{"id":122,"date":"2022-04-13T08:24:18","date_gmt":"2022-04-13T08:24:18","guid":{"rendered":"http:\/\/9thclass.deltapublications.in\/?page_id=122"},"modified":"2025-10-31T09:47:06","modified_gmt":"2025-10-31T09:47:06","slug":"g-11-hypotenuse-leg-theorem","status":"publish","type":"page","link":"https:\/\/9thclass.deltapublications.in\/index.php\/g-11-hypotenuse-leg-theorem\/","title":{"rendered":"G.11 Hypotenuse-Leg Theorem"},"content":{"rendered":"\n<h2 class=\"wp-block-heading has-text-align-center has-text-color\" style=\"color:#00056d\"><strong>Hypotenuse-Leg Theorem<\/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: 56.6038%;\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\/DAG3WVXppWo\/BJQg699iaBzRmLSwcqkE2w\/watch?embed\" allowfullscreen=\"allowfullscreen\" allow=\"fullscreen\">\n  <\/iframe>\n<\/div>\n<a href=\"https:&#x2F;&#x2F;www.canva.com&#x2F;design&#x2F;DAG3WVXppWo&#x2F;BJQg699iaBzRmLSwcqkE2w&#x2F;watch?utm_content=DAG3WVXppWo&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-af35bb0737526a185ed20476bb6b38ad\" style=\"color:#000060\">\ud83d\udd39 <strong>What is the Hypotenuse\u2013Leg Theorem?<\/strong><\/h3>\n\n\n\n<p>\ud83d\udc49 The <strong>HL Theorem<\/strong> applies only to <strong>right triangles<\/strong> \u23eb.<br>It states that:<\/p>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p>\ud83e\udde0 <em>If the hypotenuse and one leg of a right triangle are congruent to the hypotenuse and one leg of another right triangle, then the two triangles are congruent.<\/em><\/p>\n<\/blockquote>\n\n\n\n<p>\u2733\ufe0f <strong>In short:<\/strong><br>If two right triangles have the same hypotenuse and one leg equal \u27a1\ufe0f they are <strong>congruent triangles<\/strong>! \ud83d\udd3a=\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-c8ee5a5098d3d9d36b16c69cca0bd5c9\" style=\"color:#000060\">\ud83d\udd39 <strong>Conditions for HL Theorem<\/strong> \ud83e\udde9<\/h3>\n\n\n\n<p>To use the HL Theorem, make sure these are true:<br>\u2705 Both triangles are <strong>right triangles<\/strong><br>\u2705 The <strong>hypotenuse<\/strong> of one is congruent to the hypotenuse of the other<br>\u2705 One <strong>leg<\/strong> of one triangle is congruent to one <strong>leg<\/strong> of the other<\/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-985a25cae9b1103aee0c509b31a6ee75\" style=\"color:#000060\">\ud83d\udd39 <strong>Why It Works<\/strong> \ud83e\udde0<\/h3>\n\n\n\n<p>The HL Theorem is actually a <strong>special case<\/strong> of the <strong>SSS (Side-Side-Side)<\/strong> or <strong>SAS (Side-Angle-Side)<\/strong> postulates because the right angle is already known (90\u00b0). \u299c<\/p>\n\n\n\n<p>\ud83d\udfe9 The right angle acts as the included angle! \ud83c\udfaf<\/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-fbfecc5ba0bff9d25657a8525c862bb0\" style=\"color:#000060\">\ud83d\udd39 <strong>Formula Reminder<\/strong> \ud83d\udca1<\/h3>\n\n\n\n<p>There\u2019s <strong>no direct formula<\/strong>, but remember:<\/p>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p>HL \u2248 \u201cRight triangle + Hypotenuse + Leg = Congruent Triangles\u201d<\/p>\n<\/blockquote>\n\n\n\n<p>Example:<br>If in \u25b3ABC and \u25b3DEF,<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u2220C and \u2220F are <strong>right angles<\/strong><\/li>\n\n\n\n<li><strong>AB \u2245 DE<\/strong> (hypotenuse)<\/li>\n\n\n\n<li><strong>BC \u2245 EF<\/strong> (leg)<br>\u2705 Then, <strong>\u25b3ABC \u2245 \u25b3DEF<\/strong> by <strong>HL Theorem<\/strong> \ud83c\udf89<\/li>\n<\/ul>\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-25d33081546929a23a931023de06c970\" style=\"color:#000060\">\ud83d\udd39 <strong>Diagram Tip<\/strong> \ud83d\udd8a\ufe0f<\/h3>\n\n\n\n<p>Draw two right triangles side by side:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Mark one side as the <strong>leg<\/strong> \ud83e\uddb5<\/li>\n\n\n\n<li>Mark the longest side as the <strong>hypotenuse<\/strong> \ud83d\udd3a<\/li>\n\n\n\n<li>Show congruency with \u201c\u2245\u201d marks to visualize the theorem! \ud83d\udc40<\/li>\n<\/ul>\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-f5be68acef6e03053d4bf0eaab5e5fe8\" style=\"color:#000060\">\ud83d\udd39 <strong>Real-Life Connection<\/strong> \ud83c\udf0d<\/h3>\n\n\n\n<p>The HL Theorem helps in:<br>\ud83c\udfd7\ufe0f Designing ramps or right-angled supports<br>\ud83d\udcd0 Verifying structures like ladders leaning against walls<br>\ud83c\udfaf Engineering and architecture designs that involve right triangles<\/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-f011061fc57d38be1946c53e19754d8f\" style=\"color:#000060\">\ud83d\udd39 <strong>Fun Memory Trick!<\/strong> \ud83c\udfb2<\/h3>\n\n\n\n<p>\ud83d\udcac \u201c<strong>HL = Hypotenuse + Leg = Right Triangles Match!<\/strong>\u201d<br>\ud83e\udde1 HL \u2192 &#8220;Helpful in Right triangles&#8221; \ud83d\ude04<\/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-7e8a367617bf0246cb3dee0219662de7\" style=\"color:#000060\">\ud83d\udcd8 <strong>Summary:<\/strong><\/h3>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-fixed-layout\"><thead><tr><th>\ud83d\udd39Condition<\/th><th>\ud83d\udd39Meaning<\/th><\/tr><\/thead><tbody><tr><td>Both are right triangles \u299c<\/td><td>Each has a 90\u00b0 angle<\/td><\/tr><tr><td>Hypotenuse congruent<\/td><td>Longest sides equal<\/td><\/tr><tr><td>One leg congruent<\/td><td>One shorter side equal<\/td><\/tr><tr><td>\ud83d\udff0<\/td><td>Triangles are congruent (HL)<\/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-1737c2fcb333e7be076d0722f12d544c\" style=\"color:#000060\">\u2728 <strong>Example Question:<\/strong><\/h3>\n\n\n\n<p>If \u25b3ABC and \u25b3DEF are right triangles with<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Hypotenuse AB = DE<\/li>\n\n\n\n<li>Leg BC = EF<br>\u27a1\ufe0f Then, <strong>\u25b3ABC \u2245 \u25b3DEF by HL Theorem<\/strong> \u2705<\/li>\n<\/ul>\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-primary-color has-text-color has-background has-large-font-size\" style=\"background-color:#f9f0af\"><div class=\"wp-block-group__inner-container is-layout-flow wp-block-group-is-layout-flow\">\n<div class=\"wp-block-group has-background-background-color has-background\"><div class=\"wp-block-group__inner-container is-layout-flow wp-block-group-is-layout-flow\">\n<p class=\"has-text-color\" style=\"color:#b00012\"><strong>What&nbsp;values of&nbsp;v&nbsp;and&nbsp;w&nbsp;make&nbsp;\u25b3FGH \u2245 \u25b3DCE?<\/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\/2024\/01\/Untitled-design-16.png\" alt=\"\" class=\"wp-image-11752\" style=\"aspect-ratio:1.6666666666666667;width:524px;height:auto\" srcset=\"https:\/\/9thclass.deltapublications.in\/wp-content\/uploads\/2024\/01\/Untitled-design-16.png 500w, https:\/\/9thclass.deltapublications.in\/wp-content\/uploads\/2024\/01\/Untitled-design-16-300x300.png 300w, https:\/\/9thclass.deltapublications.in\/wp-content\/uploads\/2024\/01\/Untitled-design-16-150x150.png 150w\" sizes=\"auto, (max-width: 500px) 100vw, 500px\" \/><\/figure><\/div>\n\n\n<p>v = _____<\/p>\n\n\n\n<p>w = _____<\/p>\n<\/div><\/div>\n\n\n\n<p>Use&nbsp;the Hypotenuse-Leg Theorem to solve for the values of&nbsp;v&nbsp;and&nbsp;w&nbsp;that make the triangles&nbsp;congruent.<\/p>\n\n\n\n<p>First,&nbsp;look at the labelled pair of corresponding legs. The first leg&nbsp;FH&nbsp;has a length of&nbsp;v&nbsp;and the second leg&nbsp;DE&nbsp;has a length of&nbsp;18.&nbsp;For the triangles to be congruent, the leg lengths must be equal. So,&nbsp;v&nbsp;must equal&nbsp;18.<\/p>\n\n\n\n<p>Now,&nbsp;look at the hypotenuses. The first hypotenuse&nbsp;FG&nbsp;has a length of&nbsp;42&nbsp;and the second hypotenuse&nbsp;DC&nbsp;has a length of&nbsp;w.&nbsp;For the triangles to be congruent, the hypotenuse lengths must be equal. So,&nbsp;w&nbsp;must equal&nbsp;42.<\/p>\n\n\n\n<p>The&nbsp;values of&nbsp;v&nbsp;and&nbsp;w&nbsp;that make&nbsp;\u25b3FGH \u2245 \u25b3DCE&nbsp;are&nbsp;v=18&nbsp;and&nbsp;w=42.<\/p>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-group has-background has-large-font-size\" style=\"background-color:#cbf8f8\"><div class=\"wp-block-group__inner-container is-layout-flow wp-block-group-is-layout-flow\">\n<div class=\"wp-block-group has-background-background-color has-background\"><div class=\"wp-block-group__inner-container is-layout-flow wp-block-group-is-layout-flow\">\n<p class=\"has-text-color\" style=\"color:#b00012\"><strong>What&nbsp;values of&nbsp;x&nbsp;and&nbsp;y&nbsp;make&nbsp;\u25b3GHI \u2245 \u25b3EDF?<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"670\" height=\"300\" src=\"https:\/\/9thclass.deltapublications.in\/wp-content\/uploads\/2024\/01\/Untitled-design-17.png\" alt=\"\" class=\"wp-image-11753\" style=\"width:500px;height:300px\" srcset=\"https:\/\/9thclass.deltapublications.in\/wp-content\/uploads\/2024\/01\/Untitled-design-17.png 670w, https:\/\/9thclass.deltapublications.in\/wp-content\/uploads\/2024\/01\/Untitled-design-17-300x134.png 300w\" sizes=\"auto, (max-width: 670px) 100vw, 670px\" \/><\/figure>\n\n\n\n<p>X =<\/p>\n\n\n\n<p>Y =<\/p>\n<\/div><\/div>\n\n\n\n<p>Use&nbsp;the Hypotenuse-Leg Theorem to solve for the values of&nbsp;x&nbsp;and&nbsp;y&nbsp;that make the triangles&nbsp;congruent.<\/p>\n\n\n\n<p>First,&nbsp;look at the labelled pair of corresponding legs. The first leg&nbsp;HI&nbsp;has a length of&nbsp;x&nbsp;and the second leg&nbsp;DF&nbsp;has a length of&nbsp;44.&nbsp;For the triangles to be congruent, the leg lengths must be equal. So,&nbsp;x&nbsp;must equal&nbsp;44.<\/p>\n\n\n\n<p>Now,&nbsp;look at the hypotenuses. The first hypotenuse&nbsp;GI&nbsp;has a length of&nbsp;49&nbsp;and the second hypotenuse&nbsp;EF&nbsp;has a length of&nbsp;y.&nbsp;For the triangles to be congruent, the hypotenuse lengths must be equal. So,&nbsp;y&nbsp;must equal&nbsp;49.<\/p>\n\n\n\n<p>The&nbsp;values of&nbsp;x&nbsp;and&nbsp;y&nbsp;that make&nbsp;\u25b3GHI\u2245\u25b3EDF&nbsp;are&nbsp;x=44&nbsp;and&nbsp;y=49.<\/p>\n<\/div><\/div>\n\n\n\n<p class=\"has-text-color has-large-font-size\" style=\"color:#d90000\"><strong>Let&#8217;s try some problems!\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\/81361\/432\/156\"><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\/81132\/228\/807\"><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>Hypotenuse-Leg Theorem Design by Delta publications key notes : \ud83d\udd39 What is the Hypotenuse\u2013Leg Theorem? \ud83d\udc49 The HL Theorem applies only to right triangles \u23eb.It states that: \ud83e\udde0 If the hypotenuse and one leg of a right triangle are congruent to the hypotenuse and one leg of another right triangle, then the two triangles are<a class=\"more-link\" href=\"https:\/\/9thclass.deltapublications.in\/index.php\/g-11-hypotenuse-leg-theorem\/\">Continue reading <span class=\"screen-reader-text\">&#8220;G.11 Hypotenuse-Leg Theorem&#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-122","page","type-page","status-publish","hentry","entry"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/9thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages\/122","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=122"}],"version-history":[{"count":25,"href":"https:\/\/9thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages\/122\/revisions"}],"predecessor-version":[{"id":18163,"href":"https:\/\/9thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages\/122\/revisions\/18163"}],"wp:attachment":[{"href":"https:\/\/9thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/media?parent=122"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}