background-background-color has-background\"><div class=\"wp-block-group__inner-container is-layout-constrained wp-block-group-is-layout-constrained\">\n<p class=\"has-text-color\" style=\"color:#b00012\"><strong>Which&nbsp;rule explains why these triangles are&nbsp;congruent?<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"670\" height=\"200\" src=\"https:\/\/9thclass.deltapublications.in\/wp-content\/uploads\/2023\/06\/Untitled-design-2-3.png\" alt=\"\" class=\"wp-image-8511\" style=\"width:520px;height:auto\" srcset=\"https:\/\/9thclass.deltapublications.in\/wp-content\/uploads\/2023\/06\/Untitled-design-2-3.png 670w, https:\/\/9thclass.deltapublications.in\/wp-content\/uploads\/2023\/06\/Untitled-design-2-3-300x90.png 300w\" sizes=\"auto, (max-width: 670px) 100vw, 670px\" \/><\/figure>\n\n\n\n<ul class=\"wp-block-list\">\n<li>SSS<\/li>\n\n\n\n<li>SAS<\/li>\n\n\n\n<li>ASA<\/li>\n\n\n\n<li>These triangles cannot be proven congruent.<\/li>\n<\/ul>\n<\/div><\/div>\n\n\n\n<p>First,&nbsp;look for congruent sides and&nbsp;angles.<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"670\" height=\"200\" src=\"https:\/\/9thclass.deltapublications.in\/wp-content\/uploads\/2023\/06\/Untitled_design__2_-removebg-preview-5.png\" alt=\"\" class=\"wp-image-8512\" style=\"aspect-ratio:3.35;width:583px;height:auto\" srcset=\"https:\/\/9thclass.deltapublications.in\/wp-content\/uploads\/2023\/06\/Untitled_design__2_-removebg-preview-5.png 670w, https:\/\/9thclass.deltapublications.in\/wp-content\/uploads\/2023\/06\/Untitled_design__2_-removebg-preview-5-300x90.png 300w\" sizes=\"auto, (max-width: 670px) 100vw, 670px\" \/><\/figure><\/div>\n\n\n<p>Notice that there are no pairs of congruent sides. Since all of the congruency theorems call for at least one pair of congruent sides, there is not enough information to prove that the triangles are congruent.<\/p>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-group has-background has-large-font-size\" style=\"background-color:#f7eeca\"><div class=\"wp-block-group__inner-container is-layout-constrained wp-block-group-is-layout-constrained\">\n<div class=\"wp-block-group has-background-background-color has-background\"><div class=\"wp-block-group__inner-container is-layout-constrained wp-block-group-is-layout-constrained\">\n<p class=\"has-text-color\" style=\"color:#b00012\"><strong>Which&nbsp;rule explains why these triangles are&nbsp;congruent?<\/strong><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"500\" height=\"500\" src=\"https:\/\/9thclass.deltapublications.in\/wp-content\/uploads\/2023\/06\/Untitled-design-3-3.png\" alt=\"\" class=\"wp-image-8513\" style=\"aspect-ratio:1;width:413px;height:auto\" srcset=\"https:\/\/9thclass.deltapublications.in\/wp-content\/uploads\/2023\/06\/Untitled-design-3-3.png 500w, https:\/\/9thclass.deltapublications.in\/wp-content\/uploads\/2023\/06\/Untitled-design-3-3-300x300.png 300w, https:\/\/9thclass.deltapublications.in\/wp-content\/uploads\/2023\/06\/Untitled-design-3-3-150x150.png 150w\" sizes=\"auto, (max-width: 500px) 100vw, 500px\" \/><\/figure><\/div>\n\n\n<ul class=\"wp-block-list\">\n<li>ASA<\/li>\n\n\n\n<li>SAS<\/li>\n\n\n\n<li>SSS<\/li>\n\n\n\n<li>These triangles cannot be proven congruent.<\/li>\n<\/ul>\n<\/div><\/div>\n\n\n\n<p>First,&nbsp;look for congruent sides and&nbsp;angles.<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"500\" height=\"500\" src=\"https:\/\/9thclass.deltapublications.in\/wp-content\/uploads\/2023\/06\/Untitled_design__3_-removebg-preview-1.png\" alt=\"\" class=\"wp-image-8514\" style=\"aspect-ratio:1;width:386px;height:auto\" srcset=\"https:\/\/9thclass.deltapublications.in\/wp-content\/uploads\/2023\/06\/Untitled_design__3_-removebg-preview-1.png 500w, https:\/\/9thclass.deltapublications.in\/wp-content\/uploads\/2023\/06\/Untitled_design__3_-removebg-preview-1-300x300.png 300w, https:\/\/9thclass.deltapublications.in\/wp-content\/uploads\/2023\/06\/Untitled_design__3_-removebg-preview-1-150x150.png 150w\" sizes=\"auto, (max-width: 500px) 100vw, 500px\" \/><\/figure><\/div>\n\n\n<p>Notice that there are no pairs of congruent sides. Since all of the congruency theorems call for at least one pair of congruent sides, there is not enough information to prove that the triangles are congruent.<\/p>\n<\/div><\/div>\n\n\n\n<p class=\"has-text-color has-large-font-size\" style=\"color:#d90000\"><strong>Let&#8217;s practice!\u270d\ufe0f<\/strong><\/p>\n\n\n\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-9d6595d7 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<figure class=\"wp-block-image size-full\"><a href=\"https:\/\/wordwall.net\/play\/81354\/072\/456\"><img loading=\"lazy\" decoding=\"async\" width=\"500\" height=\"500\" src=\"https:\/\/9thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-3-34.png\" alt=\"\" class=\"wp-image-7757\" srcset=\"https:\/\/9thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-3-34.png 500w, https:\/\/9thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-3-34-300x300.png 300w, https:\/\/9thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-3-34-150x150.png 150w\" sizes=\"auto, (max-width: 500px) 100vw, 500px\" \/><\/a><\/figure>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<figure class=\"wp-block-image size-full\"><a href=\"https:\/\/wordwall.net\/play\/81098\/400\/479\"><img loading=\"lazy\" decoding=\"async\" width=\"500\" height=\"500\" src=\"https:\/\/9thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-2-37.png\" alt=\"\" class=\"wp-image-7758\" srcset=\"https:\/\/9thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-2-37.png 500w, https:\/\/9thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-2-37-300x300.png 300w, https:\/\/9thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-2-37-150x150.png 150w\" sizes=\"auto, (max-width: 500px) 100vw, 500px\" \/><\/a><\/figure>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>SSS, SAS and ASA Theorems Design by Delta publications key notes : \ud83d\udd3a SSS (Side-Side-Side) Congruence Theorem \ud83d\udca1 Statement:If three sides of one triangle are equal to the three sides of another triangle, then the triangles are congruent. \ud83d\udccf Symbolically:If AB = PQ, BC = QR, and CA = RP \ud83d\udc49 then \u25b3ABC \u2245 \u25b3PQR<a class=\"more-link\" href=\"https:\/\/9thclass.deltapublications.in\/index.php\/g-8-sss-sas-and-asa-theorems\/\">Continue reading <span class=\"screen-reader-text\">&#8220;G.8 SSS, SAS and ASA Theorems&#8221;<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"om_disable_all_campaigns":false,"_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"footnotes":""},"class_list":["post-116","page","type-page","status-publish","hentry","entry"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/9thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages\/116","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/9thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/9thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/9thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/9thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/comments?post=116"}],"version-history":[{"count":22,"href":"https:\/\/9thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages\/116\/revisions"}],"predecessor-version":[{"id":18162,"href":"https:\/\/9thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages\/116\/revisions\/18162"}],"wp:attachment":[{"href":"https:\/\/9thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/media?parent=116"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}