{"id":29566,"date":"2023-05-29T04:13:59","date_gmt":"2023-05-29T04:13:59","guid":{"rendered":"https:\/\/www.splashlearn.com\/math-vocabulary\/?page_id=29566"},"modified":"2023-11-15T15:43:28","modified_gmt":"2023-11-15T15:43:28","slug":"angle-bisector-theorem-definition-formula-proof-examples","status":"publish","type":"post","link":"https:\/\/www.splashlearn.com\/math-vocabulary\/angle-bisector-theorem","title":{"rendered":"Angle Bisector Theorem: Definition, Formula, Proof, Examples"},"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-16bc52e2-9267-4338-98a6-ab2d8fcf0bf9\" 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\/angle-bisector-theorem#0-what-is-angle-bisector-theorem>What Is Angle Bisector Theorem?<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/angle-bisector-theorem#2-interior-angle-bisector-theorem>Interior Angle Bisector Theorem<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/angle-bisector-theorem#3-external-angle-bisector-theorem>External Angle Bisector Theorem<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/angle-bisector-theorem#8-solved-examples-on-angle-bisector-theorem>Solved Examples on Angle Bisector Theorem<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/angle-bisector-theorem#9-practice-problems-on-angle-bisector-theorem>Practice Problems on Angle Bisector Theorem<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/angle-bisector-theorem#10-frequently-asked-questions-on-angle-bisector-theorem>Frequently Asked Questions on Angle Bisector Theorem<\/a><\/li><\/ul><\/div><\/div><\/div>\n\n\n<h2 class=\"wp-block-heading\" id=\"0-what-is-angle-bisector-theorem\">What Is Angle Bisector Theorem?<\/h2>\n\n\n\n<p><strong>Angle bisector theorem states that an angle bisector divides the opposite side into two line segments that are proportional to the other two sides.<\/strong><\/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\/angle-bisector-theorem.png\" alt=\"Angle bisector theorem\" class=\"wp-image-35731\" title=\"Angle bisector theorem\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/11\/angle-bisector-theorem.png 620w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/11\/angle-bisector-theorem-300x257.png 300w\" sizes=\"auto, (max-width: 620px) 100vw, 620px\" \/><\/figure>\n\n\n\n<p>Here, in $\\Delta ABC$, the line AD is the angle bisector of $\\angle A$.<\/p>\n\n\n\n<p>AD bisects the side BC in two parts, c and d.<\/p>\n\n\n\n<p>a and b are the lengths of the other two sides.<\/p>\n\n\n\n<p>By the angle bisector theorem, we have<\/p>\n\n\n\n<p>$\\frac{a}{b} = \\frac{c}{d}$<\/p>\n\n\n\n<p>Consider an example:<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"620\" height=\"278\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/11\/example-on-application-of-angle-bisector-theorem.png\" alt=\"Example on application of angle bisector theorem\" class=\"wp-image-35732\" title=\"Example on application of angle bisector theorem\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/11\/example-on-application-of-angle-bisector-theorem.png 620w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/11\/example-on-application-of-angle-bisector-theorem-300x135.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\/add-the-angles\" 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_add_angles_pt.png\" alt=\"Add the Angles 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\">Add the Angles 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\/answer-questions-related-to-triangles\" 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_classify_tri_on_side_pt.png\" alt=\"Answer Questions Related to Triangles 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\">Answer Questions Related to Triangles 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\/classify-triangles-and-rectangles-as-closed-shape\" 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_triangle_rectangle_2_pt.png\" alt=\"Classify Triangles and Rectangles as Closed Shape 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 Triangles and Rectangles as Closed Shape 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\/classify-triangles\" 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_classify_triangles_pt.png\" alt=\"Classify Triangles 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 Triangles 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\/draw-angles-in-multiples-of-10-degrees\" 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_draw_angle_1_pt.png\" alt=\"Draw Angles in Multiples of 10 Degrees 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\">Draw Angles in Multiples of 10 Degrees 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\/draw-angles-nearest-5-and-1-degrees\" 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_draw_angle_2_pt.png\" alt=\"Draw Angles Nearest 5 and 1 Degrees 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\">Draw Angles Nearest 5 and 1 Degrees 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\/draw-angles-using-a-protractor\" 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_draw_angle_protractor_pt.png\" alt=\"Draw Angles Using a Protractor 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\">Draw Angles Using a Protractor 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-right-angles\" 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_identify_right_angle_1_pt.png\" alt=\"Find Right Angles 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 Right Angles 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-squared-and-the-rectangles\" 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_2_pt.png\" alt=\"Find the Perimeter of the Squared and the Rectangles 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 Squared and the Rectangles 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-angles-by-their-types\" 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_types_of_angle_pt.png\" alt=\"Identify Angles by their Types 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 Angles by their Types 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-angle-bisector-theorem-statement\">Angle Bisector Theorem Statement<\/h2>\n\n\n\n<p><strong>The bisector of an interior angle of a triangle divides the opposite side internally in the ratio of the <\/strong><a href=\"https:\/\/www.splashlearn.com\/math-vocabulary\/corresponding-sides\"><strong>corresponding sides<\/strong><\/a><strong> containing the <\/strong><a href=\"https:\/\/www.splashlearn.com\/math-vocabulary\/geometry\/angle\"><strong>angle<\/strong><\/a><strong>.<\/strong><\/p>\n\n\n\n<div id=\"recommended-worksheets-container-id\" class=\"recommended-games-container\"><h4 class=\"recommended-games-container-headline\">Recommended Worksheets<\/h4><div class=\"recommended-games-container-slides\"><div class=\"worksheet-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-worksheets\/count-sides-and-angles\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"worksheets_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t    <div class=\"worksheet-card-container-inner-block\">\r\n\t        <img decoding=\"async\" class=\"worksheet-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/cms_assets\/s\/math-worksheets\/count-sides-and-angles.jpeg\" alt=\"Count Sides and Angles Worksheet\">\r\n\t    <\/div>\r\n\t\t<div class=\"worksheet-card-container-inner\" >\r\n\t\t<\/div><\/a>\r\n<\/div><div class=\"worksheet-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-worksheets\/rectangles-and-triangles\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"worksheets_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t    <div class=\"worksheet-card-container-inner-block\">\r\n\t        <img decoding=\"async\" class=\"worksheet-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/cms_assets\/s\/math-worksheets\/rectangles-and-triangles.jpeg\" alt=\"Rectangles and Triangles Worksheet\">\r\n\t    <\/div>\r\n\t\t<div class=\"worksheet-card-container-inner\" >\r\n\t\t<\/div><\/a>\r\n<\/div><div class=\"worksheet-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-worksheets\/separate-out-the-triangles\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"worksheets_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t    <div class=\"worksheet-card-container-inner-block\">\r\n\t        <img decoding=\"async\" class=\"worksheet-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/cms_assets\/s\/math-worksheets\/separate-out-the-triangles.jpeg\" alt=\"Separate Out the Triangles Worksheet\">\r\n\t    <\/div>\r\n\t\t<div class=\"worksheet-card-container-inner\" >\r\n\t\t<\/div><\/a>\r\n<\/div><\/div><p class=\"recommended-games-container-desc\"><a href=\"https:\/\/www.splashlearn.com\/worksheets\">More Worksheets<\/a><\/p><button class=\"scroll-right-arrow\"><\/button><\/div><script type=\"text\/javascript\">\n        document.addEventListener(\"DOMContentLoaded\", function() {\n            const container = document.querySelector(\"#recommended-worksheets-container-id\");\n            const slidesContainer = container.querySelector(\".recommended-games-container-slides\");\n            const cards = slidesContainer.querySelectorAll(\".worksheet-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=\"2-interior-angle-bisector-theorem\">Interior Angle Bisector Theorem<\/h2>\n\n\n\n<p>The interior angle bisector theorem is simply referred to as angle bisector theorem. Hence, an interior angle bisector of a triangle divides the opposite side in two parts such that the ratio of the two line segments is proportional to the ratio of the other two sides.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"620\" height=\"325\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/11\/interior-angle-bisector-theorem.png\" alt=\"interior angle bisector theorem\" class=\"wp-image-35733\" title=\"interior angle bisector theorem\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/11\/interior-angle-bisector-theorem.png 620w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/11\/interior-angle-bisector-theorem-300x157.png 300w\" sizes=\"auto, (max-width: 620px) 100vw, 620px\" \/><\/figure>\n\n\n\n<p>Let us consider $\\Delta ABC$ as shown above. Here, AD is the angle bisector (interior angle bisector) of $\\angle BAC$, which meets the opposite side BC at point D. Hence, as per this theorem, the ratio of the BD to DC is equal to the ratio of the lengths of other sides, AB to AC.<\/p>\n\n\n\n<p>$\\frac{BD}{DC} = \\frac{AB}{AC}$.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"3-external-angle-bisector-theorem\">External Angle Bisector Theorem<\/h2>\n\n\n\n<p>A triangle&#8217;s external angle bisector divides one side externally in proportion to the ratio of the sides containing the angle. Most of the time, non-equilateral triangles have this condition.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"620\" height=\"395\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/11\/exterior-angle-bisector-theorem.png\" alt=\"Exterior angle bisector theorem\" class=\"wp-image-35734\" title=\"Exterior angle bisector theorem\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/11\/exterior-angle-bisector-theorem.png 620w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/11\/exterior-angle-bisector-theorem-300x191.png 300w\" sizes=\"auto, (max-width: 620px) 100vw, 620px\" \/><\/figure>\n\n\n\n<p>Let us consider $\\Delta ABC$ as shown above.&nbsp;&nbsp;<\/p>\n\n\n\n<p>The side AB is extended. $\\angle CAP$ is an external angle.<\/p>\n\n\n\n<p>Here, AD is the angle bisector (exterior angle bisector) of $\\angle CAP$.&nbsp;<\/p>\n\n\n\n<p>CE is drawn parallel to DA, such that it meets the line AB of a triangle at E.&nbsp;<\/p>\n\n\n\n<p>Hence, according to the external angle bisector theorem:<\/p>\n\n\n\n<p>&nbsp;$\\frac{BD}{DC} = \\frac{AB}{AC}$<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"4-angle-bisector-theorem-proof\">Angle Bisector Theorem Proof<\/h2>\n\n\n\n<p><strong>Statement:<\/strong> The angle bisector of a triangle divides the opposite side in such a way that the ratio of the two line segments is proportional to the ratio of the other two sides.<\/p>\n\n\n\n<p>Let&#8217;s prove this statement.<\/p>\n\n\n\n<p><strong>Given:<\/strong> Let us consider $\\Delta ABC$, in which AD is the angle bisector of $\\angle BAC$, which meets BC at point D.<\/p>\n\n\n\n<p>To prove: $\\frac{BD}{DC} = \\frac{AB}{AC}$.<\/p>\n\n\n\n<p><strong>Construction:<\/strong> Draw a ray CE parallel to DA. Extend BA such that it intersects CE at E.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"620\" height=\"406\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/11\/angle-bisector-theorem-proof.png\" alt=\"Angle bisector theorem proof\" class=\"wp-image-35735\" title=\"Angle bisector theorem proof\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/11\/angle-bisector-theorem-proof.png 620w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/11\/angle-bisector-theorem-proof-300x196.png 300w\" sizes=\"auto, (max-width: 620px) 100vw, 620px\" \/><\/figure>\n\n\n\n<p>Since $CE || DA$, AC acts as the transversal.&nbsp;<\/p>\n\n\n\n<p>$\\angle DAC = \\angle ACE$ (alternate interior angles) \u2026(1)<\/p>\n\n\n\n<p>$\\angle BAD = \\angle AEC$ (corresponding angles) \u2026(2)<\/p>\n\n\n\n<p>As we know that AD is the angle bisector of $\\angle BAC$,<\/p>\n\n\n\n<p>$\\angle BAD = \\angle DAC$ \u2026(3)<\/p>\n\n\n\n<p>From (1), (2) and (3), we have<\/p>\n\n\n\n<p>$\\angle ACE = \\angle AEC$ this implies that $\\Delta ACE$ is an isosceles triangle.&nbsp;<\/p>\n\n\n\n<p>Therefore, opposite sides of equal angles are equal.<\/p>\n\n\n\n<p>Thus for $\\Delta ACE$, we can write,<\/p>\n\n\n\n<p>$AE = AC$ \u2026(4)<\/p>\n\n\n\n<p>Now, in $\\Delta BCE$ we have, $CE || DA$.<\/p>\n\n\n\n<p>By triangle proportionality theorem, a line parallel to one side of a triangle intersects the other two sides of the triangle, then the line divides these two sides proportionally.<\/p>\n\n\n\n<p>$\\frac{BD}{DC} = \\frac{AB}{AE}$ \u2026(5)<\/p>\n\n\n\n<p>From equation (4) we have $AE = AC$, so on replacing AE by AC in equation (5), we get<\/p>\n\n\n\n<p>$\\frac{BD}{DC} = \\frac{AB}{AC}$<\/p>\n\n\n\n<p>Hence, the angle bisector theorem is proved.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"5-converse-of-angle-bisector-theorem\">Converse of Angle Bisector Theorem<\/h2>\n\n\n\n<p><strong>The converse of the angle bisector theorem states that if the interior point of an angle of a triangle is equidistant from the two sides (arms) of a triangle, then that point lies on the angle bisector of the angle.<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"620\" height=\"501\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/11\/converse-of-angle-bisector-theorem.png\" alt=\"Converse of angle bisector theorem\" class=\"wp-image-35736\" title=\"Converse of angle bisector theorem\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/11\/converse-of-angle-bisector-theorem.png 620w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/11\/converse-of-angle-bisector-theorem-300x242.png 300w\" sizes=\"auto, (max-width: 620px) 100vw, 620px\" \/><\/figure>\n\n\n\n<p>D is a point in the interior of $\\angle BAC$.&nbsp;<\/p>\n\n\n\n<p>If the perpendicular distances BD and DC are equal then, the line AD is the angle bisector of angle $\\angle BAC$.<\/p>\n\n\n\n<p>If $BD = CD$, then $\\angle BAD = \\angle CAD$.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"620\" height=\"251\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/11\/converse-of-angle-bisector-theorem-.png\" alt=\"Converse of angle bisector theorem\" class=\"wp-image-35737\" title=\"Converse of angle bisector theorem\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/11\/converse-of-angle-bisector-theorem-.png 620w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/11\/converse-of-angle-bisector-theorem--300x121.png 300w\" sizes=\"auto, (max-width: 620px) 100vw, 620px\" \/><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"6-facts-about-angle-bisector-theorem\">Facts about Angle Bisector Theorem<\/h2>\n\n\n\n<div style=\"color:#0369a1;background-color:#e0f2fe\" class=\"wp-block-roelmagdaleno-callout-block has-text-color has-background is-layout-flex wp-container-roelmagdaleno-callout-block-is-layout-8cf370e7 wp-block-roelmagdaleno-callout-block-is-layout-flex\"><div>\n<ul class=\"wp-block-list\">\n<li>The points on an angle bisector are all equidistant from the arms of the angle.<\/li>\n\n\n\n<li>In real life, angle bisectors have applications in games of billiards, rugby, football, architecture, etc.<\/li>\n\n\n\n<li>Angle bisector theorem helps to find unknown lengths of sides of a triangle.<\/li>\n\n\n\n<li>The point of concurrency of the angle bisectors of a triangle is known as the incenter of a triangle.<\/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 angle bisector theorem, its proof, interior angle bisector and exterior angle bisector theorem, the converse of angle bisector theorem, and application of these theorems in problem-solving as well in real life.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"8-solved-examples-on-angle-bisector-theorem\">Solved Examples on Angle Bisector Theorem<\/h2>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>In the figure given below, find the ratio AD:DC.<\/strong><\/li>\n<\/ol>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"144\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/11\/angle-bisector-theorem-example.png\" alt=\"Angle bisector theorem example\" class=\"wp-image-35738\" title=\"Angle bisector theorem example\"\/><\/figure>\n\n\n\n<p><strong>Solution:<\/strong><\/p>\n\n\n\n<p>$AB = 6,\\; BC = 3$.<\/p>\n\n\n\n<p>Also, BD is the angle bisector.<\/p>\n\n\n\n<p>According to the angle bisector theorem, BD divides AC in the ratio proportional to the ratio of the other two sides.&nbsp;<\/p>\n\n\n\n<p>Thus, the ratio of AD to DC is the same as the ratio of AB to BC.<\/p>\n\n\n\n<p>$\\frac{AB}{BC} = \\frac{AD}{DC}$<\/p>\n\n\n\n<p>$\\Rightarrow \\frac{6}{3} = \\frac{AD}{DC}$<\/p>\n\n\n\n<p>$\\Rightarrow AD:DC = 6:3$<\/p>\n\n\n\n<p>$\\Rightarrow AD:DC = 2:1$<\/p>\n\n\n\n<ol class=\"wp-block-list\" start=\"2\">\n<li><strong>&nbsp;In the figure given below, if PS is the angle bisector, find the length of segments RS and QS.<\/strong><\/li>\n<\/ol>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"250\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/11\/converse-of-angle-bisector-theorem-example.png\" alt=\"Converse of angle bisector theorem example\" class=\"wp-image-35739\" title=\"Converse of angle bisector theorem example\"\/><\/figure>\n\n\n\n<p><strong>Solution:<\/strong><\/p>\n\n\n\n<p>RS $= 3x \\;-\\; 8,\\; SQ = 2x + 2$<\/p>\n\n\n\n<p>From the figure, point S lies on the bisector of angle QPR.<\/p>\n\n\n\n<p>Thus, the point S is equidistant from the sides PR and PQ.&nbsp;<\/p>\n\n\n\n<p>RS $= SQ$<\/p>\n\n\n\n<p>$\\Rightarrow 3x\\;-\\;8 = 2x + 2$<\/p>\n\n\n\n<p>$\\Rightarrow 3x\\;-\\;2x = 2 + 8$<\/p>\n\n\n\n<p>$x = 10$<\/p>\n\n\n\n<p>RS $= 3x\\;-\\;8 = 3(10)\\;-\\;8 = 22$<\/p>\n\n\n\n<p>SQ $= 2x + 2 = 2(10) + 2 = 22$<\/p>\n\n\n\n<p>$\\therefore$ RS $= SQ = 22$ units.<\/p>\n\n\n\n<ol class=\"wp-block-list\" start=\"3\">\n<li><strong>In <\/strong>$\\Delta PQR$<strong> as shown below, PS is the angle bisector of <\/strong>$\\angle P$<strong>. Find the lengths of sides of \u0394PQR.<\/strong><\/li>\n<\/ol>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"214\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/11\/angle-bisector-theorem-example-.png\" alt=\"Angle bisector theorem example\" class=\"wp-image-35740\" title=\"Angle bisector theorem example\"\/><\/figure>\n\n\n\n<p><strong>Solution:<\/strong><\/p>\n\n\n\n<p>According to the angle bisector theorem, $\\frac{PQ}{PR} = \\frac{QS}{SR}$<\/p>\n\n\n\n<p>$\\Rightarrow \\frac{x}{x \\;-\\; 2} = \\frac{x + 2}{x \\;-\\; 1}$<\/p>\n\n\n\n<p>$\\Rightarrow x(x\\;-\\;1) = (x\\;-\\;2)(x + 2)$<\/p>\n\n\n\n<p>$\\Rightarrow x^2\\;-\\;x = x^2\\;-\\;4$<\/p>\n\n\n\n<p>$\\Rightarrow \\;-\\;x = \\;-\\;4$<\/p>\n\n\n\n<p>$\\Rightarrow x = 4$<\/p>\n\n\n\n<p>Hence, length of the sides are&nbsp;<\/p>\n\n\n\n<p>PQ $= x = 4$,&nbsp;<\/p>\n\n\n\n<p>PR $= x\\;-\\;2 = 4\\;-\\;2 = 2$,&nbsp;<\/p>\n\n\n\n<p>QR $= (x + 2) + (x\\;-\\;1) = 2x + 1 = 2(4) + 1 = 9$.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"9-practice-problems-on-angle-bisector-theorem\">Practice Problems on Angle Bisector Theorem<\/h2>\n\n\n\n<div class=\"spq_wrapper\"><h2 style=\"display:none;\">Angle Bisector Theorem: Definition, Formula, Proof, Examples<\/h2><p style=\"display:none;\">Attend this quiz & Test your knowledge.<\/p><div class=\"spq_question_wrapper\" data-answer=\"0\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">1<\/span><h3 class=\"sqp_question_text\">If the time on a clock is exactly 04:00, then where should the bisector of the hour hand and minute-hand point?<\/h3><\/span><div class=\"spq_answer_block\" data-value=\"0\">At 12<\/div><div class=\"spq_answer_block\" data-value=\"1\">At 1<\/div><div class=\"spq_answer_block\" data-value=\"2\">At 2<\/div><div class=\"spq_answer_block\" data-value=\"3\">At 6<\/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: At 12<br\/>If the time on a clock is exactly 04:00, then the angle bisector and it points to 2.<\/span><div class=\"spq_explanation_image\"><img decoding=\"async\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/05\/Practice-Problems-1.png\"><\/div><\/div><\/div><\/div><div class=\"spq_question_wrapper\" data-answer=\"2\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">2<\/span><h3 class=\"sqp_question_text\">Which of the following options represent the statement of angle bisector theorem for the given triangle?<\/h3><\/span><div class=\"spq_question_image\"><img decoding=\"async\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/05\/Practice-Problems-2-1.png\"><\/div><div class=\"spq_answer_block\" data-value=\"0\">$\\frac{a}{x} = \\frac{y}{b}$<\/div><div class=\"spq_answer_block\" data-value=\"1\">$\\frac{a}{x} = \\frac{2b}{y}$<\/div><div class=\"spq_answer_block\" data-value=\"2\">$\\frac{a}{b} = \\frac{x}{y}$<\/div><div class=\"spq_answer_block\" data-value=\"3\">None of the above<\/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: $\\frac{a}{b} = \\frac{x}{y}$<br\/>According to the angle bisector theorem, $\\frac{a}{b} = \\frac{x}{y}$.<\/span><\/div><\/div><\/div><div class=\"spq_question_wrapper\" data-answer=\"0\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">3<\/span><h3 class=\"sqp_question_text\">The __________ of a triangle divides the opposite sides in the ratio equal to the ratio of the remaining two sides.<\/h3><\/span><div class=\"spq_answer_block\" data-value=\"0\">angle bisector<\/div><div class=\"spq_answer_block\" data-value=\"1\">median<\/div><div class=\"spq_answer_block\" data-value=\"2\">Perpendicular bisector<\/div><div class=\"spq_answer_block\" data-value=\"3\">height<\/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: angle bisector<br\/>An angle bisector divides the opposite side into two line segments that are proportional to the remianing two sides.<\/span><\/div><\/div><\/div><div class=\"spq_question_wrapper\" data-answer=\"0\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">4<\/span><h3 class=\"sqp_question_text\">The bisectors of the angles of a triangle are _________.<\/h3><\/span><div class=\"spq_answer_block\" data-value=\"0\">concurrent<\/div><div class=\"spq_answer_block\" data-value=\"1\">congruent<\/div><div class=\"spq_answer_block\" data-value=\"2\">perpendicular<\/div><div class=\"spq_answer_block\" data-value=\"3\">None of the above<\/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: concurrent<br\/>The bisectors of the angles of a triangle pass through a common point. Hence, bisectors of the angles of a triangle are concurrent.<\/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\": \"Angle Bisector Theorem: Definition, Formula, Proof, Examples\",        \n        \"about\": {\n                \"@type\": \"Thing\",\n                \"name\": \"Angle Bisector Theorem\"\n        },  \n        \"hasPart\": [{\n                    \"@type\": \"Question\",   \n                    \"eduQuestionType\": \"Multiple choice\",\n                    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triangle divides the opposite sides in the ratio equal to the ratio of the remaining two sides.\",\n                    \"text\": \"The __________ of a triangle divides the opposite sides in the ratio equal to the ratio of the remaining two sides.\",\n                    \"comment\": {\n                      \"@type\": \"Comment\",\n                      \"text\": \"An angle bisector divides the opposite side into two line segments that are proportional to the remianing two sides.\"\n                    },\n                    \"encodingFormat\": \"text\/html\",\n                    \"suggestedAnswer\": [ {\n                                \"@type\": \"Answer\",\n                                \"position\": 1,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"median\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"An angle bisector divides the opposite side into two line segments that are proportional to the remianing two sides.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 2,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"Perpendicular bisector\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"An angle bisector divides the opposite side into two line segments that are proportional to the remianing two sides.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 3,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"height\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"An angle bisector divides the opposite side into two line segments that are proportional to the remianing two sides.\"\n                                    }\n                                }],\n                    \"acceptedAnswer\": {\n                      \"@type\": \"Answer\",\n                      \"position\": 0,\n                      \"encodingFormat\": \"text\/html\",\n                      \"text\": \"angle bisector\",\n                      \"comment\": {\n                          \"@type\": \"Comment\",\n                          \"text\": \"An angle bisector divides the opposite side into two line segments that are proportional to the remianing two sides.\"\n                        },\n                      \"answerExplanation\": {\n                        \"@type\": \"Comment\",\n                        \"text\": \"An angle bisector divides the opposite side into two line segments that are proportional to the remianing two sides.\"\n                      }\n                    } \n\n                    },{\n                    \"@type\": \"Question\",   \n                    \"eduQuestionType\": \"Multiple choice\",\n                    \"learningResourceType\": \"Practice problem\",\n                    \"name\": \"The bisectors of the angles of a triangle are _________.\",\n                    \"text\": \"The bisectors of the angles of a triangle are _________.\",\n                    \"comment\": {\n                      \"@type\": \"Comment\",\n                      \"text\": \"The bisectors of the angles of a triangle pass through a common point. Hence, bisectors of the angles of a triangle are concurrent.\"\n                    },\n                    \"encodingFormat\": \"text\/html\",\n                    \"suggestedAnswer\": [ {\n                                \"@type\": \"Answer\",\n                                \"position\": 1,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"congruent\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"The bisectors of the angles of a triangle pass through a common point. Hence, bisectors of the angles of a triangle are concurrent.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 2,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"perpendicular\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"The bisectors of the angles of a triangle pass through a common point. Hence, bisectors of the angles of a triangle are concurrent.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 3,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"None of the above\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"The bisectors of the angles of a triangle pass through a common point. Hence, bisectors of the angles of a triangle are concurrent.\"\n                                    }\n                                }],\n                    \"acceptedAnswer\": {\n                      \"@type\": \"Answer\",\n                      \"position\": 0,\n                      \"encodingFormat\": \"text\/html\",\n                      \"text\": \"concurrent\",\n                      \"comment\": {\n                          \"@type\": \"Comment\",\n                          \"text\": \"The bisectors of the angles of a triangle pass through a common point. Hence, bisectors of the angles of a triangle are concurrent.\"\n                        },\n                      \"answerExplanation\": {\n                        \"@type\": \"Comment\",\n                        \"text\": \"The bisectors of the angles of a triangle pass through a common point. Hence, bisectors of the angles of a triangle are concurrent.\"\n                      }\n                    } \n\n                    }]}<\/script>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"10-frequently-asked-questions-on-angle-bisector-theorem\">Frequently Asked Questions on Angle Bisector Theorem<\/h2>\n\n\n<div class=\"wp-block-ub-content-toggle\" id=\"ub-content-toggle-8fe66445-289f-4431-9a17-dd609d4f3e13\" 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-0-8fe66445-289f-4431-9a17-dd609d4f3e13\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-8fe66445-289f-4431-9a17-dd609d4f3e13\"><strong>What is the chord bisector theorem?<\/strong><\/p><div class=\"wp-block-ub-content-toggle-accordion-toggle-wrap right\"><span class=\"wp-block-ub-content-toggle-accordion-state-indicator wp-block-ub-chevron-down\"><\/span>\n                    <\/div><\/div><div role=\"region\" class=\"wp-block-ub-content-toggle-accordion-content-wrap ub-hide\" id=\"ub-content-toggle-panel-0-8fe66445-289f-4431-9a17-dd609d4f3e13\">\n\n<p>The perpendicular drawn from the center of a circle to a chord bisects the chord.<\/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-1-8fe66445-289f-4431-9a17-dd609d4f3e13\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-8fe66445-289f-4431-9a17-dd609d4f3e13\"><strong>Is the angle bisector theorem and triangle angle bisector theorem the same?<\/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-1-8fe66445-289f-4431-9a17-dd609d4f3e13\">\n\n<p>Yes, the angle bisector theorem is also known as the triangle angle bisector theorem.<\/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-2-8fe66445-289f-4431-9a17-dd609d4f3e13\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-8fe66445-289f-4431-9a17-dd609d4f3e13\"><strong>What is the exterior angle bisector theorem?<\/strong><\/p><div class=\"wp-block-ub-content-toggle-accordion-toggle-wrap right\"><span class=\"wp-block-ub-content-toggle-accordion-state-indicator wp-block-ub-chevron-down\"><\/span>\n                    <\/div><\/div><div role=\"region\" class=\"wp-block-ub-content-toggle-accordion-content-wrap ub-hide\" id=\"ub-content-toggle-panel-2-8fe66445-289f-4431-9a17-dd609d4f3e13\">\n\n<p>According to the exterior angle bisector theorem, a triangle&#8217;s external angle bisector divides one side externally in proportion to the ratio of the sides containing the angle.<\/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-3-8fe66445-289f-4431-9a17-dd609d4f3e13\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-8fe66445-289f-4431-9a17-dd609d4f3e13\"><strong>What is the converse of the angle bisector theorem?<\/strong><\/p><div class=\"wp-block-ub-content-toggle-accordion-toggle-wrap right\"><span class=\"wp-block-ub-content-toggle-accordion-state-indicator wp-block-ub-chevron-down\"><\/span>\n                    <\/div><\/div><div role=\"region\" class=\"wp-block-ub-content-toggle-accordion-content-wrap ub-hide\" id=\"ub-content-toggle-panel-3-8fe66445-289f-4431-9a17-dd609d4f3e13\">\n\n<p>The converse of the angle bisector theorem states that the point on the opposite side of an angle is assumed to be on its angle bisector if the sides of a triangle satisfy the following condition: &#8220;If a line drawn from a vertex of a triangle divides the opposite side into two parts such that they are proportional to the other two sides of the triangle.&#8221;<\/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-4-8fe66445-289f-4431-9a17-dd609d4f3e13\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-8fe66445-289f-4431-9a17-dd609d4f3e13\"><strong>How to apply the triangle angle bisector theorem?<\/strong><\/p><div class=\"wp-block-ub-content-toggle-accordion-toggle-wrap right\"><span class=\"wp-block-ub-content-toggle-accordion-state-indicator wp-block-ub-chevron-down\"><\/span>\n                    <\/div><\/div><div role=\"region\" class=\"wp-block-ub-content-toggle-accordion-content-wrap ub-hide\" id=\"ub-content-toggle-panel-4-8fe66445-289f-4431-9a17-dd609d4f3e13\">\n\n<p>The missing lengths of a triangle&#8217;s sides can be determined by applying the triangle angle bisector theorem. Let $\\angle A$ be divided by the bisector AD. Hence, according to the triangle angle bisector theorem $\\frac{BD}{DC} = \\frac{AB}{AC}$.<\/p>\n\n<\/div><\/div>\n<\/div>","protected":false},"excerpt":{"rendered":"<p>What Is Angle Bisector Theorem? Angle bisector theorem states that an angle bisector divides the opposite side into two line segments that are proportional to the other two sides. Here, in $\\Delta ABC$, the line AD is the angle bisector of $\\angle A$. AD bisects the side BC in two parts, c and d. a &#8230; <a title=\"Angle Bisector Theorem: Definition, Formula, Proof, Examples\" class=\"read-more\" href=\"https:\/\/www.splashlearn.com\/math-vocabulary\/angle-bisector-theorem\" aria-label=\"More on Angle Bisector Theorem: Definition, Formula, Proof, Examples\">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-29566","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\/29566","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=29566"}],"version-history":[{"count":9,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/posts\/29566\/revisions"}],"predecessor-version":[{"id":35741,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/posts\/29566\/revisions\/35741"}],"wp:attachment":[{"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/media?parent=29566"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/categories?post=29566"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/tags?post=29566"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}