{"id":33556,"date":"2023-08-29T13:26:19","date_gmt":"2023-08-29T13:26:19","guid":{"rendered":"https:\/\/www.splashlearn.com\/math-vocabulary\/?page_id=33556"},"modified":"2023-09-03T15:20:37","modified_gmt":"2023-09-03T15:20:37","slug":"hypotenuse-leg-theorem-definition-proof-examples-faqs","status":"publish","type":"post","link":"https:\/\/www.splashlearn.com\/math-vocabulary\/hypotenuse-leg-theorem","title":{"rendered":"Hypotenuse Leg Theorem \u2013 Definition, Proof, Examples, FAQs"},"content":{"rendered":"<div class=\"aioseo-breadcrumbs\"><span class=\"aioseo-breadcrumb\">\n\tMath-Vocabulary\n<\/span><\/div>\n\n<div class=\"ub_table-of-contents\" data-showtext=\"show\" data-hidetext=\"hide\" data-scrolltype=\"auto\" id=\"ub_table-of-contents-41f46012-2629-45a1-b8a7-eb0c482c7d49\" data-initiallyhideonmobile=\"false\"\n                    data-initiallyshow=\"true\"><div class=\"ub_table-of-contents-extra-container\"><div class=\"ub_table-of-contents-container ub_table-of-contents-1-column \"><ul><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/hypotenuse-leg-theorem#0-what-is-the-hypotenuse-leg-theorem>What Is the Hypotenuse Leg Theorem?<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/hypotenuse-leg-theorem#1-hypotenuse-leg-theorem-statement>Hypotenuse Leg Theorem Statement<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/hypotenuse-leg-theorem#2-hypotenuse-leg-theorem-proof>Hypotenuse Leg Theorem Proof<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/hypotenuse-leg-theorem#5-solved-examples-on-hl-theorem-hypotenuse-leg-theorem>Solved Examples on HL Theorem (Hypotenuse Leg Theorem)<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/hypotenuse-leg-theorem#6-practice-problems-on-hl-theorem-hypotenuse-leg-theorem>Practice Problems on HL Theorem (Hypotenuse Leg Theorem)<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/hypotenuse-leg-theorem#7-frequently-asked-questions-on-hl-theorem-hypotenuse-leg-theorem>Frequently Asked Questions on HL Theorem (Hypotenuse Leg Theorem)<\/a><\/li><\/ul><\/div><\/div><\/div>\n\n\n<h2 class=\"wp-block-heading\" id=\"0-what-is-the-hypotenuse-leg-theorem\">What Is the Hypotenuse Leg Theorem?<\/h2>\n\n\n\n<p><strong>Hypotenuse Leg Theorem (HL theorem) is used to prove the congruence of two right angled triangles. It is also known as the RHS (Right angle-Hypotenuse-Side) congruence rule.<\/strong>&nbsp;<\/p>\n\n\n\n<p>In a right triangle, there is one right angle ($90^{\\circ}$ angle) and two acute angles. The side opposite to the right angle is called hypotenuse. It is the longest side of the right triangle. The other two sides are called \u2018legs\u2019 of the right triangle.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"620\" height=\"455\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/09\/right-triangle-legs-and-hypotenuse.png\" alt=\"Right triangle (legs and hypotenuse)\" class=\"wp-image-33558\" title=\"Right triangle (legs and hypotenuse)\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/09\/right-triangle-legs-and-hypotenuse.png 620w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/09\/right-triangle-legs-and-hypotenuse-300x220.png 300w\" sizes=\"auto, (max-width: 620px) 100vw, 620px\" \/><\/figure>\n\n\n\n<p>According to the HL theorem, if the hypotenuse and one leg of a right triangle are congruent to the hypotenuse and one leg of the other right triangle, then the two triangles are congruent.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"648\" height=\"401\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/09\/hl-theorem-congruence-example.png\" alt=\"HL theorem congruence example\" class=\"wp-image-33559\" title=\"HL theorem congruence example\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/09\/hl-theorem-congruence-example.png 648w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/09\/hl-theorem-congruence-example-300x186.png 300w\" sizes=\"auto, (max-width: 648px) 100vw, 648px\" \/><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"1-hypotenuse-leg-theorem-statement\">Hypotenuse Leg Theorem Statement<\/h2>\n\n\n\n<p><strong>HL triangle congruence theorem: If the hypotenuse and one leg of one right triangle are congruent to the hypotenuse and leg of the other right triangle, then the two right triangles are said to be congruent.<\/strong>&nbsp;<\/p>\n\n\n\n<p>In other words, two right triangles are congruent if the corresponding lengths of the hypotenuse and one leg are equal. &nbsp;<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"2-hypotenuse-leg-theorem-proof\">Hypotenuse Leg Theorem Proof<\/h2>\n\n\n\n<p><strong>Given:<\/strong> In right triangles \u25b3ABC \u2245\u25b3XYZ, AB = XY and AC = XZ<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"620\" height=\"414\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/09\/right-triangles-abc-and-xyz.png\" alt=\"Right triangles ABC and XYZ\u00a0\" class=\"wp-image-33560\" title=\"Right triangles ABC and XYZ\u00a0\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/09\/right-triangles-abc-and-xyz.png 620w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/09\/right-triangles-abc-and-xyz-300x200.png 300w\" sizes=\"auto, (max-width: 620px) 100vw, 620px\" \/><\/figure>\n\n\n\n<p><strong>To prove: <\/strong>\u25b3ABC \u2245\u25b3XYZ<\/p>\n\n\n\n<p><strong>Proof:<\/strong><strong>&nbsp;<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"wj-table-class\"><thead><tr><th class=\"has-text-align-left\" data-align=\"left\"><strong>Statement<\/strong><\/th><th class=\"has-text-align-left\" data-align=\"left\"><strong>Reason<\/strong><\/th><\/tr><\/thead><tbody><tr><td class=\"has-text-align-left\" data-align=\"left\">AB<sup>2<\/sup> + BC<sup>2<\/sup> = AC<sup>2<\/sup><\/td><td class=\"has-text-align-left\" data-align=\"left\">Pythagoras theorem<\/td><\/tr><tr><td class=\"has-text-align-left\" data-align=\"left\">XY<sup>2<\/sup> + YZ<sup>2<\/sup> = XZ<sup>2<\/sup><\/td><td class=\"has-text-align-left\" data-align=\"left\">Pythagoras theorem<\/td><\/tr><tr><td class=\"has-text-align-left\" data-align=\"left\">XZ = AC&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; \u2026(1)<\/td><td class=\"has-text-align-left\" data-align=\"left\">Given&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<\/td><\/tr><tr><td class=\"has-text-align-left\" data-align=\"left\">AB<sup>2<\/sup> + BC<sup>2 <\/sup>= XY<sup>2<\/sup> + YZ<sup>2&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; <\/sup>\u2026(2)<\/td><td class=\"has-text-align-left\" data-align=\"left\">From (1)<\/td><\/tr><tr><td class=\"has-text-align-left\" data-align=\"left\">AB = XY&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; \u2026(3)<\/td><td class=\"has-text-align-left\" data-align=\"left\">Given<\/td><\/tr><tr><td class=\"has-text-align-left\" data-align=\"left\">XY<sup>2<\/sup> + BC<sup>2 <\/sup>= XY<sup>2<\/sup> + YZ<sup>2<\/sup><\/td><td class=\"has-text-align-left\" data-align=\"left\">From (2) and (3)<\/td><\/tr><tr><td class=\"has-text-align-left\" data-align=\"left\">BC<sup>2<\/sup> = YZ<sup>2<\/sup><\/td><td class=\"has-text-align-left\" data-align=\"left\">Canceling out XY<sup>2<\/sup><\/td><\/tr><tr><td class=\"has-text-align-left\" data-align=\"left\">BC = XY<\/td><td class=\"has-text-align-left\" data-align=\"left\">Taking positive square root<\/td><\/tr><tr><td class=\"has-text-align-left\" data-align=\"left\">\u25b3ABC \u2245\u25b3XYZ<\/td><td class=\"has-text-align-left\" data-align=\"left\">By SSS criterion<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"3-facts-about-hypotenuse-leg-theorem\">Facts about Hypotenuse Leg Theorem<\/h2>\n\n\n\n<div style=\"color:#0369a1;background-color:#e0f2fe\" class=\"wp-block-roelmagdaleno-callout-block has-text-color has-background is-layout-flex wp-container-roelmagdaleno-callout-block-is-layout-8cf370e7 wp-block-roelmagdaleno-callout-block-is-layout-flex\"><div>\n<ul class=\"wp-block-list\">\n<li>HL theorem (Hypotenuse Leg Theorem or Hypotenuse Leg Congruence Theorem) is also known as the RHS (Right angle-Hypotenuse-Side) congruence rule.<\/li>\n\n\n\n<li>The LA Theorem (Leg- Acute angle Theorem): If the leg and an acute angle of one right triangle are both congruent to the corresponding leg and acute angle of another right triangle, the two triangles are congruent.<\/li>\n<\/ul>\n<\/div><\/div>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"4-conclusion\">Conclusion<\/h2>\n\n\n\n<p>In this article, we learned about HL theorem or HL congruence criterion to prove congruence of two right triangles. We learned the statement and also proved the theorem. Let\u2019s use the HL theorem to solve a few examples and practice problems.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"5-solved-examples-on-hl-theorem-hypotenuse-leg-theorem\">Solved Examples on HL Theorem (Hypotenuse Leg Theorem)<\/h2>\n\n\n\n<p><strong>1. Is \u25b3PQR <\/strong>\u2245<strong> \u25b3NLM?&nbsp;<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"561\" height=\"284\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/09\/right-triangles-pqr-and-lmn.png\" alt=\"Right triangles PQR and LMN\" class=\"wp-image-33561\" title=\"Right triangles PQR and LMN\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/09\/right-triangles-pqr-and-lmn.png 561w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/09\/right-triangles-pqr-and-lmn-300x152.png 300w\" sizes=\"auto, (max-width: 561px) 100vw, 561px\" \/><\/figure>\n\n\n\n<p><strong>Solution:&nbsp;<\/strong><\/p>\n\n\n\n<p>To prove the congruence of right triangles, we will use the HL theorem.<\/p>\n\n\n\n<p>In right triangles \u25b3PQR and \u25b3LMN, we have<\/p>\n\n\n\n<p>\u2220Q = \u2220L = 90\u00b0<\/p>\n\n\n\n<p>PR = MN = 13 units<\/p>\n\n\n\n<p>QR = LM = 5 units<\/p>\n\n\n\n<p>By HL Theorem, one leg and hypotenuse of \u25b3PQR are congruent to one leg and hypotenuse of \u25b3LMN.<\/p>\n\n\n\n<p>Thus, \u25b3PQR \u2245 \u25b3NLM<\/p>\n\n\n\n<p><strong>2. In the \u25b3PQR, if PS is the perpendicular bisector of the side QR, then prove that \u2220QPS = \u2220RPS.<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"328\" height=\"268\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/09\/two-right-triangles-with-one-common-leg.png\" alt=\"Two right triangles with one common leg\" class=\"wp-image-33562\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/09\/two-right-triangles-with-one-common-leg.png 328w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/09\/two-right-triangles-with-one-common-leg-300x245.png 300w\" sizes=\"auto, (max-width: 328px) 100vw, 328px\" \/><\/figure>\n\n\n\n<p><strong>Solution:&nbsp;<\/strong><\/p>\n\n\n\n<p>In \u25b3PQR, PS is the perpendicular bisector of QR.<\/p>\n\n\n\n<p>$\\Rightarrow$ QS = SR \u2026(1)<\/p>\n\n\n\n<p>In \u25b3PSQ and \u25b3PSR<\/p>\n\n\n\n<p>\u2220PSQ = \u2220PSR = 90\u00b0<\/p>\n\n\n\n<p>PQ = PR = 3 units<\/p>\n\n\n\n<p>QS = SR \u2026from (1)<\/p>\n\n\n\n<p>Thus, \u25b3PSQ \u2245 \u25b3PSR \u2026by HL Theorem<\/p>\n\n\n\n<p>$\\Rightarrow$ \u2220QPS = \u2220RPS \u2026CPCTC (Corresponding parts of congruent triangles are congruent.)<\/p>\n\n\n\n<p><strong>3. In the following figure, find the value of JK and MN.<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"540\" height=\"284\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/09\/right-triangles-ijk-and-mno.png\" alt=\"Right triangles IJK and MNO\" class=\"wp-image-33563\" title=\"Right triangles IJK and MNO\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/09\/right-triangles-ijk-and-mno.png 540w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/09\/right-triangles-ijk-and-mno-300x158.png 300w\" sizes=\"auto, (max-width: 540px) 100vw, 540px\" \/><\/figure>\n\n\n\n<p><strong>Solution:<\/strong>&nbsp;<\/p>\n\n\n\n<p>In \u25b3IJK and \u25b3ONM,<\/p>\n\n\n\n<p>\u2220IJK = \u2220ONM = 90\u00b0<\/p>\n\n\n\n<p>IK = MO = 5 units<\/p>\n\n\n\n<p>IJ = ON = 4 units<\/p>\n\n\n\n<p>By HL Theorem,<\/p>\n\n\n\n<p>\u25b3IJK \u2245 \u25b3ONM<\/p>\n\n\n\n<p>$\\Rightarrow$KJ = MN                    (CPCTC)<\/p>\n\n\n\n<p>KJ = MN = $\\sqrt{5^{2}\\;-\\;4^{2}}$           (By Pythagoras theorem)<\/p>\n\n\n\n<p>KJ = MN = $\\sqrt{25 \\;-\\; 16}<\/p>\n\n\n\n<p>KJ = MN = $\\sqrt{9}$<\/p>\n\n\n\n<p>KJ = MN = 3 units<\/p>\n\n\n\n<p><strong>4. In the following figure, prove that \u25b3DAB <\/strong>\u2245<strong> \u25b3CBA.<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"546\" height=\"263\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/09\/hl-theorem-example.png\" alt=\"HL theorem example\" class=\"wp-image-33564\" title=\"HL theorem example\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/09\/hl-theorem-example.png 546w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/09\/hl-theorem-example-300x145.png 300w\" sizes=\"auto, (max-width: 546px) 100vw, 546px\" \/><\/figure>\n\n\n\n<p><strong>Solution:<\/strong>&nbsp;<\/p>\n\n\n\n<p>In \u25b3DAB and \u25b3CBA, we have<\/p>\n\n\n\n<p>\u2220ADB = \u2220ACB = 90\u00b0<\/p>\n\n\n\n<p>AB = AB              (common side)<\/p>\n\n\n\n<p>AD = BC = 4 units<\/p>\n\n\n\n<p>By HL Theorem,<\/p>\n\n\n\n<p>\u25b3DAB \u2245 \u25b3CBA<\/p>\n\n\n\n<p><strong>5. In the following figure, prove that \u25b3ABC <\/strong>\u2245<strong> \u25b3EDC.<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"513\" height=\"308\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/09\/hl-theorem-problem.png\" alt=\"HL theorem problem\" class=\"wp-image-33565\" title=\"HL theorem problem\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/09\/hl-theorem-problem.png 513w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/09\/hl-theorem-problem-300x180.png 300w\" sizes=\"auto, (max-width: 513px) 100vw, 513px\" \/><\/figure>\n\n\n\n<p><strong>Solution:<\/strong>&nbsp;<\/p>\n\n\n\n<p>CD = BD \u2013 BC = 14 \u2013 8 = 6 units<\/p>\n\n\n\n<p>In \u25b3ABC, we have<\/p>\n\n\n\n<p>$AC^{2} = AB^{2} + BC^{2}$<\/p>\n\n\n\n<p>$AC^{2} = 6^{2} + 8^{2} = 36 + 64 = 100$<\/p>\n\n\n\n<p>AC = 10 inches<\/p>\n\n\n\n<p>In \u25b3ABC and \u25b3EDC,<\/p>\n\n\n\n<p>\u2220ABC = \u2220EDC = 90\u00b0<\/p>\n\n\n\n<p>AC = CE = 10 units \u2026hypotenuse<\/p>\n\n\n\n<p>AB = CD = 6 units \u2026leg<\/p>\n\n\n\n<p>By HL Theorem,<\/p>\n\n\n\n<p>\u25b3ABC \u2245 \u25b3EDC<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"6-practice-problems-on-hl-theorem-hypotenuse-leg-theorem\">Practice Problems on HL Theorem (Hypotenuse Leg Theorem)<\/h2>\n\n\n\n<div class=\"spq_wrapper\"><h2 style=\"display:none;\">Hypotenuse Leg Theorem \u2013 Definition, Proof, Examples, FAQs<\/h2><p style=\"display:none;\">Attend this quiz & Test your knowledge.<\/p><div class=\"spq_question_wrapper\" data-answer=\"2\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">1<\/span><h3 class=\"sqp_question_text\">Which information is missing to prove \u25b3ABC \u2245 \u25b3DEF by HL theorem?<\/h3><\/span><div class=\"spq_question_image\"><img decoding=\"async\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/09\/hl-theorem-practice-problem-i.png\"><\/div><div class=\"spq_answer_block\" data-value=\"0\">\u2220BAC = \u2220EDF<\/div><div class=\"spq_answer_block\" data-value=\"1\">AB = ED<\/div><div class=\"spq_answer_block\" data-value=\"2\">AC = DF<\/div><div class=\"spq_answer_block\" data-value=\"3\">\u2220ACB = \u2220EFD<\/div><div class=\"sqp_question_hint\"><div class=\"sqp_question_hint__header\"><span class=\"spq_correct\">Correct<\/span><span class=\"spq_incorrect\">Incorrect<\/span><\/div><div class=\"sqp_question_hint__content\"><span>Correct answer is: AC = DF<br\/>In the given right triangle, BC = EF are two congruent legs.<br>\r\nIf the two hypotenuses are also equal, we can apply the HL theorem of congruence.<br>\r\nMissing information is AC = DF.<\/span><\/div><\/div><\/div><div class=\"spq_question_wrapper\" data-answer=\"1\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">2<\/span><h3 class=\"sqp_question_text\">Which congruence theorem is used for proving \u25b3XYZ \u2245 \u25b3UZY?<\/h3><\/span><div class=\"spq_question_image\"><img decoding=\"async\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/09\/hl-theorem-practice-problem-ii.png\"><\/div><div class=\"spq_answer_block\" data-value=\"0\">SAS<\/div><div class=\"spq_answer_block\" data-value=\"1\">HL<\/div><div class=\"spq_answer_block\" data-value=\"2\">AAS<\/div><div class=\"spq_answer_block\" data-value=\"3\">SSS<\/div><div class=\"sqp_question_hint\"><div class=\"sqp_question_hint__header\"><span class=\"spq_correct\">Correct<\/span><span class=\"spq_incorrect\">Incorrect<\/span><\/div><div class=\"sqp_question_hint__content\"><span>Correct answer is: HL<br\/>In \u25b3XYZ and \u25b3UZY<br>\r\n\u2220XYZ = \u2220UZY = 90\u00b0<br>\r\nXZ = YU (given)<br>\r\nYZ = ZY (commons side)<br>\r\nBy HL theorem,<br>\r\n\u25b3XYZ \u2245 \u25b3UZY<\/span><\/div><\/div><\/div><div class=\"spq_question_wrapper\" data-answer=\"1\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">3<\/span><h3 class=\"sqp_question_text\">Find the values of a and b that make \u25b3WXY \u2245 \u25b3GFH<\/h3><\/span><div class=\"spq_question_image\"><img decoding=\"async\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/09\/hl-theorem-practice-problem-iii.png\"><\/div><div class=\"spq_answer_block\" data-value=\"0\">a = 9, b = 5<\/div><div class=\"spq_answer_block\" data-value=\"1\">a = 5, b = 9<\/div><div class=\"spq_answer_block\" data-value=\"2\">a = 4, b = 9<\/div><div class=\"spq_answer_block\" data-value=\"3\">a = 5, b = 4<\/div><div class=\"sqp_question_hint\"><div class=\"sqp_question_hint__header\"><span class=\"spq_correct\">Correct<\/span><span class=\"spq_incorrect\">Incorrect<\/span><\/div><div class=\"sqp_question_hint__content\"><span>Correct answer is: a = 5, b = 9<br\/>For \u25b3WXY \u2245 \u25b3GFH, one leg should be congruent.<br>\r\nWX = GF<br>\r\n2b + 20 = b + 25<br>\r\nb = 5<br>\r\nAlso, we need to have two congruent hypotenuses.<br>\r\nWY = GH<br>\r\n11a - 27 = 8a<br>\r\n3a = 27<br>\r\na = 9<\/span><\/div><\/div><\/div><\/div>  <script type=\"application\/ld+json\">{\n        \"@context\": \"https:\/\/schema.org\/\", \n        \"@type\": \"Quiz\", \n        \"typicalAgeRange\": \"3-11\",\n        \"educationalLevel\":  \"beginner\",\n        \"assesses\" : \"Attend this quiz & Test your knowledge.\",\n        \"educationalAlignment\": [\n              {\n                \"@type\": \"AlignmentObject\",\n                \"alignmentType\": \"educationalSubject\",\n                \"targetName\": \"Math\"\n              }] ,\n        \"name\": \"Hypotenuse Leg Theorem \u2013 Definition, Proof, Examples, FAQs\",        \n        \"about\": {\n                \"@type\": \"Thing\",\n                \"name\": \"Hypotenuse Leg Theorem\"\n        },  \n        \"hasPart\": [{\n                    \"@type\": \"Question\",   \n                    \"eduQuestionType\": \"Multiple choice\",\n                    \"learningResourceType\": \"Practice problem\",\n                    \"name\": \"Which information is missing to prove \u25b3ABC \u2245 \u25b3DEF by HL theorem?\",\n                    \"text\": \"Which information is missing to prove \u25b3ABC \u2245 \u25b3DEF by HL theorem? <img src=\\\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/09\/hl-theorem-practice-problem-i.png\\\"\/>\",\n                    \"comment\": {\n                      \"@type\": \"Comment\",\n                      \"text\": \"In the given right triangle, BC = EF are two congruent legs.<br>\r\nIf the two hypotenuses are also equal, we can apply the HL theorem of congruence.<br>\r\nMissing information is AC = DF.\"\n                    },\n                    \"encodingFormat\": \"text\/html\",\n                    \"suggestedAnswer\": [ {\n                                \"@type\": \"Answer\",\n                                \"position\": 0,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"\u2220BAC = \u2220EDF\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"In the given right triangle, BC = EF are two congruent legs.<br>\r\nIf the two hypotenuses are also equal, we can apply the HL theorem of congruence.<br>\r\nMissing information is AC = DF.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 1,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"AB = ED\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"In the given right triangle, BC = EF are two congruent legs.<br>\r\nIf the two hypotenuses are also equal, we can apply the HL theorem of congruence.<br>\r\nMissing information is AC = DF.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 3,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"\u2220ACB = \u2220EFD\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"In the given right triangle, BC = EF are two congruent legs.<br>\r\nIf the two hypotenuses are also equal, we can apply the HL theorem of congruence.<br>\r\nMissing information is AC = DF.\"\n                                    }\n                                }],\n                    \"acceptedAnswer\": {\n                      \"@type\": \"Answer\",\n                      \"position\": 2,\n                      \"encodingFormat\": \"text\/html\",\n                      \"text\": \"AC = DF\",\n                      \"comment\": {\n                          \"@type\": \"Comment\",\n                          \"text\": \"In the given right triangle, BC = EF are two congruent legs.<br>\r\nIf the two hypotenuses are also equal, we can apply the HL theorem of congruence.<br>\r\nMissing information is AC = DF.\"\n                        },\n                      \"answerExplanation\": {\n                        \"@type\": \"Comment\",\n                        \"text\": \"In the given right triangle, BC = EF are two congruent legs.<br>\r\nIf the two hypotenuses are also equal, we can apply the HL theorem of congruence.<br>\r\nMissing information is AC = DF.\"\n                      }\n                    } \n\n                    },{\n                    \"@type\": \"Question\",   \n                    \"eduQuestionType\": \"Multiple choice\",\n                    \"learningResourceType\": \"Practice problem\",\n                    \"name\": \"Which congruence theorem is used for proving \u25b3XYZ \u2245 \u25b3UZY?\",\n                    \"text\": \"Which congruence theorem is used for proving \u25b3XYZ \u2245 \u25b3UZY? <img src=\\\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/09\/hl-theorem-practice-problem-ii.png\\\"\/>\",\n                    \"comment\": {\n                      \"@type\": \"Comment\",\n                      \"text\": \"In \u25b3XYZ and \u25b3UZY<br>\r\n\u2220XYZ = \u2220UZY = 90\u00b0<br>\r\nXZ = YU (given)<br>\r\nYZ = ZY (commons side)<br>\r\nBy HL theorem,<br>\r\n\u25b3XYZ \u2245 \u25b3UZY\"\n                    },\n                    \"encodingFormat\": \"text\/html\",\n                    \"suggestedAnswer\": [ {\n                                \"@type\": \"Answer\",\n                                \"position\": 0,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"SAS\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"In \u25b3XYZ and \u25b3UZY<br>\r\n\u2220XYZ = \u2220UZY = 90\u00b0<br>\r\nXZ = YU (given)<br>\r\nYZ = ZY (commons side)<br>\r\nBy HL theorem,<br>\r\n\u25b3XYZ \u2245 \u25b3UZY\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 2,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"AAS\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"In \u25b3XYZ and \u25b3UZY<br>\r\n\u2220XYZ = \u2220UZY = 90\u00b0<br>\r\nXZ = YU (given)<br>\r\nYZ = ZY (commons side)<br>\r\nBy HL theorem,<br>\r\n\u25b3XYZ \u2245 \u25b3UZY\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 3,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"SSS\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"In \u25b3XYZ and \u25b3UZY<br>\r\n\u2220XYZ = \u2220UZY = 90\u00b0<br>\r\nXZ = YU (given)<br>\r\nYZ = ZY (commons side)<br>\r\nBy HL theorem,<br>\r\n\u25b3XYZ \u2245 \u25b3UZY\"\n                                    }\n                                }],\n                    \"acceptedAnswer\": {\n                      \"@type\": \"Answer\",\n                      \"position\": 1,\n                      \"encodingFormat\": \"text\/html\",\n                      \"text\": \"HL\",\n                      \"comment\": {\n                          \"@type\": \"Comment\",\n                          \"text\": \"In \u25b3XYZ and \u25b3UZY<br>\r\n\u2220XYZ = \u2220UZY = 90\u00b0<br>\r\nXZ = YU (given)<br>\r\nYZ = ZY (commons side)<br>\r\nBy HL theorem,<br>\r\n\u25b3XYZ \u2245 \u25b3UZY\"\n                        },\n                      \"answerExplanation\": {\n                        \"@type\": \"Comment\",\n                        \"text\": \"In \u25b3XYZ and \u25b3UZY<br>\r\n\u2220XYZ = \u2220UZY = 90\u00b0<br>\r\nXZ = YU (given)<br>\r\nYZ = ZY (commons side)<br>\r\nBy HL theorem,<br>\r\n\u25b3XYZ \u2245 \u25b3UZY\"\n                      }\n                    } \n\n                    },{\n                    \"@type\": \"Question\",   \n                    \"eduQuestionType\": \"Multiple choice\",\n                    \"learningResourceType\": \"Practice problem\",\n                    \"name\": \"Find the values of a and b that make \u25b3WXY \u2245 \u25b3GFH\",\n                    \"text\": \"Find the values of a and b that make \u25b3WXY \u2245 \u25b3GFH <img src=\\\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/09\/hl-theorem-practice-problem-iii.png\\\"\/>\",\n                    \"comment\": {\n                      \"@type\": \"Comment\",\n                      \"text\": \"For \u25b3WXY \u2245 \u25b3GFH, one leg should be congruent.<br>\r\nWX = GF<br>\r\n2b + 20 = b + 25<br>\r\nb = 5<br>\r\nAlso, we need to have two congruent hypotenuses.<br>\r\nWY = GH<br>\r\n11a - 27 = 8a<br>\r\n3a = 27<br>\r\na = 9\"\n                    },\n                    \"encodingFormat\": \"text\/html\",\n                    \"suggestedAnswer\": [ {\n                                \"@type\": \"Answer\",\n                                \"position\": 0,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"a = 9, b = 5\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"For \u25b3WXY \u2245 \u25b3GFH, one leg should be congruent.<br>\r\nWX = GF<br>\r\n2b + 20 = b + 25<br>\r\nb = 5<br>\r\nAlso, we need to have two congruent hypotenuses.<br>\r\nWY = GH<br>\r\n11a - 27 = 8a<br>\r\n3a = 27<br>\r\na = 9\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 2,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"a = 4, b = 9\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"For \u25b3WXY \u2245 \u25b3GFH, one leg should be congruent.<br>\r\nWX = GF<br>\r\n2b + 20 = b + 25<br>\r\nb = 5<br>\r\nAlso, we need to have two congruent hypotenuses.<br>\r\nWY = GH<br>\r\n11a - 27 = 8a<br>\r\n3a = 27<br>\r\na = 9\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 3,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"a = 5, b = 4\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"For \u25b3WXY \u2245 \u25b3GFH, one leg should be congruent.<br>\r\nWX = GF<br>\r\n2b + 20 = b + 25<br>\r\nb = 5<br>\r\nAlso, we need to have two congruent hypotenuses.<br>\r\nWY = GH<br>\r\n11a - 27 = 8a<br>\r\n3a = 27<br>\r\na = 9\"\n                                    }\n                                }],\n                    \"acceptedAnswer\": {\n                      \"@type\": \"Answer\",\n                      \"position\": 1,\n                      \"encodingFormat\": \"text\/html\",\n                      \"text\": \"a = 5, b = 9\",\n                      \"comment\": {\n                          \"@type\": \"Comment\",\n                          \"text\": \"For \u25b3WXY \u2245 \u25b3GFH, one leg should be congruent.<br>\r\nWX = GF<br>\r\n2b + 20 = b + 25<br>\r\nb = 5<br>\r\nAlso, we need to have two congruent hypotenuses.<br>\r\nWY = GH<br>\r\n11a - 27 = 8a<br>\r\n3a = 27<br>\r\na = 9\"\n                        },\n                      \"answerExplanation\": {\n                        \"@type\": \"Comment\",\n                        \"text\": \"For \u25b3WXY \u2245 \u25b3GFH, one leg should be congruent.<br>\r\nWX = GF<br>\r\n2b + 20 = b + 25<br>\r\nb = 5<br>\r\nAlso, we need to have two congruent hypotenuses.<br>\r\nWY = GH<br>\r\n11a - 27 = 8a<br>\r\n3a = 27<br>\r\na = 9\"\n                      }\n                    } \n\n                    }]}<\/script>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"7-frequently-asked-questions-on-hl-theorem-hypotenuse-leg-theorem\">Frequently Asked Questions on HL Theorem (Hypotenuse Leg Theorem)<\/h2>\n\n\n<div class=\"wp-block-ub-content-toggle\" id=\"ub-content-toggle-6debe335-bfc6-480b-9fb7-5902fc043b40\" data-mobilecollapse=\"true\" data-desktopcollapse=\"true\">\n<div class=\"wp-block-ub-content-toggle-accordion\">\n                <div class=\"wp-block-ub-content-toggle-accordion-title-wrap\" aria-expanded=\"false\" aria-controls=\"ub-content-toggle-panel-0-6debe335-bfc6-480b-9fb7-5902fc043b40\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-6debe335-bfc6-480b-9fb7-5902fc043b40\"><strong>What is the Pythagorean Theorem?<\/strong><\/p><div class=\"wp-block-ub-content-toggle-accordion-toggle-wrap right\"><span class=\"wp-block-ub-content-toggle-accordion-state-indicator wp-block-ub-chevron-down\"><\/span>\n                    <\/div><\/div><div role=\"region\" class=\"wp-block-ub-content-toggle-accordion-content-wrap ub-hide\" id=\"ub-content-toggle-panel-0-6debe335-bfc6-480b-9fb7-5902fc043b40\">\n\n<p>According to the Pythagorean theorem, in a <a href=\"https:\/\/www.splashlearn.com\/math-vocabulary\/geometry\/right-triangle\">right-angled triangle<\/a>, the square of the hypotenuse is equal to the sum of the squares of the other two sides.<\/p>\n\n\n\n<p>Hypotenuse\u00b2 = Base\u00b2 + Perpendicular\u00b2.<\/p>\n\n<\/div><\/div>\n\n<div class=\"wp-block-ub-content-toggle-accordion\">\n                <div class=\"wp-block-ub-content-toggle-accordion-title-wrap\" aria-expanded=\"false\" aria-controls=\"ub-content-toggle-panel-1-6debe335-bfc6-480b-9fb7-5902fc043b40\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-6debe335-bfc6-480b-9fb7-5902fc043b40\"><strong>What is the leg leg theorem (leg leg congruence)?<\/strong><\/p><div class=\"wp-block-ub-content-toggle-accordion-toggle-wrap right\"><span class=\"wp-block-ub-content-toggle-accordion-state-indicator wp-block-ub-chevron-down\"><\/span>\n                    <\/div><\/div><div role=\"region\" class=\"wp-block-ub-content-toggle-accordion-content-wrap ub-hide\" id=\"ub-content-toggle-panel-1-6debe335-bfc6-480b-9fb7-5902fc043b40\">\n\n<p>According to the leg-leg theorem, if the legs of one right triangle are congruent to the legs of another right triangle, then the triangles are said to be congruent.<\/p>\n\n<\/div><\/div>\n\n<div class=\"wp-block-ub-content-toggle-accordion\">\n                <div class=\"wp-block-ub-content-toggle-accordion-title-wrap\" aria-expanded=\"false\" aria-controls=\"ub-content-toggle-panel-2-6debe335-bfc6-480b-9fb7-5902fc043b40\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-6debe335-bfc6-480b-9fb7-5902fc043b40\"><strong>What is the HL similarity theorem?<\/strong><\/p><div class=\"wp-block-ub-content-toggle-accordion-toggle-wrap right\"><span class=\"wp-block-ub-content-toggle-accordion-state-indicator wp-block-ub-chevron-down\"><\/span>\n                    <\/div><\/div><div role=\"region\" class=\"wp-block-ub-content-toggle-accordion-content-wrap ub-hide\" id=\"ub-content-toggle-panel-2-6debe335-bfc6-480b-9fb7-5902fc043b40\">\n\n<p>If the hypotenuse and one leg (H and L) of a right triangle is proportional to the hypotenuse and leg of the other right triangle, the triangles are said to be similar.<\/p>\n\n<\/div><\/div>\n<\/div>","protected":false},"excerpt":{"rendered":"<p>What Is the Hypotenuse Leg Theorem? Hypotenuse Leg Theorem (HL theorem) is used to prove the congruence of two right angled triangles. It is also known as the RHS (Right angle-Hypotenuse-Side) congruence rule.&nbsp; In a right triangle, there is one right angle ($90^{\\circ}$ angle) and two acute angles. The side opposite to the right angle &#8230; <a title=\"Hypotenuse Leg Theorem \u2013 Definition, Proof, Examples, FAQs\" class=\"read-more\" href=\"https:\/\/www.splashlearn.com\/math-vocabulary\/hypotenuse-leg-theorem\" aria-label=\"More on Hypotenuse Leg Theorem \u2013 Definition, Proof, Examples, FAQs\">Read more<\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"ub_ctt_via":"","footnotes":""},"categories":[1],"tags":[],"class_list":["post-33556","post","type-post","status-publish","format-standard","hentry","category-all"],"featured_image_src":null,"author_info":{"display_name":"Mithun Jhawar","author_link":"https:\/\/www.splashlearn.com\/math-vocabulary\/author\/mithun-jhawarsplashlearn-com\/"},"acf":[],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/posts\/33556","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/comments?post=33556"}],"version-history":[{"count":9,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/posts\/33556\/revisions"}],"predecessor-version":[{"id":33629,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/posts\/33556\/revisions\/33629"}],"wp:attachment":[{"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/media?parent=33556"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/categories?post=33556"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/tags?post=33556"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}