{"id":30981,"date":"2023-06-21T07:39:40","date_gmt":"2023-06-21T07:39:40","guid":{"rendered":"https:\/\/www.splashlearn.com\/math-vocabulary\/?page_id=30981"},"modified":"2023-11-15T17:55:59","modified_gmt":"2023-11-15T17:55:59","slug":"cyclic-quadrilateral-definition-theorem-examples-faqs","status":"publish","type":"post","link":"https:\/\/www.splashlearn.com\/math-vocabulary\/cyclic-quadrilaterals","title":{"rendered":"Cyclic Quadrilateral \u2013 Definition, Theorem, 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-fb38185a-ea55-4ceb-80cb-3bfcde653c20\" 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\/cyclic-quadrilaterals#0-what-is-a-cyclic-quadrilateral>What Is a Cyclic Quadrilateral?<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/cyclic-quadrilaterals#4-properties-of-a-cyclic-quadrilateral>Properties of a Cyclic Quadrilateral<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/cyclic-quadrilaterals#5-cyclic-quadrilateral-formulas>Cyclic Quadrilateral Formulas<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/cyclic-quadrilaterals#8-solved-examples-on-cyclic-quadrilateral>Solved Examples on Cyclic Quadrilateral<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/cyclic-quadrilaterals#9-practice-problems-on-cyclic-quadrilateral>Practice Problems on Cyclic Quadrilateral<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/cyclic-quadrilaterals#10-frequently-asked-questions-on-cyclic-quadrilateral>Frequently Asked Questions on Cyclic Quadrilateral<\/a><\/li><\/ul><\/div><\/div><\/div>\n\n\n<h2 class=\"wp-block-heading\" id=\"0-what-is-a-cyclic-quadrilateral\">What Is a Cyclic Quadrilateral?<\/h2>\n\n\n\n<p><strong>A cyclic quadrilateral is a <\/strong><a href=\"https:\/\/www.splashlearn.com\/math-vocabulary\/geometry\/quadrilateral\"><strong>quadrilateral<\/strong><\/a><strong> whose all four vertices lie on a circle. It is also called an inscribed quadrilateral.<\/strong>&nbsp;<\/p>\n\n\n\n<p>The <a href=\"https:\/\/www.splashlearn.com\/math-vocabulary\/geometry\/circle\">circle<\/a> that passes through the four vertices of a quadrilateral is called a circumcircle. The vertices of a cyclic quadrilateral are said to be concyclic since they lie on a circle.&nbsp;<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"620\" height=\"494\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/11\/cyclic-quadrilateral-examples.png\" alt=\"Cyclic quadrilateral examples\" class=\"wp-image-35961\" title=\"Cyclic quadrilateral examples\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/11\/cyclic-quadrilateral-examples.png 620w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/11\/cyclic-quadrilateral-examples-300x239.png 300w\" sizes=\"auto, (max-width: 620px) 100vw, 620px\" \/><\/figure>\n\n\n\n<p>In the image given below, the quadrilateral on the right is not cyclic since its one vertex does not lie on the circumference of the circle.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"620\" height=\"531\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/11\/example-and-non-example-of-a-cyclic-quadrilateral.png\" alt=\"Example and non-example of a cyclic quadrilateral\" class=\"wp-image-35962\" title=\"Example and non-example of a cyclic quadrilateral\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/11\/example-and-non-example-of-a-cyclic-quadrilateral.png 620w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/11\/example-and-non-example-of-a-cyclic-quadrilateral-300x257.png 300w\" sizes=\"auto, (max-width: 620px) 100vw, 620px\" \/><\/figure>\n\n\n\n<div id=\"recommended-games-container-id\" class=\"recommended-games-container\"><h4 class=\"recommended-games-container-headline\">Recommended Games<\/h4><div class=\"recommended-games-container-slides\"><div class=\"game-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-games\/classify-quadrilaterals-in-different-orientation\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"games_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t\t<div class=\"game-card-container-inner-block\">\r\n\t\t\t<img decoding=\"async\" class=\"game-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/curriculum_uploads\/images\/playables\/geometry_rhombus_parallelogram_orientation_pt.png\" alt=\"Classify Quadrilaterals in Different Orientation Game\">\r\n\t\t<\/div>\r\n\t\r\n\t\t<div class=\"game-card-container-inner\">\r\n\t\t\t<div class=\"game-card-container-inner-name\">Classify Quadrilaterals in Different Orientation Game<\/div>\r\n\t\t\t<span class=\"game-card-container-inner-cta\">Play<\/span>\r\n\t\t<\/div>\r\n\t<\/a>\r\n<\/div><div class=\"game-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-games\/find-the-perimeter-of-the-quadrilaterals\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"games_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t\t<div class=\"game-card-container-inner-block\">\r\n\t\t\t<img decoding=\"async\" class=\"game-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/curriculum_uploads\/images\/playables\/geo_meas_peri_rectangle_3_pt.png\" alt=\"Find the Perimeter of the Quadrilaterals Game\">\r\n\t\t<\/div>\r\n\t\r\n\t\t<div class=\"game-card-container-inner\">\r\n\t\t\t<div class=\"game-card-container-inner-name\">Find the Perimeter of the Quadrilaterals Game<\/div>\r\n\t\t\t<span class=\"game-card-container-inner-cta\">Play<\/span>\r\n\t\t<\/div>\r\n\t<\/a>\r\n<\/div><div class=\"game-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-games\/identify-polygons-and-quadrilaterals\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"games_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t\t<div class=\"game-card-container-inner-block\">\r\n\t\t\t<img decoding=\"async\" class=\"game-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/curriculum_uploads\/images\/playables\/geometry_sort_shapes_broad_cat_pt.png\" alt=\"Identify Polygons and Quadrilaterals Game\">\r\n\t\t<\/div>\r\n\t\r\n\t\t<div class=\"game-card-container-inner\">\r\n\t\t\t<div class=\"game-card-container-inner-name\">Identify Polygons and Quadrilaterals Game<\/div>\r\n\t\t\t<span class=\"game-card-container-inner-cta\">Play<\/span>\r\n\t\t<\/div>\r\n\t<\/a>\r\n<\/div><div class=\"game-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-games\/identify-quadrilaterals-in-different-orientation\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"games_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t\t<div class=\"game-card-container-inner-block\">\r\n\t\t\t<img decoding=\"async\" class=\"game-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/curriculum_uploads\/images\/playables\/geometry_square_rectangle_orientation_pt.png\" alt=\"Identify Quadrilaterals in Different Orientation Game\">\r\n\t\t<\/div>\r\n\t\r\n\t\t<div class=\"game-card-container-inner\">\r\n\t\t\t<div class=\"game-card-container-inner-name\">Identify Quadrilaterals in Different Orientation Game<\/div>\r\n\t\t\t<span class=\"game-card-container-inner-cta\">Play<\/span>\r\n\t\t<\/div>\r\n\t<\/a>\r\n<\/div><div class=\"game-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-games\/identify-the-lines-of-symmetry-in-quadrilaterals\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"games_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t\t<div class=\"game-card-container-inner-block\">\r\n\t\t\t<img decoding=\"async\" class=\"game-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/curriculum_uploads\/images\/playables\/geometry_symm_regular_shape_pt.png\" alt=\"Identify the LInes of Symmetry in Quadrilaterals Game\">\r\n\t\t<\/div>\r\n\t\r\n\t\t<div class=\"game-card-container-inner\">\r\n\t\t\t<div class=\"game-card-container-inner-name\">Identify the LInes of Symmetry in Quadrilaterals Game<\/div>\r\n\t\t\t<span class=\"game-card-container-inner-cta\">Play<\/span>\r\n\t\t<\/div>\r\n\t<\/a>\r\n<\/div><\/div><p class=\"recommended-games-container-desc\"><a href=\"https:\/\/www.splashlearn.com\/games\">More Games<\/a><\/p><button class=\"scroll-right-arrow\"><\/button><\/div><script type=\"text\/javascript\">\n        document.addEventListener(\"DOMContentLoaded\", function() {\n            const container = document.querySelector(\"#recommended-games-container-id\");\n            const slidesContainer = container.querySelector(\".recommended-games-container-slides\");\n            const cards = slidesContainer.querySelectorAll(\".game-card-container-outer\");\n            const scrollRightArrow = container.querySelector(\".scroll-right-arrow\");\n\n            function adjustContainerStyles() {\n                const numCards = cards.length;\n\n                if (numCards === 1) {\n                    container.style.maxWidth = \"30%\";\n                    container.style.textAlign = \"center\";\n                } else if (numCards === 2) {\n                    container.style.maxWidth = \"50%\";\n                    container.style.textAlign = \"center\";\n                } else if (numCards === 3) {\n                    container.style.maxWidth = \"75%\";\n                    container.style.textAlign = \"center\";\n                } else {\n                    container.style.maxWidth = \"\";\n                    container.style.textAlign = \"\";\n                }\n            }\n\n            function checkScrollPosition() {\n                const maxScrollLeft = slidesContainer.scrollWidth - slidesContainer.clientWidth;\n                if ((slidesContainer.scrollLeft + 10) >= maxScrollLeft) {\n                    scrollRightArrow.style.display = \"none\"; \/\/ Hide the arrow if fully scrolled\n                } else {\n                    scrollRightArrow.style.display = \"block\"; \/\/ Show the arrow if not fully scrolled\n                }\n            }\n\n            scrollRightArrow.addEventListener(\"click\", function() {\n                const scrollAmount = 300; \/\/ Adjust based on the container's width and your needs\n                slidesContainer.scrollLeft += scrollAmount;\n                setTimeout(checkScrollPosition, 100); \/\/ Delay to allow scroll update\n            });\n\n            adjustContainerStyles();\n            checkScrollPosition();\n            slidesContainer.addEventListener(\"scroll\", checkScrollPosition);\n        });\n    <\/script><h2 class=\"wp-block-heading\" id=\"1-cyclic-quadrilateral-definition\">Cyclic Quadrilateral Definition<\/h2>\n\n\n\n<p><strong>A cyclic quadrilateral can be defined as a quadrilateral inscribed in a circle. It is a four-sided <\/strong><a href=\"https:\/\/www.splashlearn.com\/math-vocabulary\/geometry\/polygon\"><strong>polygon<\/strong><\/a><strong> around which a circle can be drawn such that the vertices lie on the circumference of a circle.<\/strong><\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"2-cyclic-quadrilateral-theorems\">Cyclic Quadrilateral Theorems<\/h2>\n\n\n\n<p>One important question is: how do you prove that a quadrilateral is cyclic? Let\u2019s understand how to find cyclic quadrilaterals with the help of theorems.<\/p>\n\n\n\n<p><strong>Ptolemy Theorem of Cyclic Quadrilateral<\/strong><\/p>\n\n\n<div class=\"ub-styled-box ub-notification-box\" id=\"ub-styled-box-99221cac-63fe-4fc1-bcd5-af44020d5386\">\n\n\n<p><strong>The Ptolemy theorem states that the sum of the product of the opposite sides of a cyclic quadrilateral equals the product of diagonals.<\/strong><\/p>\n\n\n<\/div>\n\n\n<p>Consider a cyclic quadrilateral ABCD with successive vertices A, B, C, and D, sides given by $a = AB,\\; b = BC,\\; c = CD,\\; d = DA$, and diagonals $p = AC,\\; q = BD$.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"620\" height=\"460\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/11\/cyclic-quadrilateral-ABCD.png\" alt=\"Cyclic quadrilateral ABCD\" class=\"wp-image-35963\" title=\"Cyclic quadrilateral ABCD\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/11\/cyclic-quadrilateral-ABCD.png 620w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/11\/cyclic-quadrilateral-ABCD-300x223.png 300w\" sizes=\"auto, (max-width: 620px) 100vw, 620px\" \/><\/figure>\n\n\n\n<p>We can express diagonals in terms of the sides as:&nbsp;<\/p>\n\n\n\n<p>$p \\times q = (a \\times c) + (b \\times d)$<\/p>\n\n\n\n<p>$pq = ac + bd$<\/p>\n\n\n\n<p><strong>Cyclic Quadrilateral Angles<\/strong><\/p>\n\n\n\n<p><strong>To Prove: Opposite Angles of a Cyclic Quadrilateral are Supplementary.<\/strong><\/p>\n\n\n\n<p><strong>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Or&nbsp;<\/strong><\/p>\n\n\n\n<p><strong>&nbsp;&nbsp;&nbsp;&nbsp;<\/strong>$\\angle A +\\angle C = 180^{\\circ}$ and $\\angle B + \\angle D = 180^{\\circ}$<strong>.<\/strong><\/p>\n\n\n\n<p><strong>Construction: <\/strong>Join the vertices A and C with the center of the circle O.<strong><\/strong><\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"620\" height=\"532\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/11\/construction-for-cyclic-quadrilateral-theorem-on-opposite-angles.webp\" alt=\"Construction for cyclic quadrilateral theorem on opposite angles\" class=\"wp-image-35965\" title=\"Construction for cyclic quadrilateral theorem on opposite angles\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/11\/construction-for-cyclic-quadrilateral-theorem-on-opposite-angles.webp 620w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/11\/construction-for-cyclic-quadrilateral-theorem-on-opposite-angles-300x257.webp 300w\" sizes=\"auto, (max-width: 620px) 100vw, 620px\" \/><\/figure>\n\n\n\n<p>Let x and y be the <a href=\"https:\/\/www.splashlearn.com\/math-vocabulary\/geometry\/angle\">angles<\/a> subtended by the minor arc BAD and the major arc BCD respectively.<\/p>\n\n\n\n<p>$x^{\\circ} = 2 \\angle BCD = 2 \\angle C$ \u2026Inscribed angle theorem (angle subtended by the same arc is half the angle subtended at the center)<\/p>\n\n\n\n<p>$y^{\\circ} = 2 \\angle BAD = 2 \\angle A$ \u2026Inscribed angle theorem<\/p>\n\n\n\n<p>Angles $x^{\\circ}$ and $y^{\\circ}$ together form a full angle.<\/p>\n\n\n\n<p>$x^{\\circ} + y^{\\circ} = &nbsp;360^{\\circ}$<\/p>\n\n\n\n<p>$2 \\angle C + 2 \\angle A = 360^{\\circ}$<\/p>\n\n\n\n<p>$\\angle C + \\angle A = 180^{\\circ}$<\/p>\n\n\n\n<p>Hence, proved.<\/p>\n\n\n\n<p>The converse of the above theorem is also true.<\/p>\n\n\n<div class=\"ub-styled-box ub-notification-box\" id=\"ub-styled-box-d5565ac9-0bb6-4fb8-87dd-598a8732bbfc\">\n\n\n<p><strong>Converse: <\/strong>If opposite angles of a quadrilateral are supplementary, then the quadrilateral is cyclic.<\/p>\n\n\n<\/div>\n\n\n<p><strong>Brahmagupta Theorem of Cyclic Quadrilateral<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-table wj-custom-table\"><table class=\"wj-table-class\"><tbody><tr><td><strong>The Brahmagupta theorem states that the area A of a cyclic quadrilateral whose sides are a, b, c, d is given by:&nbsp;<\/strong><br>$Area = \\sqrt{(s\\;-\\;a)(s\\;-\\;b)(s\\;-\\;c)(s\\;-\\;d)}$<br>Here, s is the semi-perimeter given by $s = \\frac{a + b + c + d}{2}$<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"3-isosceles-trapezoid-theorem\">Isosceles Trapezoid Theorem<\/h2>\n\n\n<div class=\"ub-styled-box ub-notification-box\" id=\"ub-styled-box-1275851d-1e95-4ed0-9b92-7e47a84e57bf\">\n\n\n<p><strong>The isosceles trapezoid theorem states that a trapezoid is cyclic quadrilateral if and only if it is an isosceles trapezoid.&nbsp;<\/strong><\/p>\n\n\n<\/div>\n\n\n<p>An isosceles trapezoid is a trapezoid where the two legs of the trapezoid are equal in length.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"4-properties-of-a-cyclic-quadrilateral\">Properties of a Cyclic Quadrilateral<\/h2>\n\n\n\n<ul class=\"wp-block-list\">\n<li>All the four vertices of the inscribed quadrilateral lie on the circumference or boundary of the circle. They are said to be concyclic.<\/li>\n<\/ul>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The four sides of the inscribed quadrilateral are chords of the circle as their endpoints lie on the boundary of the circle.&nbsp;<\/li>\n<\/ul>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The measure of an exterior angle at a vertex equals the opposite interior angle.<\/li>\n<\/ul>\n\n\n\n<ul class=\"wp-block-list\">\n<li>In a cyclic quadrilateral, the sum of the product of the opposite sides equals the product of diagonals.<\/li>\n<\/ul>\n\n\n\n<ul class=\"wp-block-list\">\n<li>In a cyclic quadrilateral, the perpendicular bisectors of the sides are always concurrent and they meet at the center O.<\/li>\n<\/ul>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The sum of a pair of opposite angles is always supplementary.<\/li>\n<\/ul>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The sum of all four angles of a cyclic quadrilateral is $360^{\\circ}$.<\/li>\n<\/ul>\n\n\n\n<ul class=\"wp-block-list\">\n<li>A cyclic quadrilateral has the maximum area possible with the given side lengths. In other words, a quadrilateral that is inscribed in a circle represents the maximum area possible with those side lengths.&nbsp;<\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"5-cyclic-quadrilateral-formulas\">Cyclic Quadrilateral Formulas<\/h2>\n\n\n\n<p>Let\u2019s look at the formulas associated with the cyclic quadrilaterals.<\/p>\n\n\n\n<p><strong>Radius of a Cyclic Quadrilateral<\/strong><\/p>\n\n\n\n<p>If a, b, c and d are the successive sides of a cyclic quadrilateral, and s is the semi-perimeter, then the radius is given by&nbsp;<\/p>\n\n\n\n<p>$R = \\frac{1}{4} \\sqrt{\\frac{(ab + cd)(ac + bd)(ad + bc)}{(s\\;-\\;a)(s\\;-\\;b)(s\\;-\\;c)(s\\;-\\;d)}}$<\/p>\n\n\n\n<p><strong>Diagonals of a Cyclic Quadrilateral<\/strong><\/p>\n\n\n\n<p>Suppose a, b, c, and d are the sides of a cyclic quadrilateral and p &amp; q are the diagonals, then the formula for the length of diagonals is given by:<\/p>\n\n\n\n<p>$p = \\sqrt{\\frac{(ac+bd)(ad+bc)}{ab+cd}}$ and $q = \\sqrt{\\frac{(ac + bd)(ab + cd)}{ad + bc}}$<\/p>\n\n\n\n<p><strong>Area of a Cyclic Quadrilateral<\/strong><\/p>\n\n\n\n<p>As mentioned earlier, if the sides of the inscribed quadrilateral are a, b, c, and d and \u201cs\u201d is the semiperimeter, then the area of a cyclic quadrilateral is given by the Brahmagupta theorem:<\/p>\n\n\n\n<p>$Area = \\sqrt{(s\\;-\\;a)(s\\;-\\;b)(s\\;-\\;c)(s\\;-\\;d)}$<\/p>\n\n\n\n<p>where $s = \\frac{a + b + c + d}{2}$<\/p>\n\n\n\n<p>Heron&#8217;s formula for the area of a triangle is derived from this equation.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"6-facts-about-cyclic-quadrilateral\">Facts about Cyclic Quadrilateral<\/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>A cyclic quadrilateral has the maximum area possible with the given side lengths.<\/li>\n\n\n\n<li>The circle that has all the vertices of a polygon on its circumference is called the circumcircle or circumscribed circle.<\/li>\n<\/ul>\n<\/div><\/div>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"7-conclusion\">Conclusion<\/h2>\n\n\n\n<p>In this article, we learned about cyclic quadrilaterals, their properties, important theorems, and formulas associated with them. Now, let\u2019s understand how to solve cyclic quadrilateral problems with the help of these properties and theorems.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"8-solved-examples-on-cyclic-quadrilateral\">Solved Examples on Cyclic Quadrilateral<\/h2>\n\n\n\n<p><strong>1. Find the values of x and y.<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"267\" height=\"267\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/11\/a-cyclic-quadrilateral-with-two-adjacent-missing-angles.png\" alt=\"A cyclic quadrilateral with two adjacent missing angles\" class=\"wp-image-35966\" title=\"A cyclic quadrilateral with two adjacent missing angles\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/11\/a-cyclic-quadrilateral-with-two-adjacent-missing-angles.png 267w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/11\/a-cyclic-quadrilateral-with-two-adjacent-missing-angles-150x150.png 150w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/11\/a-cyclic-quadrilateral-with-two-adjacent-missing-angles-250x250.png 250w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/11\/a-cyclic-quadrilateral-with-two-adjacent-missing-angles-120x120.png 120w\" sizes=\"auto, (max-width: 267px) 100vw, 267px\" \/><\/figure>\n\n\n\n<p><strong>Solution:<\/strong>&nbsp;<\/p>\n\n\n\n<p>We know that opposite angles of a cyclic quadrilateral are supplementary.<\/p>\n\n\n\n<p>So, $100^{\\circ} + x = 180^{\\circ}$<\/p>\n\n\n\n<p>$x = 180^{\\circ}\\;-\\;100^{\\circ}$<\/p>\n\n\n\n<p>$x = 80^{\\circ}$<\/p>\n\n\n\n<p>Similarly, $70^{\\circ} + y = 180^{\\circ}$<\/p>\n\n\n\n<p>$y = 180^{\\circ} \\;-\\; 70^{\\circ}$<\/p>\n\n\n\n<p>$y = 110^{\\circ}$<\/p>\n\n\n\n<p><strong>2. Find the value of x and y in the following figure.<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"270\" height=\"283\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/11\/a-cyclic-quadrilateral-PQRS-with-angle-measures-3x-y-x-2y.png\" alt=\"A cyclic quadrilateral PQRS with angle measures 3x, y, x, 2y\" class=\"wp-image-35967\" title=\"A cyclic quadrilateral PQRS with angle measures 3x, y, x, 2y\"\/><\/figure>\n\n\n\n<p><strong>Solution:&nbsp;<\/strong><\/p>\n\n\n\n<p>We know that opposite angles of a cyclic quadrilateral are supplementary.<\/p>\n\n\n\n<p>So, $\\angle P + \\angle R = 180^{\\circ}$ and $\\angle S + \\angle Q = 180^{\\circ}$<\/p>\n\n\n\n<p>$3x + x = 180^{\\circ}$ and $2y + y = 180^{\\circ}$<\/p>\n\n\n\n<p>$4x = 180^{\\circ}$ and $3y = 180^{\\circ}$<\/p>\n\n\n\n<p>$\\Rightarrow x = \\frac{180^{\\circ}}{4} = 45^{\\circ}$ and $y = \\frac{180^{\\circ}}{3} = 60^{\\circ}$<\/p>\n\n\n\n<p>So, $x = 45^{\\circ}$ and $y = 60^{\\circ}$<\/p>\n\n\n\n<p><strong>3. Find the value of <\/strong>$\\angle RQS$<strong> in the following figure.<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"259\" height=\"268\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/11\/cyclic-quadrilateral-PQRS-with-missing-angles.png\" alt=\"Cyclic quadrilateral PQRS with missing angles\" class=\"wp-image-35968\" title=\"Cyclic quadrilateral PQRS with missing angles\"\/><\/figure>\n\n\n\n<p><strong>Solution:<\/strong>&nbsp;<\/p>\n\n\n\n<p>We know that opposite angles of a cyclic quadrilateral are supplementary.<\/p>\n\n\n\n<p>So, $\\angle PSR + \\angle RQP = 180^{\\circ}$<\/p>\n\n\n\n<p>$82^{\\circ} + \\angle RQP = 180^{\\circ}$<\/p>\n\n\n\n<p>$\\angle RQP = 180^{\\circ} \\;\u2013\\; 82^{\\circ} = 98^{\\circ}$<\/p>\n\n\n\n<p>Now,&nbsp;<\/p>\n\n\n\n<p>$\\angle PQS + \\angle RQS = \\angle RQP$<\/p>\n\n\n\n<p>$55^{\\circ} + \\angle RQS = 98^{\\circ}$<\/p>\n\n\n\n<p>$\\angle RQS = 98^{\\circ} \\;\u2013\\; 55^{\\circ}$<\/p>\n\n\n\n<p>$\\angle RQS = 43^{\\circ}$<\/p>\n\n\n\n<p><strong>4. Find the value of <\/strong>$\\angle BCD$<strong> in the following diagram.&nbsp;<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"275\" height=\"265\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/11\/cyclic-quadrilateral-ABCD-with-missing-angles.png\" alt=\"Cyclic quadrilateral ABCD with missing angles\" class=\"wp-image-35969\" title=\"Cyclic quadrilateral ABCD with missing angles\"\/><\/figure>\n\n\n\n<p><strong>Solution:<\/strong><\/p>\n\n\n\n<p>$\\angle DAC = \\angle DBC$ (Angles in the same segment are equal)<\/p>\n\n\n\n<p>Thus, $\\angle DAC = 70^{\\circ}$<\/p>\n\n\n\n<p>$\\angle DAB = \\angle DAC + \\angle BAC$<\/p>\n\n\n\n<p>$\\angle DAB = 70^{\\circ} + 30^{\\circ} = 100^{\\circ}$<\/p>\n\n\n\n<p>We know that opposite angles of a cyclic quadrilateral are supplementary.<\/p>\n\n\n\n<p>$\\angle DAB + \\angle DCB = 180^{\\circ}$<\/p>\n\n\n\n<p>$100^{\\circ} + \\angle DCB = 180^{\\circ}$<\/p>\n\n\n\n<p>$\\angle DCB = 180^{\\circ}\\; \u2013 \\;100^{\\circ}$<\/p>\n\n\n\n<p>$\\angle DCB = 80^{\\circ}$<\/p>\n\n\n\n<p><strong>5. Find the area of a cyclic quadrilateral whose sides are 2 inches, 4 inches, 10 inches, and 12 inches.<\/strong><\/p>\n\n\n\n<p><strong>Solution:<\/strong><\/p>\n\n\n\n<p>Let ABCD be the cyclic quadrilateral.<\/p>\n\n\n\n<p>$a = 2,\\; b = 4,\\; c = 10,\\; d = 12$<\/p>\n\n\n\n<p>Semi-perimeter $= s = \\frac{a + b + c + d}{2} = \\frac{2 + 4 + 10 + 12}{2} = \\frac{28}{2} = 14$<\/p>\n\n\n\n<p>$Area = \\sqrt{(s\\;-\\;a)(s\\;-\\;b)(s\\;-\\;c)(s\\;-\\;d)}$<\/p>\n\n\n\n<p>$= \\sqrt{(14\\;-\\;2)(14\\;-\\;4)(14\\;-\\;10)(14\\;-\\;12)}$<\/p>\n\n\n\n<p>$= \\sqrt{12\\times10\\times4\\times2}$<\/p>\n\n\n\n<p>$= \\sqrt{2\\times2\\times3\\times2\\times5\\times2\\times2\\times2}$<\/p>\n\n\n\n<p>$= 8\\sqrt{15}$ square inches<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"9-practice-problems-on-cyclic-quadrilateral\">Practice Problems on Cyclic Quadrilateral<\/h2>\n\n\n\n<div class=\"spq_wrapper\"><h2 style=\"display:none;\">Cyclic Quadrilateral \u2013 Definition, Theorem, Examples, FAQs<\/h2><p style=\"display:none;\">Attend this quiz & Test your knowledge.<\/p><div class=\"spq_question_wrapper\" data-answer=\"1\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">1<\/span><h3 class=\"sqp_question_text\">Find the value of x in the following figure.<\/h3><\/span><div class=\"spq_question_image\"><img decoding=\"async\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/06\/Practice-Problems-1-2.png\"><\/div><div class=\"spq_answer_block\" data-value=\"0\">$27^{\\circ}$<\/div><div class=\"spq_answer_block\" data-value=\"1\">$107^{\\circ}$<\/div><div class=\"spq_answer_block\" data-value=\"2\">$180^{\\circ}$<\/div><div class=\"spq_answer_block\" data-value=\"3\">$187^{\\circ}$<\/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: $107^{\\circ}$<br\/>We know that opposite angles of a cyclic quadrilateral are supplementary.<br>\r\nSo, $\\angle ABC + \\angle ADC = 180^{\\circ}$<br>\r\n$x + 73^{\\circ} = 180^{\\circ}$<br>\r\n$\\angle DCB = 180^{\\circ} \\;\u2013\\; 73^{\\circ} = 107^{\\circ}$<\/span><\/div><\/div><\/div><div class=\"spq_question_wrapper\" data-answer=\"0\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">2<\/span><h3 class=\"sqp_question_text\">What is the value of $\\angle BCD$ in the following figure?<\/h3><\/span><div class=\"spq_question_image\"><img decoding=\"async\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/06\/Practice-Problems-2-2.png\"><\/div><div class=\"spq_answer_block\" data-value=\"0\">$80^{\\circ}$<\/div><div class=\"spq_answer_block\" data-value=\"1\">$100^{\\circ}$<\/div><div class=\"spq_answer_block\" data-value=\"2\">$120^{\\circ}$<\/div><div class=\"spq_answer_block\" data-value=\"3\">$10^{\\circ}$<\/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: $80^{\\circ}$<br\/>$\\angle CBD = \\angle CAD$ (Angles in the same segment are equal)<br>\r\n$\\angle CAD = 40^{\\circ}$<br>\r\n$\\angle BAD = \\angle BAC + \\angle CAD$<br>\r\n$\\angle BAD = 60^{\\circ} + 40^{\\circ}  = 100^{\\circ}$<br>\r\nWe know that opposite angles of a cyclic quadrilateral are supplementary.<br>\r\n$\\angle BAD + \\angle BCD = 180^{\\circ}$<br>\r\n$100^{\\circ} + \\angle BCD = 180^{\\circ}$<br>\r\n$\\angle BCD = 180^{\\circ}\\;\u2013\\;100^{\\circ} = 80^{\\circ}$<br>\r\n$\\angle  BCD = 180^{\\circ}\\;\u2013\\;110^{\\circ} = 70^{\\circ}$<\/span><\/div><\/div><\/div><div class=\"spq_question_wrapper\" data-answer=\"2\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">3<\/span><h3 class=\"sqp_question_text\">What is the value of x in the following figure?<\/h3><\/span><div class=\"spq_question_image\"><img decoding=\"async\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/06\/Practice-Problems-4.png\"><\/div><div class=\"spq_answer_block\" data-value=\"0\">$6^{\\circ}$<\/div><div class=\"spq_answer_block\" data-value=\"1\">$26^{\\circ}$<\/div><div class=\"spq_answer_block\" data-value=\"2\">$16^{\\circ}$<\/div><div class=\"spq_answer_block\" data-value=\"3\">$56^{\\circ}$<\/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: $16^{\\circ}$<br\/>We know that opposite angles of a cyclic quadrilateral are supplementary.<br>\r\n$\\angle SPQ + \\angle SRQ = 180^{\\circ}$<br>\r\n$4x + 124^{\\circ} = 180^{\\circ}$<br>\r\n$4x = 180^{\\circ} \\;\u2013\\; 124^{\\circ}$<br>\r\n$4x = 56^{\\circ}$<br>\r\n$x = 16^{\\circ}$<\/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\": \"Cyclic Quadrilateral \u2013 Definition, Theorem, Examples, FAQs\",        \n        \"about\": {\n                \"@type\": \"Thing\",\n                \"name\": \"Cyclic Quadrilateral\"\n        },  \n        \"hasPart\": [{\n                    \"@type\": \"Question\",   \n                    \"eduQuestionType\": \"Multiple choice\",\n                    \"learningResourceType\": \"Practice problem\",\n                    \"name\": \"Find the value of x in the following figure.\",\n                    \"text\": \"Find the value of x in the following figure. <img src=\\\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/06\/Practice-Problems-1-2.png\\\"\/>\",\n                    \"comment\": {\n                      \"@type\": \"Comment\",\n                      \"text\": \"We know that opposite angles of a cyclic quadrilateral are supplementary.<br>\r\nSo, $$\\\\angle ABC + \\\\angle ADC = 180^{\\\\circ}$$<br>\r\n$$x + 73^{\\\\circ} = 180^{\\\\circ}$$<br>\r\n$$\\\\angle DCB = 180^{\\\\circ} \\\\;\u2013\\\\; 73^{\\\\circ} = 107^{\\\\circ}$$\"\n                    },\n                    \"encodingFormat\": \"text\/html\",\n                    \"suggestedAnswer\": [ {\n                                \"@type\": \"Answer\",\n                                \"position\": 0,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"$$27^{\\\\circ}$$\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"We know that opposite angles of a cyclic quadrilateral are supplementary.<br>\r\nSo, $$\\\\angle ABC + \\\\angle ADC = 180^{\\\\circ}$$<br>\r\n$$x + 73^{\\\\circ} = 180^{\\\\circ}$$<br>\r\n$$\\\\angle DCB = 180^{\\\\circ} \\\\;\u2013\\\\; 73^{\\\\circ} = 107^{\\\\circ}$$\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 2,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"$$180^{\\\\circ}$$\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"We know that opposite angles of a cyclic quadrilateral are supplementary.<br>\r\nSo, $$\\\\angle ABC + \\\\angle ADC = 180^{\\\\circ}$$<br>\r\n$$x + 73^{\\\\circ} = 180^{\\\\circ}$$<br>\r\n$$\\\\angle DCB = 180^{\\\\circ} \\\\;\u2013\\\\; 73^{\\\\circ} = 107^{\\\\circ}$$\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 3,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"$$187^{\\\\circ}$$\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"We know that opposite angles of a cyclic quadrilateral are supplementary.<br>\r\nSo, $$\\\\angle ABC + \\\\angle ADC = 180^{\\\\circ}$$<br>\r\n$$x + 73^{\\\\circ} = 180^{\\\\circ}$$<br>\r\n$$\\\\angle DCB = 180^{\\\\circ} \\\\;\u2013\\\\; 73^{\\\\circ} = 107^{\\\\circ}$$\"\n                                    }\n                                }],\n                    \"acceptedAnswer\": {\n                      \"@type\": \"Answer\",\n                      \"position\": 1,\n                      \"encodingFormat\": \"text\/html\",\n                      \"text\": \"$$107^{\\\\circ}$$\",\n                      \"comment\": {\n                          \"@type\": \"Comment\",\n                          \"text\": \"We know that opposite angles of a cyclic quadrilateral are supplementary.<br>\r\nSo, $$\\\\angle ABC + \\\\angle ADC = 180^{\\\\circ}$$<br>\r\n$$x + 73^{\\\\circ} = 180^{\\\\circ}$$<br>\r\n$$\\\\angle DCB = 180^{\\\\circ} \\\\;\u2013\\\\; 73^{\\\\circ} = 107^{\\\\circ}$$\"\n                        },\n                      \"answerExplanation\": {\n                        \"@type\": \"Comment\",\n                        \"text\": \"We know that opposite angles of a cyclic quadrilateral are supplementary.<br>\r\nSo, $$\\\\angle ABC + \\\\angle ADC = 180^{\\\\circ}$$<br>\r\n$$x + 73^{\\\\circ} = 180^{\\\\circ}$$<br>\r\n$$\\\\angle DCB = 180^{\\\\circ} \\\\;\u2013\\\\; 73^{\\\\circ} = 107^{\\\\circ}$$\"\n                      }\n                    } \n\n                    },{\n                    \"@type\": \"Question\",   \n                    \"eduQuestionType\": \"Multiple choice\",\n                    \"learningResourceType\": \"Practice problem\",\n                    \"name\": \"What is the value of $$\\\\angle BCD$$ in the following figure?\",\n                    \"text\": \"What is the value of $$\\\\angle BCD$$ in the following figure? <img src=\\\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/06\/Practice-Problems-2-2.png\\\"\/>\",\n                    \"comment\": {\n                      \"@type\": \"Comment\",\n                      \"text\": \"$$\\\\angle CBD = \\\\angle CAD$$ (Angles in the same segment are equal)<br>\r\n$$\\\\angle CAD = 40^{\\\\circ}$$<br>\r\n$$\\\\angle BAD = \\\\angle BAC + \\\\angle CAD$$<br>\r\n$$\\\\angle BAD = 60^{\\\\circ} + 40^{\\\\circ}  = 100^{\\\\circ}$$<br>\r\nWe know that opposite angles of a cyclic quadrilateral are supplementary.<br>\r\n$$\\\\angle BAD + \\\\angle BCD = 180^{\\\\circ}$$<br>\r\n$$100^{\\\\circ} + \\\\angle BCD = 180^{\\\\circ}$$<br>\r\n$$\\\\angle BCD = 180^{\\\\circ}\\\\;\u2013\\\\;100^{\\\\circ} = 80^{\\\\circ}$$<br>\r\n$$\\\\angle  BCD = 180^{\\\\circ}\\\\;\u2013\\\\;110^{\\\\circ} = 70^{\\\\circ}$$\"\n                    },\n                    \"encodingFormat\": \"text\/html\",\n                    \"suggestedAnswer\": [ {\n                                \"@type\": \"Answer\",\n                                \"position\": 1,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"$$100^{\\\\circ}$$\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"$$\\\\angle CBD = \\\\angle CAD$$ (Angles in the same segment are equal)<br>\r\n$$\\\\angle CAD = 40^{\\\\circ}$$<br>\r\n$$\\\\angle BAD = \\\\angle BAC + \\\\angle CAD$$<br>\r\n$$\\\\angle BAD = 60^{\\\\circ} + 40^{\\\\circ}  = 100^{\\\\circ}$$<br>\r\nWe know that opposite angles of a cyclic quadrilateral are supplementary.<br>\r\n$$\\\\angle BAD + \\\\angle BCD = 180^{\\\\circ}$$<br>\r\n$$100^{\\\\circ} + \\\\angle BCD = 180^{\\\\circ}$$<br>\r\n$$\\\\angle BCD = 180^{\\\\circ}\\\\;\u2013\\\\;100^{\\\\circ} = 80^{\\\\circ}$$<br>\r\n$$\\\\angle  BCD = 180^{\\\\circ}\\\\;\u2013\\\\;110^{\\\\circ} = 70^{\\\\circ}$$\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 2,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"$$120^{\\\\circ}$$\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"$$\\\\angle CBD = \\\\angle CAD$$ (Angles in the same segment are equal)<br>\r\n$$\\\\angle CAD = 40^{\\\\circ}$$<br>\r\n$$\\\\angle BAD = \\\\angle BAC + \\\\angle CAD$$<br>\r\n$$\\\\angle BAD = 60^{\\\\circ} + 40^{\\\\circ}  = 100^{\\\\circ}$$<br>\r\nWe know that opposite angles of a cyclic quadrilateral are supplementary.<br>\r\n$$\\\\angle BAD + \\\\angle BCD = 180^{\\\\circ}$$<br>\r\n$$100^{\\\\circ} + \\\\angle BCD = 180^{\\\\circ}$$<br>\r\n$$\\\\angle BCD = 180^{\\\\circ}\\\\;\u2013\\\\;100^{\\\\circ} = 80^{\\\\circ}$$<br>\r\n$$\\\\angle  BCD = 180^{\\\\circ}\\\\;\u2013\\\\;110^{\\\\circ} = 70^{\\\\circ}$$\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 3,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"$$10^{\\\\circ}$$\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"$$\\\\angle CBD = \\\\angle CAD$$ (Angles in the same segment are equal)<br>\r\n$$\\\\angle CAD = 40^{\\\\circ}$$<br>\r\n$$\\\\angle BAD = \\\\angle BAC + \\\\angle CAD$$<br>\r\n$$\\\\angle BAD = 60^{\\\\circ} + 40^{\\\\circ}  = 100^{\\\\circ}$$<br>\r\nWe know that opposite angles of a cyclic quadrilateral are supplementary.<br>\r\n$$\\\\angle BAD + \\\\angle BCD = 180^{\\\\circ}$$<br>\r\n$$100^{\\\\circ} + \\\\angle BCD = 180^{\\\\circ}$$<br>\r\n$$\\\\angle BCD = 180^{\\\\circ}\\\\;\u2013\\\\;100^{\\\\circ} = 80^{\\\\circ}$$<br>\r\n$$\\\\angle  BCD = 180^{\\\\circ}\\\\;\u2013\\\\;110^{\\\\circ} = 70^{\\\\circ}$$\"\n                                    }\n                                }],\n                    \"acceptedAnswer\": {\n                      \"@type\": \"Answer\",\n                      \"position\": 0,\n                      \"encodingFormat\": \"text\/html\",\n                      \"text\": \"$$80^{\\\\circ}$$\",\n                      \"comment\": {\n                          \"@type\": \"Comment\",\n                          \"text\": \"$$\\\\angle CBD = \\\\angle CAD$$ (Angles in the same segment are equal)<br>\r\n$$\\\\angle CAD = 40^{\\\\circ}$$<br>\r\n$$\\\\angle BAD = \\\\angle BAC + \\\\angle CAD$$<br>\r\n$$\\\\angle BAD = 60^{\\\\circ} + 40^{\\\\circ}  = 100^{\\\\circ}$$<br>\r\nWe know that opposite angles of a cyclic quadrilateral are supplementary.<br>\r\n$$\\\\angle BAD + \\\\angle BCD = 180^{\\\\circ}$$<br>\r\n$$100^{\\\\circ} + \\\\angle BCD = 180^{\\\\circ}$$<br>\r\n$$\\\\angle BCD = 180^{\\\\circ}\\\\;\u2013\\\\;100^{\\\\circ} = 80^{\\\\circ}$$<br>\r\n$$\\\\angle  BCD = 180^{\\\\circ}\\\\;\u2013\\\\;110^{\\\\circ} = 70^{\\\\circ}$$\"\n                        },\n                      \"answerExplanation\": {\n                        \"@type\": \"Comment\",\n                        \"text\": \"$$\\\\angle CBD = \\\\angle CAD$$ (Angles in the same segment are equal)<br>\r\n$$\\\\angle CAD = 40^{\\\\circ}$$<br>\r\n$$\\\\angle BAD = \\\\angle BAC + \\\\angle CAD$$<br>\r\n$$\\\\angle BAD = 60^{\\\\circ} + 40^{\\\\circ}  = 100^{\\\\circ}$$<br>\r\nWe know that opposite angles of a cyclic quadrilateral are supplementary.<br>\r\n$$\\\\angle BAD + \\\\angle BCD = 180^{\\\\circ}$$<br>\r\n$$100^{\\\\circ} + \\\\angle BCD = 180^{\\\\circ}$$<br>\r\n$$\\\\angle BCD = 180^{\\\\circ}\\\\;\u2013\\\\;100^{\\\\circ} = 80^{\\\\circ}$$<br>\r\n$$\\\\angle  BCD = 180^{\\\\circ}\\\\;\u2013\\\\;110^{\\\\circ} = 70^{\\\\circ}$$\"\n                      }\n                    } \n\n                    },{\n                    \"@type\": \"Question\",   \n                    \"eduQuestionType\": \"Multiple choice\",\n                    \"learningResourceType\": \"Practice problem\",\n                    \"name\": \"What is the value of x in the following figure?\",\n                    \"text\": \"What is the value of x in the following figure? <img src=\\\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/06\/Practice-Problems-4.png\\\"\/>\",\n                    \"comment\": {\n                      \"@type\": \"Comment\",\n                      \"text\": \"We know that opposite angles of a cyclic quadrilateral are supplementary.<br>\r\n$$\\\\angle SPQ + \\\\angle SRQ = 180^{\\\\circ}$$<br>\r\n$$4x + 124^{\\\\circ} = 180^{\\\\circ}$$<br>\r\n$$4x = 180^{\\\\circ} \\\\;\u2013\\\\; 124^{\\\\circ}$$<br>\r\n$$4x = 56^{\\\\circ}$$<br>\r\n$$x = 16^{\\\\circ}$$\"\n                    },\n                    \"encodingFormat\": \"text\/html\",\n                    \"suggestedAnswer\": [ {\n                                \"@type\": \"Answer\",\n                                \"position\": 0,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"$$6^{\\\\circ}$$\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"We know that opposite angles of a cyclic quadrilateral are supplementary.<br>\r\n$$\\\\angle SPQ + \\\\angle SRQ = 180^{\\\\circ}$$<br>\r\n$$4x + 124^{\\\\circ} = 180^{\\\\circ}$$<br>\r\n$$4x = 180^{\\\\circ} \\\\;\u2013\\\\; 124^{\\\\circ}$$<br>\r\n$$4x = 56^{\\\\circ}$$<br>\r\n$$x = 16^{\\\\circ}$$\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 1,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"$$26^{\\\\circ}$$\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"We know that opposite angles of a cyclic quadrilateral are supplementary.<br>\r\n$$\\\\angle SPQ + \\\\angle SRQ = 180^{\\\\circ}$$<br>\r\n$$4x + 124^{\\\\circ} = 180^{\\\\circ}$$<br>\r\n$$4x = 180^{\\\\circ} \\\\;\u2013\\\\; 124^{\\\\circ}$$<br>\r\n$$4x = 56^{\\\\circ}$$<br>\r\n$$x = 16^{\\\\circ}$$\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 3,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"$$56^{\\\\circ}$$\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"We know that opposite angles of a cyclic quadrilateral are supplementary.<br>\r\n$$\\\\angle SPQ + \\\\angle SRQ = 180^{\\\\circ}$$<br>\r\n$$4x + 124^{\\\\circ} = 180^{\\\\circ}$$<br>\r\n$$4x = 180^{\\\\circ} \\\\;\u2013\\\\; 124^{\\\\circ}$$<br>\r\n$$4x = 56^{\\\\circ}$$<br>\r\n$$x = 16^{\\\\circ}$$\"\n                                    }\n                                }],\n                    \"acceptedAnswer\": {\n                      \"@type\": \"Answer\",\n                      \"position\": 2,\n                      \"encodingFormat\": \"text\/html\",\n                      \"text\": \"$$16^{\\\\circ}$$\",\n                      \"comment\": {\n                          \"@type\": \"Comment\",\n                          \"text\": \"We know that opposite angles of a cyclic quadrilateral are supplementary.<br>\r\n$$\\\\angle SPQ + \\\\angle SRQ = 180^{\\\\circ}$$<br>\r\n$$4x + 124^{\\\\circ} = 180^{\\\\circ}$$<br>\r\n$$4x = 180^{\\\\circ} \\\\;\u2013\\\\; 124^{\\\\circ}$$<br>\r\n$$4x = 56^{\\\\circ}$$<br>\r\n$$x = 16^{\\\\circ}$$\"\n                        },\n                      \"answerExplanation\": {\n                        \"@type\": \"Comment\",\n                        \"text\": \"We know that opposite angles of a cyclic quadrilateral are supplementary.<br>\r\n$$\\\\angle SPQ + \\\\angle SRQ = 180^{\\\\circ}$$<br>\r\n$$4x + 124^{\\\\circ} = 180^{\\\\circ}$$<br>\r\n$$4x = 180^{\\\\circ} \\\\;\u2013\\\\; 124^{\\\\circ}$$<br>\r\n$$4x = 56^{\\\\circ}$$<br>\r\n$$x = 16^{\\\\circ}$$\"\n                      }\n                    } \n\n                    }]}<\/script>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"10-frequently-asked-questions-on-cyclic-quadrilateral\">Frequently Asked Questions on Cyclic Quadrilateral<\/h2>\n\n\n<div class=\"wp-block-ub-content-toggle\" id=\"ub-content-toggle-380888c5-1212-4b92-a096-f869109a9f74\" 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-0-380888c5-1212-4b92-a096-f869109a9f74\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-380888c5-1212-4b92-a096-f869109a9f74\"><strong>Is every quadrilateral cyclic?<\/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-0-380888c5-1212-4b92-a096-f869109a9f74\">\n\n<p>No. Not every quadrilateral is cyclic. Examples of non-cyclic quadrilaterals are non-square rhombus (<a href=\"https:\/\/www.splashlearn.com\/math-vocabulary\/geometry\/rhombus\">rhombus<\/a> that\u2019s not a square) and non-rectangular parallelogram.<\/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-1-380888c5-1212-4b92-a096-f869109a9f74\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-380888c5-1212-4b92-a096-f869109a9f74\"><strong>Is a parallelogram a cyclic quadrilateral?<\/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-1-380888c5-1212-4b92-a096-f869109a9f74\">\n\n<p>If a parallelogram is cyclic, it must be a rectangle. A cyclic parallelogram is always a rectangle. A non-rectangular parallelogram is not cyclic.<\/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-2-380888c5-1212-4b92-a096-f869109a9f74\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-380888c5-1212-4b92-a096-f869109a9f74\"><strong>Do angles in a cyclic quadrilateral add up to <\/strong>$360^{\\circ}$<strong>?<\/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-2-380888c5-1212-4b92-a096-f869109a9f74\">\n\n<p>Yes, angles in a cyclic quadrilateral add up to $360^{\\circ}$ as the sum of opposite angles is $180^{\\circ}$. So, $180^{\\circ} + 180^{\\circ} = 360^{\\circ}$.<\/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-3-380888c5-1212-4b92-a096-f869109a9f74\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-380888c5-1212-4b92-a096-f869109a9f74\"><strong>Is a rhombus a cyclic quadrilateral?<\/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-3-380888c5-1212-4b92-a096-f869109a9f74\">\n\n<p>A non-square rhombus is not cyclic. A rhombus cannot be a cyclic quadrilateral because the opposite angles of a cyclic quadrilateral are supplementary, but in the case of a rhombus, the opposite angles are equal. A cyclic rhombus is a <a href=\"https:\/\/www.splashlearn.com\/math-vocabulary\/geometry\/square\">square<\/a>.<\/p>\n\n<\/div><\/div>\n<\/div>","protected":false},"excerpt":{"rendered":"<p>What Is a Cyclic Quadrilateral? A cyclic quadrilateral is a quadrilateral whose all four vertices lie on a circle. It is also called an inscribed quadrilateral.&nbsp; The circle that passes through the four vertices of a quadrilateral is called a circumcircle. The vertices of a cyclic quadrilateral are said to be concyclic since they lie &#8230; <a title=\"Cyclic Quadrilateral \u2013 Definition, Theorem, Examples, FAQs\" class=\"read-more\" href=\"https:\/\/www.splashlearn.com\/math-vocabulary\/cyclic-quadrilaterals\" aria-label=\"More on Cyclic Quadrilateral \u2013 Definition, Theorem, 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-30981","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\/30981","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=30981"}],"version-history":[{"count":19,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/posts\/30981\/revisions"}],"predecessor-version":[{"id":35970,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/posts\/30981\/revisions\/35970"}],"wp:attachment":[{"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/media?parent=30981"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/categories?post=30981"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/tags?post=30981"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}