{"id":348,"date":"2022-04-13T10:45:05","date_gmt":"2022-04-13T10:45:05","guid":{"rendered":"http:\/\/10thclass.deltapublications.in\/?page_id=348"},"modified":"2025-01-10T10:02:49","modified_gmt":"2025-01-10T10:02:49","slug":"r-9-inscribed-angles","status":"publish","type":"page","link":"https:\/\/10thclass.deltapublications.in\/index.php\/r-9-inscribed-angles\/","title":{"rendered":"R.9 Inscribed angles"},"content":{"rendered":"\n<h2 class=\"wp-block-heading has-text-align-center has-text-color\" style=\"color:#00056d\"><strong>Inscribed angles<\/strong><\/h2>\n\n\n\n<p class=\"has-text-color has-link-color has-huge-font-size wp-elements-869a5c0b6b78055316c8d0186252dcbd\" style=\"color:#74008b\"><strong>key notes :<\/strong><\/p>\n\n\n\n<p class=\"has-text-color has-link-color has-large-font-size wp-elements-d71284a60442a3f98cb42225be763ffc\" style=\"color:#000060\">1. <strong>Definition of Inscribed Angle:<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list has-large-font-size\">\n<li>An <strong>inscribed angle<\/strong> is an angle formed by two chords in a circle that have a common endpoint. This common endpoint is the vertex of the angle, and the sides of the angle are the chords of the circle.<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n\n\n\n<h3 class=\"wp-block-heading has-text-color has-link-color has-large-font-size wp-elements-184575e83eb4f47cab6aa2637a632649\" style=\"color:#000060\">2. <strong>Relationship with the Arc:<\/strong><\/h3>\n\n\n\n<ul class=\"wp-block-list has-large-font-size\">\n<li>The <strong>inscribed angle<\/strong> intercepts a part of the circle&#8217;s circumference called the <strong>arc<\/strong>.<\/li>\n\n\n\n<li>The measure of an <strong>inscribed angle<\/strong> is <strong>half<\/strong> the measure of the intercepted arc.<\/li>\n<\/ul>\n\n\n\n<p class=\"has-large-font-size\"><strong>Formula:<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"597\" height=\"55\" src=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2025\/01\/image-4.png\" alt=\"\" class=\"wp-image-17393\" srcset=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2025\/01\/image-4.png 597w, https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2025\/01\/image-4-300x28.png 300w\" sizes=\"auto, (max-width: 597px) 100vw, 597px\" \/><\/figure>\n\n\n\n<p class=\"has-large-font-size\"><strong>Example:<\/strong> If the intercepted arc is 60\u00b0, the inscribed angle will be 30\u00b0.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n\n\n\n<h3 class=\"wp-block-heading has-text-color has-link-color has-large-font-size wp-elements-d2f1ecd85261bfd98dc9b64d4245014b\" style=\"color:#000060\">3. <strong>Inscribed Angles Subtended by the Same Arc:<\/strong><\/h3>\n\n\n\n<ul class=\"wp-block-list has-large-font-size\">\n<li><strong>Angles subtended by the same arc<\/strong> on the circle are <strong>equal<\/strong>.<\/li>\n\n\n\n<li>This means that any two inscribed angles that intercept the same arc will have the same measure.<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n\n\n\n<h3 class=\"wp-block-heading has-text-color has-link-color has-large-font-size wp-elements-9c9ba4a0f9b4f9f419b711b4f1100f1f\" style=\"color:#000060\">4. <strong>Inscribed Angle in a Semicircle:<\/strong><\/h3>\n\n\n\n<ul class=\"wp-block-list has-large-font-size\">\n<li>An <strong>inscribed angle<\/strong> that subtends a <strong>diameter<\/strong> of the circle is always a <strong>right angle<\/strong> (90\u00b0).<\/li>\n\n\n\n<li>This is known as <strong>Thales&#8217; Theorem<\/strong>.<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n\n\n\n<h3 class=\"wp-block-heading has-text-color has-link-color has-large-font-size wp-elements-5f9012c5388ba32356ea7cbdecf7d0ff\" style=\"color:#000060\">5. <strong>Cyclic Quadrilaterals:<\/strong><\/h3>\n\n\n\n<ul class=\"wp-block-list has-large-font-size\">\n<li>A <strong>cyclic quadrilateral<\/strong> is a quadrilateral whose vertices lie on the circumference of a circle.<\/li>\n\n\n\n<li>The opposite angles of a cyclic quadrilateral are supplementary (i.e., their sum is 180\u00b0).<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n\n\n\n<h3 class=\"wp-block-heading has-text-color has-link-color has-large-font-size wp-elements-477f7fc784ae8bb886eefb402bde678c\" style=\"color:#000060\">6. <strong>Application of Inscribed Angles:<\/strong><\/h3>\n\n\n\n<ul class=\"wp-block-list has-large-font-size\">\n<li>To find the angle between two chords.<\/li>\n\n\n\n<li>To calculate unknown angles in cyclic quadrilaterals.<\/li>\n\n\n\n<li>To solve problems involving tangents and secants.<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n\n\n\n<h3 class=\"wp-block-heading has-text-color has-link-color has-large-font-size wp-elements-2c2b1c40ef75f8c2e0e321c80c4e58cb\" style=\"color:#000060\">7. <strong>Important Theorems:<\/strong><\/h3>\n\n\n\n<ul class=\"wp-block-list has-large-font-size\">\n<li><strong>Angle at the Center vs. Angle at the Circumference:<\/strong>\n<ul class=\"wp-block-list\">\n<li>The <strong>angle at the center<\/strong> of a circle is twice the <strong>angle at the circumference<\/strong> subtended by the same arc.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"463\" height=\"50\" src=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2025\/01\/image-5.png\" alt=\"\" class=\"wp-image-17395\" srcset=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2025\/01\/image-5.png 463w, https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2025\/01\/image-5-300x32.png 300w\" sizes=\"auto, (max-width: 463px) 100vw, 463px\" \/><\/figure>\n\n\n\n<ul class=\"wp-block-list\">\n<li class=\"has-large-font-size\"><strong>Angle between a Chord and a Tangent:<\/strong>\n<ul class=\"wp-block-list\">\n<li>The <strong>angle between a tangent and a chord<\/strong> through the point of contact is equal to the <strong>inscribed angle<\/strong> subtended by the chord on the opposite side of the tangent.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n\n\n\n<h3 class=\"wp-block-heading has-text-color has-link-color has-large-font-size wp-elements-7f4c20fa9f3aada9dad6c234f1fa3ef4\" style=\"color:#000060\">8. <strong>Examples:<\/strong><\/h3>\n\n\n\n<ul class=\"wp-block-list has-large-font-size\">\n<li>If an inscribed angle intercepts an arc of 120\u00b0, the angle will measure 60\u00b0.<\/li>\n\n\n\n<li>In a circle with a diameter as a chord, the inscribed angle will always be a right angle (90\u00b0).<\/li>\n<\/ul>\n\n\n\n<p class=\"has-large-font-size\"><a href=\"https:\/\/www.ixl.com\/math\/geometry\/inscribed-angles\"><\/a><\/p>\n\n\n\n<p class=\"has-text-align-center has-text-color has-huge-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-d99337844886c8fe971516063c7555f8\" style=\"background-color:#d1efd5\"><div class=\"wp-block-group__inner-container is-layout-constrained wp-block-group-is-layout-constrained\">\n<div class=\"wp-block-group has-primary-color has-background-background-color has-text-color has-background has-link-color has-large-font-size wp-elements-cd50955d4c394423114939bc224f4b8c\"><div class=\"wp-block-group__inner-container is-layout-constrained wp-block-group-is-layout-constrained\">\n<p class=\"has-text-color has-link-color wp-elements-2849a6155e82a91cfd16cc1f5a00ec4d\" style=\"color:#b00012\"><strong>What&nbsp;is \u2220HGI?<\/strong><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"559\" height=\"521\" src=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2024\/01\/image-38.png\" alt=\"\" class=\"wp-image-11864\" style=\"width:292px;height:auto\" srcset=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2024\/01\/image-38.png 559w, https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2024\/01\/image-38-300x280.png 300w\" sizes=\"auto, (max-width: 559px) 100vw, 559px\" \/><\/figure><\/div>\n\n\n<p>\u2220HGI= _____\u2218<\/p>\n<\/div><\/div>\n\n\n\n<p>Look&nbsp;at the&nbsp;diagram:<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"520\" height=\"479\" src=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2024\/01\/image-removebg-preview-23.png\" alt=\"\" class=\"wp-image-11871\" style=\"width:284px;height:auto\" srcset=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2024\/01\/image-removebg-preview-23.png 520w, https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2024\/01\/image-removebg-preview-23-300x276.png 300w\" sizes=\"auto, (max-width: 520px) 100vw, 520px\" \/><\/figure><\/div>\n\n\n<p>\u2220HGI is an inscribed angle that intercepts the same arc as the central angle \u2220J, so use the Inscribed Angle Theorem.<\/p>\n\n\n\n<h1 class=\"wp-block-heading has-large-font-size\">\u2220HGI =1\/2 . \u2220j<\/h1>\n\n\n\n<p>         =1\/2 (122\u00b0) plug \u2220J=122\u00b0<\/p>\n\n\n\n<p>         =61\u00b0<\/p>\n\n\n\n<p>\u2220HGI is 61\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-3e7c334c74a1c6db0e0c3ead72a76117\" style=\"background-color:#f6f6b6\"><div class=\"wp-block-group__inner-container is-layout-constrained wp-block-group-is-layout-constrained\">\n<div class=\"wp-block-group has-primary-color has-background-background-color has-text-color has-background has-link-color has-large-font-size wp-elements-8b113a3ab508ac39c1ffa28557a87f58\"><div class=\"wp-block-group__inner-container is-layout-constrained wp-block-group-is-layout-constrained\">\n<p class=\"has-text-color has-link-color has-large-font-size wp-elements-8fa88ccf51fd1fc3517b82fd2401c8d1\" style=\"color:#b00012\"><strong>What is \u2220F?<\/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:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2024\/01\/d-9.png\" alt=\"\" class=\"wp-image-11884\" style=\"width:259px;height:auto\" srcset=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2024\/01\/d-9.png 500w, https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2024\/01\/d-9-300x300.png 300w, https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2024\/01\/d-9-150x150.png 150w\" sizes=\"auto, (max-width: 500px) 100vw, 500px\" \/><\/figure><\/div>\n\n\n<p>\u2220F= ________\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\">\n<figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"500\" height=\"500\" src=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2024\/01\/image-removebg-preview-24.png\" alt=\"\" class=\"wp-image-11892\" style=\"width:264px;height:auto\" srcset=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2024\/01\/image-removebg-preview-24.png 500w, https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2024\/01\/image-removebg-preview-24-300x300.png 300w, https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2024\/01\/image-removebg-preview-24-150x150.png 150w\" sizes=\"auto, (max-width: 500px) 100vw, 500px\" \/><\/figure><\/div>\n\n\n<p>\u2220GHI is an inscribed angle that intercepts the same arc as the central angle \u2220F, so use the Inscribed Angle Theorem.<\/p>\n\n\n\n<p>\u2220F= 2 . \u2220GHI     Inscribed Angle Theorem<\/p>\n\n\n\n<p>= 2 .(47\u00b0)       Plug in   \u2220GHI=47\u00b0<\/p>\n\n\n\n<p>= 94\u00b0 Multiply<\/p>\n\n\n\n<p>\u2220F is 94\u00b0.<\/p>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-group has-background has-large-font-size\" style=\"background-color:#d1e6f7\"><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 has-link-color wp-elements-736844ee4de2e05ecafa37fdbef647f6\" style=\"color:#b00012\"><strong>What is \u2220J?<\/strong><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"581\" height=\"515\" src=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2024\/01\/image-41.png\" alt=\"\" class=\"wp-image-11895\" style=\"width:326px;height:auto\" srcset=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2024\/01\/image-41.png 581w, https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2024\/01\/image-41-300x266.png 300w\" sizes=\"auto, (max-width: 581px) 100vw, 581px\" \/><\/figure><\/div>\n\n\n<p>\u2220J=______ \u00b0<\/p>\n<\/div><\/div>\n\n\n\n<p>Look at the diagram:<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"519\" height=\"480\" src=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2024\/01\/image-removebg-preview-28.png\" alt=\"\" class=\"wp-image-11896\" style=\"width:297px;height:auto\" srcset=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2024\/01\/image-removebg-preview-28.png 519w, https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2024\/01\/image-removebg-preview-28-300x277.png 300w\" sizes=\"auto, (max-width: 519px) 100vw, 519px\" \/><\/figure><\/div>\n\n\n<p>\u2220GHI is an inscribed angle that intercepts the same arc as the central angle \u2220J, so use the Inscribed Angle Theorem.<br>\u2220J = 2 . \u2220GHI    Inscribed Angle Theorem<br>= 2 . (65\u00b0)       Plug in \u2220GHI=65\u00b0<br>= 130\u00b0         Multiply<br>\u2220J is 130\u00b0.<\/p>\n<\/div><\/div>\n\n\n\n<p class=\"has-text-color has-huge-font-size\" style=\"color:#d90000\">let&#8217;s practice!<\/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\/84469\/589\/129\"><img loading=\"lazy\" decoding=\"async\" width=\"500\" height=\"500\" src=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-3-120.png\" alt=\"\" class=\"wp-image-7348\" srcset=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-3-120.png 500w, https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-3-120-300x300.png 300w, https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-3-120-150x150.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\/84597\/483\/312\"><img loading=\"lazy\" decoding=\"async\" width=\"500\" height=\"500\" src=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-2-139.png\" alt=\"\" class=\"wp-image-7349\" srcset=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-2-139.png 500w, https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-2-139-300x300.png 300w, https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-2-139-150x150.png 150w\" sizes=\"auto, (max-width: 500px) 100vw, 500px\" \/><\/a><\/figure>\n<\/div>\n<\/div>\n\n\n\n<p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Inscribed angles key notes : 1. Definition of Inscribed Angle: 2. Relationship with the Arc: Formula: Example: If the intercepted arc is 60\u00b0, the inscribed angle will be 30\u00b0. 3. Inscribed Angles Subtended by the Same Arc: 4. Inscribed Angle in a Semicircle: 5. Cyclic Quadrilaterals: 6. Application of Inscribed Angles: 7. Important Theorems: 8.<a class=\"more-link\" href=\"https:\/\/10thclass.deltapublications.in\/index.php\/r-9-inscribed-angles\/\">Continue reading <span class=\"screen-reader-text\">&#8220;R.9 Inscribed angles&#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-348","page","type-page","status-publish","hentry","entry"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/10thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages\/348","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/10thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/10thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/10thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/10thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/comments?post=348"}],"version-history":[{"count":14,"href":"https:\/\/10thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages\/348\/revisions"}],"predecessor-version":[{"id":17398,"href":"https:\/\/10thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages\/348\/revisions\/17398"}],"wp:attachment":[{"href":"https:\/\/10thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/media?parent=348"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}