{"id":214,"date":"2022-04-13T08:43:58","date_gmt":"2022-04-13T08:43:58","guid":{"rendered":"http:\/\/9thclass.deltapublications.in\/?page_id=214"},"modified":"2025-12-04T12:23:36","modified_gmt":"2025-12-04T12:23:36","slug":"l-11-angles-in-inscribed-quadrilaterals","status":"publish","type":"page","link":"https:\/\/9thclass.deltapublications.in\/index.php\/l-11-angles-in-inscribed-quadrilaterals\/","title":{"rendered":"L.11 Angles in inscribed quadrilaterals"},"content":{"rendered":"\n<h2 class=\"wp-block-heading has-text-align-center has-text-color\" style=\"color:#00056d;text-transform:uppercase\"><strong>Angles in inscribed quadrilaterals<\/strong><\/h2>\n\n\n\n<p class=\"has-text-color has-link-color has-huge-font-size wp-elements-aae0d72b0df05b1a7d05750bd97517d2\" style=\"color:#74008b\"><strong>Key Notes :<\/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-a589028dd729b7151945fe718fba00e0\" style=\"color:#000060\">\ud83d\udd35 <strong>What is an Inscribed Quadrilateral?<\/strong><\/h3>\n\n\n\n<ul class=\"wp-block-list has-normal-font-size\">\n<li>An <strong>inscribed quadrilateral<\/strong> is a four-sided shape where <strong>all vertices (corners)<\/strong> lie on a <strong>single circle<\/strong> \u2014 this circle is called the <strong>circumscribed circle<\/strong>.<\/li>\n\n\n\n<li>Each side acts like a <strong>chord<\/strong> of the circle.<\/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-90f35c06932ec65b3b22744c51af8996\" style=\"color:#000060\">\ud83d\udccc<strong>Key Theorems and Properties<\/strong><\/h3>\n\n\n\n<p class=\"has-text-color has-link-color has-large-font-size wp-elements-d6b505d811b7bfb0312e70b1a8108422\" style=\"color:#000060\">\u2705 <strong>Property 1: Opposite Angles are Supplementary<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list has-normal-font-size\">\n<li class=\"has-large-font-size\">In an inscribed quadrilateral, the <strong>opposite angles<\/strong> add up to <strong>180\u00b0<\/strong> (they are supplementary).<\/li>\n<\/ul>\n\n\n\n<p class=\"has-text-align-center has-normal-font-size\">\u2220A + \u2220C = 180<sup>\u2218<\/sup> and \u2220B + \u2220D = 180<sup>\u2218<\/sup><\/p>\n\n\n\n<p class=\"has-text-color has-link-color has-large-font-size wp-elements-134f8df59751257356fe5dddfb6e3084\" style=\"color:#000060\">\u2705 <strong>Property 2: Exterior Angle Rule<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list has-normal-font-size\">\n<li>An <strong>exterior angle<\/strong> of an inscribed quadrilateral equals the <strong>interior opposite angle<\/strong>.<\/li>\n<\/ul>\n\n\n\n<p class=\"has-text-align-center has-normal-font-size\">\u2220E = \u2220C<\/p>\n\n\n\n<p class=\"has-normal-font-size\">(<em>where E is an exterior angle at vertex B opposite angle C<\/em>)<\/p>\n\n\n\n<p class=\"has-text-color has-link-color has-large-font-size wp-elements-a2a13bdcd1ee6a2627895a9dae7d36c3\" style=\"color:#000060\">\u2705 <strong>Property 3: Cyclic Quadrilateral Condition<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list has-normal-font-size\">\n<li>A quadrilateral can <strong>only<\/strong> be inscribed in a circle if its <strong>opposite angles<\/strong> are <strong>supplementary<\/strong>.<\/li>\n\n\n\n<li>If you check this property and it holds true, you <strong>confirm<\/strong> the quadrilateral is cyclic.<\/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-12a468890c92b3a8252512970921873d\" style=\"color:#000060\">\ud83d\udee0\ufe0f<strong>Example Problem<\/strong><\/h3>\n\n\n\n<p class=\"has-normal-font-size\">\u2705 <strong>Example 1:<\/strong><\/p>\n\n\n\n<p class=\"has-normal-font-size\">If an inscribed quadrilateral has angles \u2220A = 80\u2218 and \u2220C = 100\u2218, find \u2220B and \u2220D.<\/p>\n\n\n\n<p class=\"has-large-font-size\">\ud83d\udc49 <strong>Solution:<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list has-normal-font-size\">\n<li>Opposite angles are supplementary:<\/li>\n<\/ul>\n\n\n\n<p class=\"has-text-align-center has-normal-font-size\">\u2220A + \u2220C = 180<sup>\u2218<\/sup>\u2005\u200a\u27f9\u2005\u200a80<sup>\u2218<\/sup>+100<sup>\u2218<\/sup> = 180<sup>\u2218<\/sup>\u2705&nbsp;Correct<\/p>\n\n\n\n<p class=\"has-normal-font-size\">For the other pair: <\/p>\n\n\n\n<p class=\"has-text-align-center has-normal-font-size\">\u2220B + \u2220D = 180<sup>\u2218<\/sup><\/p>\n\n\n\n<p class=\"has-normal-font-size\">If \u2220B = 110\u2218, then: <\/p>\n\n\n\n<p class=\"has-text-align-center has-normal-font-size\">\u2220D = 180<sup>\u2218<\/sup> \u2212 110<sup>\u2218<\/sup> = 70<sup>\u2218<\/sup><\/p>\n\n\n\n<p class=\"has-normal-font-size\">\ud83d\udc49 <strong>Final Answer:<\/strong> \u2220B = 110<sup>\u2218<\/sup> , D = 70<sup>\u2218<\/sup><\/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-ca9fec65a5dff6bafe319359993d75b9\" style=\"color:#000060\">\ud83d\udd25<strong>Common Mistakes to Avoid<\/strong><\/h3>\n\n\n\n<p class=\"has-normal-font-size\">\u26a0\ufe0f <strong>1. Confusing adjacent angles with opposite angles<\/strong> \u2014 Only opposite angles are supplementary, <strong>not<\/strong> adjacent ones.<\/p>\n\n\n\n<p class=\"has-normal-font-size\">\u26a0\ufe0f <strong>2. Forgetting the exterior angle rule<\/strong> \u2014 Exterior angles are equal to the interior opposite angle, <strong>not adjacent angles<\/strong>.<\/p>\n\n\n\n<p class=\"has-normal-font-size\">\u26a0\ufe0f <strong>3. Assuming all quadrilaterals are cyclic<\/strong> \u2014 A quadrilateral is only cyclic if the <strong>opposite angles<\/strong> are <strong>supplementary<\/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-edb55183c9b0c22a0f92460d2bff68f4\" style=\"color:#000060\">\ud83e\udde0<strong>Real-Life Applications<\/strong><\/h3>\n\n\n\n<ul class=\"wp-block-list has-normal-font-size\">\n<li><strong>Architecture:<\/strong> Circular windows with decorative inscribed designs.<\/li>\n\n\n\n<li><strong>Engineering:<\/strong> Gear and wheel designs in machines.<\/li>\n\n\n\n<li><strong>Astronomy:<\/strong> Calculating angles in planetary orbits.<\/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-link-color has-large-font-size wp-elements-c2fb015a356f16136d1dbf4fc1fdcec5\" style=\"background-color:#bef1e8\"><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 has-link-color wp-elements-43c3cf747733c7a749d92aa8f30b7def\" style=\"color:#b00012\"><strong>\u27a1\ufe0f What&nbsp;is&nbsp;\u2220D?<\/strong><\/p>\n\n\n<div class=\"wp-block-image size-full\">\n<figure class=\"aligncenter is-resized\"><img decoding=\"async\" src=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2024\/01\/d-1.png\" alt=\"\" class=\"wp-image-11806\" style=\"width:370px;height:auto\"\/><\/figure><\/div>\n\n\n<p>\u2220D=_____ \u00b0<\/p>\n<\/div><\/div>\n\n\n\n<p class=\"has-primary-color has-text-color has-link-color wp-elements-c027a0ef54b0f71bc7166c4b77fe72af\">Look&nbsp;at the&nbsp;diagram:<\/p>\n\n\n<div class=\"wp-block-image size-full\">\n<figure class=\"aligncenter is-resized\"><img decoding=\"async\" src=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2024\/01\/d__1_-removebg-preview.png\" alt=\"\" class=\"wp-image-11808\" style=\"width:429px;height:auto\"\/><\/figure><\/div>\n\n\n<p>Since&nbsp;DEFG&nbsp;is an inscribed quadrilateral,&nbsp;\u2220F&nbsp;and&nbsp;\u2220D&nbsp;are supplementary. Write an equation setting the sum of their measures equal to&nbsp;180\u00b0,&nbsp;and solve for&nbsp;\u2220D.<\/p>\n\n\n\n<p>\u2220F+\u2220D=180\u00b0<\/p>\n\n\n\n<p><strong>108\u00b0<\/strong>+\u2220D=180\u00b0            Plug&nbsp;in&nbsp;\u2220F=108\u00b0<\/p>\n\n\n\n<p>\u2220D=72\u00b0                    Subtract&nbsp;108\u00b0&nbsp;from both&nbsp;sides<\/p>\n\n\n\n<p>\u2220D is 72\u00b0.<\/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-87655ec4dfb40b32e810f4abaa43c8ed\" style=\"background-color:#f1eea2\"><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 has-link-color wp-elements-8a7bd0928aac1d371c62ba882aae619f\" style=\"color:#b00012\"><strong>\u27a1\ufe0f What&nbsp;is&nbsp;\u2220H?<\/strong><\/p>\n\n\n<div class=\"wp-block-image size-full\">\n<figure class=\"aligncenter is-resized\"><img decoding=\"async\" src=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2024\/01\/d-2.png\" alt=\"\" class=\"wp-image-11811\" style=\"width:389px;height:auto\"\/><\/figure><\/div>\n\n\n<p>\u2220H=______ \u00b0<\/p>\n<\/div><\/div>\n\n\n\n<p>Look&nbsp;at the&nbsp;diagram:<\/p>\n\n\n<div class=\"wp-block-image size-full\">\n<figure class=\"aligncenter is-resized\"><img decoding=\"async\" src=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2024\/01\/image-removebg-preview-17.png\" alt=\"\" class=\"wp-image-11812\" style=\"width:394px;height:auto\"\/><\/figure><\/div>\n\n\n<p>Since&nbsp;FGHI&nbsp;is an inscribed quadrilateral,&nbsp;\u2220F&nbsp;and&nbsp;\u2220H&nbsp;are supplementary. Write an equation setting the sum of their measures equal to&nbsp;180\u00b0,&nbsp;and solve for&nbsp;\u2220H.<\/p>\n\n\n\n<p>\u2220F+\u2220H=180\u00b0<\/p>\n\n\n\n<p><strong>86\u00b0<\/strong>+\u2220H=180\u00b0             Plug&nbsp;in&nbsp;\u2220F=86\u00b0<\/p>\n\n\n\n<p>\u2220H=94\u00b0                   Subtract&nbsp;86\u00b0&nbsp;from both&nbsp;sides<\/p>\n\n\n\n<p>\u2220H is 94\u00b0.<\/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-92a3747b00f31de0696a616609d09be6\" style=\"background-color:#c7faf5\"><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 has-link-color wp-elements-74f9e376786cc43d9f8db1670e0a0784\" style=\"color:#b00012\"><strong>\u27a1\ufe0f What&nbsp;is&nbsp;\u2220G?<\/strong><\/p>\n\n\n<div class=\"wp-block-image size-full\">\n<figure class=\"aligncenter is-resized\"><img decoding=\"async\" src=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2024\/01\/d-3.png\" alt=\"\" class=\"wp-image-11816\" style=\"width:393px;height:auto\"\/><\/figure><\/div>\n\n\n<p>\u2220G=________ \u00b0<\/p>\n<\/div><\/div>\n\n\n\n<p>Look&nbsp;at the&nbsp;diagram:<\/p>\n\n\n<div class=\"wp-block-image size-full\">\n<figure class=\"aligncenter is-resized\"><img decoding=\"async\" src=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2024\/01\/image-removebg-preview-18.png\" alt=\"\" class=\"wp-image-11817\" style=\"width:417px;height:auto\"\/><\/figure><\/div>\n\n\n<p>Since&nbsp;GHIJ&nbsp;is an inscribed quadrilateral,&nbsp;\u2220I&nbsp;and&nbsp;\u2220G&nbsp;are supplementary. Write an equation setting the sum of their measures equal to&nbsp;180\u00b0,&nbsp;and solve for&nbsp;\u2220G.<\/p>\n\n\n\n<p>\u2220I+\u2220G=180\u00b0<\/p>\n\n\n\n<p><strong>102\u00b0<\/strong>+\u2220G=180\u00b0             Plug&nbsp;in&nbsp;\u2220I=102\u00b0<\/p>\n\n\n\n<p>\u2220G=78\u00b0                     Subtract&nbsp;102\u00b0&nbsp;from both&nbsp;sides<\/p>\n\n\n\n<p>\u2220G is 78\u00b0.<\/p>\n<\/div><\/div>\n\n\n\n<p class=\"has-text-color has-huge-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\/89045\/222\/126\"><img loading=\"lazy\" decoding=\"async\" width=\"500\" height=\"500\" src=\"https:\/\/9thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-3-73.png\" alt=\"\" class=\"wp-image-7907\" srcset=\"https:\/\/9thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-3-73.png 500w, https:\/\/9thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-3-73-300x300.png 300w, https:\/\/9thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-3-73-150x150.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\/89044\/400\/617\"><img loading=\"lazy\" decoding=\"async\" width=\"500\" height=\"500\" src=\"https:\/\/9thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-2-82.png\" alt=\"\" class=\"wp-image-7908\" srcset=\"https:\/\/9thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-2-82.png 500w, https:\/\/9thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-2-82-300x300.png 300w, https:\/\/9thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-2-82-150x150.png 150w\" sizes=\"auto, (max-width: 500px) 100vw, 500px\" \/><\/a><\/figure>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Angles in inscribed quadrilaterals Key Notes : \ud83d\udd35 What is an Inscribed Quadrilateral? \ud83d\udcccKey Theorems and Properties \u2705 Property 1: Opposite Angles are Supplementary \u2220A + \u2220C = 180\u2218 and \u2220B + \u2220D = 180\u2218 \u2705 Property 2: Exterior Angle Rule \u2220E = \u2220C (where E is an exterior angle at vertex B opposite angle<a class=\"more-link\" href=\"https:\/\/9thclass.deltapublications.in\/index.php\/l-11-angles-in-inscribed-quadrilaterals\/\">Continue reading <span class=\"screen-reader-text\">&#8220;L.11 Angles in inscribed quadrilaterals&#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-214","page","type-page","status-publish","hentry","entry"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/9thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages\/214","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=214"}],"version-history":[{"count":18,"href":"https:\/\/9thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages\/214\/revisions"}],"predecessor-version":[{"id":18299,"href":"https:\/\/9thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages\/214\/revisions\/18299"}],"wp:attachment":[{"href":"https:\/\/9thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/media?parent=214"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}