{"id":110,"date":"2022-04-13T08:22:04","date_gmt":"2022-04-13T08:22:04","guid":{"rendered":"http:\/\/9thclass.deltapublications.in\/?page_id=110"},"modified":"2025-10-31T08:38:18","modified_gmt":"2025-10-31T08:38:18","slug":"g-5-angle-side-relationships-in-triangles","status":"publish","type":"page","link":"https:\/\/9thclass.deltapublications.in\/index.php\/g-5-angle-side-relationships-in-triangles\/","title":{"rendered":"G.5 Angle-side relationships in triangles"},"content":{"rendered":"\n<h2 class=\"wp-block-heading has-text-align-center has-text-color\" style=\"color:#00056d;text-transform:uppercase\"><strong>Angle-side relationships in triangles<\/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\/DAG3VEMCjnM\/ML2wYhCRhOqE5tIZieuscA\/watch?embed\" allowfullscreen=\"allowfullscreen\" allow=\"fullscreen\">\n  <\/iframe>\n<\/div>\n<a href=\"https:&#x2F;&#x2F;www.canva.com&#x2F;design&#x2F;DAG3VEMCjnM&#x2F;ML2wYhCRhOqE5tIZieuscA&#x2F;watch?utm_content=DAG3VEMCjnM&amp;utm_campaign=designshare&amp;utm_medium=embeds&amp;utm_source=link\" target=\"_blank\" rel=\"noopener\">nr<\/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-b3c298a03bc227779441af17233acfed\" style=\"color:#000060\">\ud83d\udd39 <strong>Basic Idea<\/strong><\/h3>\n\n\n\n<p>\ud83d\udc49 In a triangle, <strong>angles and sides are related<\/strong> \u2014<br>\ud83d\udccf The <strong>longer side<\/strong> is <strong>opposite the larger angle<\/strong>, and<br>\ud83d\udcd0 The <strong>shorter side<\/strong> is <strong>opposite the smaller angle<\/strong>.<\/p>\n\n\n\n<p class=\"has-text-color has-link-color has-large-font-size wp-elements-b3477c2bb0f34b0abdcd1323d69f4a15\" style=\"color:#000060\">\ud83d\udca1 <strong>Rule:<\/strong><\/p>\n\n\n\n<p>\u27a1\ufe0f <strong>Bigger Angle \u2192 Longer Opposite Side<\/strong><br>\u27a1\ufe0f <strong>Smaller Angle \u2192 Shorter Opposite Side<\/strong><\/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-4d3842108d315a33c702017fc286d950\" style=\"color:#000060\">\ud83d\udd39 <strong>Ordering Sides and Angles<\/strong><\/h3>\n\n\n\n<p>\ud83e\udde9 If you know the <strong>angle measures<\/strong>, you can order the <strong>sides<\/strong> easily!<br>Example:<br>If \u2220A &gt; \u2220B &gt; \u2220C<br>\ud83d\udc49 Then side <strong>BC &gt; AC &gt; AB<\/strong><\/p>\n\n\n\n<p>And vice versa:<br>If side <strong>BC &gt; AC &gt; AB<\/strong><br>\ud83d\udc49 Then \u2220A &gt; \u2220B &gt; \u2220C<\/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-4d3dee06bc5fb9ee88fa099aaf282d05\" style=\"color:#000060\">\ud83d\udd39 <strong>Triangle Inequality Theorem<\/strong> \ud83d\udcd0\u2728<\/h3>\n\n\n\n<p>\ud83e\udde0 The <strong>sum of the lengths of any two sides<\/strong> of a triangle is <strong>greater than<\/strong> the length of the <strong>third side<\/strong>.<\/p>\n\n\n\n<p>\u2705 a + b &gt; c<br>\u2705 a + c &gt; b<br>\u2705 b + c &gt; a<\/p>\n\n\n\n<p>\ud83d\udeab If this rule is not true, a triangle <strong>cannot be formed!<\/strong><\/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-7cffe5d999e6912d20e169b3478303d0\" style=\"color:#000060\">\ud83d\udd39 <strong>Converse of the Triangle Inequality Theorem<\/strong> \ud83d\udd04<\/h3>\n\n\n\n<p>If <strong>one side<\/strong> of a triangle is <strong>longer<\/strong>, then the <strong>angle opposite<\/strong> that side is <strong>larger<\/strong>.<br>Example:<br>If side AB &gt; side AC<br>\ud83d\udc49 Then \u2220C &gt; \u2220B<\/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-060412c77a890b082fcadae8ab27647c\" style=\"color:#000060\">\ud83d\udd39 <strong>Equality Case (Isosceles Triangle)<\/strong> \ud83d\udc96<\/h3>\n\n\n\n<p>In an <strong>isosceles triangle<\/strong>,<br>\u2728 Two sides are equal,<br>\u2728 The angles <strong>opposite those sides<\/strong> are also <strong>equal<\/strong>.<\/p>\n\n\n\n<p>\ud83d\udccf Example: If AB = AC, then \u2220B = \u2220C<\/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-c3dccea82699cfc2f5d0d1ea2912c50e\" style=\"color:#000060\">\ud83d\udd39 <strong>Practical Use \ud83e\uddee<\/strong><\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Used in <strong>geometry proofs<\/strong> and <strong>construction problems<\/strong>.<\/li>\n\n\n\n<li>Helps in <strong>ranking sides or angles<\/strong> by size.<\/li>\n\n\n\n<li>Important in <strong>real-life applications<\/strong> like design, construction, and navigation.<\/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-20198f6747f43fdcd65e655842834eb5\" style=\"color:#000060\">\ud83e\udde0 <strong>Remember These Key Patterns<\/strong> \ud83d\udcab<\/h3>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-fixed-layout\"><thead><tr><th>Relation<\/th><th>Rule<\/th><\/tr><\/thead><tbody><tr><td>Larger Angle \u2194 Longer Side<\/td><td>\u2705<\/td><\/tr><tr><td>Smaller Angle \u2194 Shorter Side<\/td><td>\u2705<\/td><\/tr><tr><td>Equal Sides \u2194 Equal Angles<\/td><td>\u2705<\/td><\/tr><tr><td>a + b &gt; c<\/td><td>Triangle can exist \u2705<\/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-478fc837195793bbae79ded23fea85bb\" style=\"color:#000060\">\ud83c\udf1f <strong>Quick Example<\/strong><\/h3>\n\n\n\n<p>In \u25b3ABC,<br>\u2220A = 40\u00b0, \u2220B = 70\u00b0, \u2220C = 70\u00b0<br>\ud83d\udc49 Side <strong>a (BC)<\/strong> is shortest \ud83e\udeb6<br>\ud83d\udc49 Sides <strong>b (AC)<\/strong> and <strong>c (AB)<\/strong> are equal and longer \ud83c\udfd7\ufe0f<\/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-13845c84330c4266bdb8df795672e42d\" style=\"color:#000060\">\ud83e\udde9 <strong>In Summary<\/strong><\/h3>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p><strong>Angles and sides in a triangle always &#8220;balance&#8221; each other!<\/strong><br>The <strong>bigger the angle, the longer the opposite side<\/strong> \u2014<br>That\u2019s the beauty of triangles! \ud83d\udc96\ud83d\udd3a<\/p>\n<\/blockquote>\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-link-color has-large-font-size wp-elements-183a3a697c9edf71d840220f7701ff95\" style=\"background-color:#daebfe\"><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>Find the largest angle of \u25b3VWX.<\/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-1-1.png\" alt=\"\" class=\"wp-image-8490\" srcset=\"https:\/\/9thclass.deltapublications.in\/wp-content\/uploads\/2023\/06\/Untitled-design-1-1.png 500w, https:\/\/9thclass.deltapublications.in\/wp-content\/uploads\/2023\/06\/Untitled-design-1-1-300x300.png 300w, https:\/\/9thclass.deltapublications.in\/wp-content\/uploads\/2023\/06\/Untitled-design-1-1-150x150.png 150w\" sizes=\"auto, (max-width: 500px) 100vw, 500px\" \/><\/figure><\/div>\n\n\n<p>\u2220 _____<\/p>\n<\/div><\/div>\n\n\n\n<p>The side lengths are VX=10 metres, VW=11 metres, and WX=13 metres. Since 10 &lt; 11 &lt; 13, VX &lt; VW &lt; WX.<\/p>\n\n\n\n<p>Their&nbsp;opposite angles are in the same order, from smallest to&nbsp;largest:<\/p>\n\n\n\n<p>\u2220W &lt; \u2220X &lt; \u2220V<\/p>\n\n\n\n<p>So, the largest angle is \u2220V.<\/p>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-group has-primary-color has-text-color has-background has-link-color has-large-font-size wp-elements-425db0898e9bca02048482b0e27d67c2\" style=\"background-color:#f7d2f9\"><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>Find the largest angle of \u25b3GHI.<\/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-2-2.png\" alt=\"\" class=\"wp-image-8493\" srcset=\"https:\/\/9thclass.deltapublications.in\/wp-content\/uploads\/2023\/06\/Untitled-design-2-2.png 500w, https:\/\/9thclass.deltapublications.in\/wp-content\/uploads\/2023\/06\/Untitled-design-2-2-300x300.png 300w, https:\/\/9thclass.deltapublications.in\/wp-content\/uploads\/2023\/06\/Untitled-design-2-2-150x150.png 150w\" sizes=\"auto, (max-width: 500px) 100vw, 500px\" \/><\/figure><\/div>\n\n\n<p>\u2220 ______<\/p>\n<\/div><\/div>\n\n\n\n<p>The side lengths are HI=35 metres, GH=43 metres, and GI=48 metres. Since 35 &lt; 43 &lt; 48, HI &lt; GH &lt; GI.<\/p>\n\n\n\n<p>Their&nbsp;opposite angles are in the same order, from smallest to&nbsp;largest:<\/p>\n\n\n\n<p>\u2220G &lt; \u2220I &lt; \u2220H<\/p>\n\n\n\n<p>So, the largest angle is \u2220H.<\/p>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-group has-primary-color has-text-color has-background has-link-color has-large-font-size wp-elements-039b9c861d98942941ded296b15828e5\" style=\"background-color:#bbf9c4\"><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>Find the largest angle of \u25b3UVW.<\/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-3-2.png\" alt=\"\" class=\"wp-image-8494\" style=\"aspect-ratio:1;width:379px;height:auto\" srcset=\"https:\/\/9thclass.deltapublications.in\/wp-content\/uploads\/2023\/06\/Untitled-design-3-2.png 500w, https:\/\/9thclass.deltapublications.in\/wp-content\/uploads\/2023\/06\/Untitled-design-3-2-300x300.png 300w, https:\/\/9thclass.deltapublications.in\/wp-content\/uploads\/2023\/06\/Untitled-design-3-2-150x150.png 150w\" sizes=\"auto, (max-width: 500px) 100vw, 500px\" \/><\/figure><\/div>\n\n\n<p>\u2220 ___<\/p>\n<\/div><\/div>\n\n\n\n<p>The side lengths are VW=35 metres, UW=52 metres, and UV=53 metres. Since 35 &lt; 52 &lt; 53, VW &lt; UW &lt; UV.<\/p>\n\n\n\n<p>Their&nbsp;opposite angles are in the same order, from smallest to&nbsp;largest:<\/p>\n\n\n\n<p>\u2220U &lt; \u2220V &lt; \u2220W<\/p>\n\n\n\n<p>So, the largest angle is \u2220W.<\/p>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-columns are-vertically-aligned-center has-background is-layout-flex wp-container-core-columns-is-layout-9d6595d7 wp-block-columns-is-layout-flex\" style=\"background-color:#ffef76\">\n<div class=\"wp-block-column is-vertically-aligned-center is-layout-flow wp-block-column-is-layout-flow\" style=\"flex-basis:33.33%\">\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"500\" height=\"500\" src=\"https:\/\/9thclass.deltapublications.in\/wp-content\/uploads\/2023\/09\/Pink_Modern_Minimalist_Microphone_Talk_Podcast_Cover-removebg-preview-10.png\" alt=\"\" class=\"wp-image-10134\" srcset=\"https:\/\/9thclass.deltapublications.in\/wp-content\/uploads\/2023\/09\/Pink_Modern_Minimalist_Microphone_Talk_Podcast_Cover-removebg-preview-10.png 500w, https:\/\/9thclass.deltapublications.in\/wp-content\/uploads\/2023\/09\/Pink_Modern_Minimalist_Microphone_Talk_Podcast_Cover-removebg-preview-10-300x300.png 300w, https:\/\/9thclass.deltapublications.in\/wp-content\/uploads\/2023\/09\/Pink_Modern_Minimalist_Microphone_Talk_Podcast_Cover-removebg-preview-10-150x150.png 150w\" sizes=\"auto, (max-width: 500px) 100vw, 500px\" \/><\/figure>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-vertically-aligned-center is-layout-flow wp-block-column-is-layout-flow\" style=\"flex-basis:66.66%\">\n<figure class=\"wp-block-audio\"><audio controls src=\"https:\/\/9thclass.deltapublications.in\/wp-content\/uploads\/2023\/09\/1695720086375vfwkc23-voicemaker.in-speech.mp3\"><\/audio><\/figure>\n<\/div>\n<\/div>\n\n\n\n<p class=\"has-text-color has-large-font-size\" style=\"color:#d90000\"><strong>Let&#8217;s practice!<\/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\/81350\/416\/693\"><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\/81097\/002\/706\"><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>Angle-side relationships in triangles nr by Delta publications key notes : \ud83d\udd39 Basic Idea \ud83d\udc49 In a triangle, angles and sides are related \u2014\ud83d\udccf The longer side is opposite the larger angle, and\ud83d\udcd0 The shorter side is opposite the smaller angle. \ud83d\udca1 Rule: \u27a1\ufe0f Bigger Angle \u2192 Longer Opposite Side\u27a1\ufe0f Smaller Angle \u2192 Shorter Opposite<a class=\"more-link\" href=\"https:\/\/9thclass.deltapublications.in\/index.php\/g-5-angle-side-relationships-in-triangles\/\">Continue reading <span class=\"screen-reader-text\">&#8220;G.5 Angle-side relationships in triangles&#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-110","page","type-page","status-publish","hentry","entry"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/9thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages\/110","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=110"}],"version-history":[{"count":28,"href":"https:\/\/9thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages\/110\/revisions"}],"predecessor-version":[{"id":18157,"href":"https:\/\/9thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages\/110\/revisions\/18157"}],"wp:attachment":[{"href":"https:\/\/9thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/media?parent=110"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}