{"id":5032,"date":"2016-05-19T11:49:00","date_gmt":"2016-05-19T06:19:00","guid":{"rendered":"http:\/\/mycbseguide.com\/blog\/ncert-solutions-class-9-maths-exercise-8-1\/"},"modified":"2018-06-18T14:38:38","modified_gmt":"2018-06-18T09:08:38","slug":"ncert-solutions-for-class-9-maths-exercise-8-1","status":"publish","type":"post","link":"https:\/\/mycbseguide.com\/blog\/ncert-solutions-for-class-9-maths-exercise-8-1\/","title":{"rendered":"NCERT Solutions for Class 9 Maths Exercise 8.1"},"content":{"rendered":"<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_76 counter-hierarchy ez-toc-counter ez-toc-grey ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\" style=\"cursor:inherit\">Table of Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Toggle Table of Content\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/span><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1 eztoc-toggle-hide-by-default' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-for-class-9-maths-exercise-8-1\/#NCERT_Solutions_for_Class_9_Mathematics_Quadrilaterals\" >NCERT Solutions for Class 9 Mathematics Quadrilaterals<\/a><ul class='ez-toc-list-level-6' ><li class='ez-toc-heading-level-6'><ul class='ez-toc-list-level-6' ><li class='ez-toc-heading-level-6'><ul class='ez-toc-list-level-6' ><li class='ez-toc-heading-level-6'><ul class='ez-toc-list-level-6' ><li class='ez-toc-heading-level-6'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-for-class-9-maths-exercise-8-1\/#1_The_angles_of_a_quadrilateral_are_in_the_ratio_3_5_9_13_Find_all_angles_of_the_quadrilateral\" >1. The angles of a quadrilateral are in the ratio 3: 5: 9: 13. Find all angles of the quadrilateral.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-6'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-for-class-9-maths-exercise-8-1\/#2_If_the_diagonals_of_a_parallelogram_are_equal_show_that_it_is_a_rectangle\" >2. If the diagonals of a parallelogram are equal, show that it is a rectangle.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-6'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-for-class-9-maths-exercise-8-1\/#3_Show_that_is_diagonals_of_a_quadrilateral_bisect_each_other_at_right_angles_then_it_is_a_rhombus\" >3. Show that is diagonals of a quadrilateral bisect each other at right angles, then it is a rhombus.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-6'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-for-class-9-maths-exercise-8-1\/#4_Show_that_the_diagonals_of_a_square_are_equal_and_bisect_each_other_at_right_angles\" >4. Show that the diagonals of a square are equal and bisect each other at right angles.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-6'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-for-class-9-maths-exercise-8-1\/#5_Show_that_if_the_diagonals_of_a_quadrilateral_are_equal_and_bisect_each_other_at_right_angles_then_it_is_a_square\" >5. Show that if the diagonals of a quadrilateral are equal and bisect each other at right angles, then it is a square.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-6'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-for-class-9-maths-exercise-8-1\/#6_Diagonal_AC_of_a_parallelogram_ABCD_bisects_A_See_figure_Show_that\" >6. Diagonal AC of a parallelogram ABCD bisects A (See figure). Show that:<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-6'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-for-class-9-maths-exercise-8-1\/#7_ABCD_is_a_rhombus_Show_that_the_diagonal_AC_bisects_A_as_well_as_C_and_diagonal_BD_bisects_B_as_well_as_D\" >7. ABCD is a rhombus. Show that the diagonal AC bisects A as well as C and diagonal BD bisects B as well as D.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-6'><a class=\"ez-toc-link ez-toc-heading-9\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-for-class-9-maths-exercise-8-1\/#8_ABCD_is_a_rectangle_in_which_diagonal_AC_bisects_A_as_well_as_C_Show_that\" >8. ABCD is a rectangle in which diagonal AC bisects A as well as C. Show that:<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-6'><a class=\"ez-toc-link ez-toc-heading-10\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-for-class-9-maths-exercise-8-1\/#9_In_parallelogram_ABCD_two_points_P_and_Q_are_taken_on_diagonal_BD_such_that_DP_BQ_See_figure_Show_that\" >9. In parallelogram ABCD, two points P and Q are taken on diagonal BD such that DP = BQ (See figure). Show that:<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-6'><a class=\"ez-toc-link ez-toc-heading-11\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-for-class-9-maths-exercise-8-1\/#10_ABCD_is_a_parallelogram_and_AP_and_CQ_are_the_perpendiculars_from_vertices_A_and_C_on_its_diagonal_BD_See_figure_Show_that\" >10. ABCD is a parallelogram and AP and CQ are the perpendiculars from vertices A and C on its diagonal BD (See figure). Show that:<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-6'><a class=\"ez-toc-link ez-toc-heading-12\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-for-class-9-maths-exercise-8-1\/#11_An_ABC_and_DEF_AB_DE_AB_DE_BC_EF_and_BC_EF_Vertices_A_B_and_C_are_joined_to_vertices_D_E_and_F_respectively_See_figure_Show_that\" >11. An ABC and DEF, AB = DE, AB  DE, BC = EF and BC  EF. Vertices A, B and C are joined to vertices D, E and F respectively (See figure). Show that:<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-6'><a class=\"ez-toc-link ez-toc-heading-13\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-for-class-9-maths-exercise-8-1\/#12_ABCD_is_a_trapezium_in_which_AB_CD_and_AD_BC_See_figure_Show_that\" >12. ABCD is a trapezium in which AB  CD and AD = BC (See figure). Show that:<\/a><\/li><\/ul><\/li><\/ul><\/li><\/ul><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-14\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-for-class-9-maths-exercise-8-1\/#NCERT_Solutions_for_Class_9_Maths_Exercise_81\" >NCERT Solutions for Class 9 Maths Exercise 8.1<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-15\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-for-class-9-maths-exercise-8-1\/#CBSE_app_for_Class_9\" >CBSE app for Class 9<\/a><\/li><\/ul><\/nav><\/div>\n<p>NCERT Solutions for Class 9 Maths Exercise 8.1 book solutions are available in PDF format for free download. These ncert book chapter wise questions and answers are very helpful for CBSE board exam. CBSE recommends NCERT books and most of the questions in CBSE exam are asked from NCERT text books. Class 9 Maths chapter wise NCERT solution for Maths Book for all the chapters can be downloaded from our website and myCBSEguide mobile app for free.<\/p>\n<p style=\"text-align: center;\"><strong>NCERT solutions for Class 9 Maths\u00a0<\/strong><strong>Quadrilateral<\/strong><strong>\u00a0<\/strong><strong><a class=\"button\" href=\"https:\/\/mycbseguide.com\/downloads\/cbse-class-09-mathematics-quadrilaterals\/1242\/ncert-solutions\/5\/\">Download as PDF<\/a><\/strong><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/media-mycbseguide.s3.ap-south-1.amazonaws.com\/images\/blog\/09_Class_Maths_Book.jpg\" alt=\"NCERT Solutions for Class 9 Maths Exercise 8.1\" width=\"189\" height=\"243\" \/><\/p>\n<h2><span class=\"ez-toc-section\" id=\"NCERT_Solutions_for_Class_9_Mathematics_Quadrilaterals\"><\/span>NCERT Solutions for Class 9 Mathematics Quadrilaterals<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h6 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"1_The_angles_of_a_quadrilateral_are_in_the_ratio_3_5_9_13_Find_all_angles_of_the_quadrilateral\"><\/span><strong>1. The angles of a quadrilateral are in the ratio 3: 5: 9: 13. Find all angles of the quadrilateral.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h6>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Let in quadrilateral ABCD, <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>A = <img decoding=\"async\" style=\"height: 21px; width: 24px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image002.png\" \/> <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>B = <img decoding=\"async\" style=\"height: 21px; width: 24px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image003.png\" \/> <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>C = <img decoding=\"async\" style=\"height: 19px; width: 21px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image004.png\" \/> and <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>D = <img decoding=\"async\" style=\"height: 19px; width: 29px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image005.png\" \/><\/p>\n<p style=\"text-align: justify;\">Since, sum of all the angles of a quadrilateral = <img decoding=\"async\" style=\"height: 19px; width: 35px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image006.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image007.png\" \/><img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>A + <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>B + <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>C + <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>D = <img decoding=\"async\" style=\"height: 19px; width: 35px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image006.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image008.png\" \/><img decoding=\"async\" style=\"height: 19px; width: 163px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image009.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image008.png\" \/> <img decoding=\"async\" style=\"height: 19px; width: 75px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image010.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image008.png\" \/> <img decoding=\"async\" style=\"height: 19px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image011.png\" \/><\/p>\n<p style=\"text-align: justify;\">Now <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>A = <img decoding=\"async\" style=\"height: 19px; width: 108px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image012.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>B = <img decoding=\"async\" style=\"height: 19px; width: 109px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image013.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>C = <img decoding=\"async\" style=\"height: 19px; width: 116px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image014.png\" \/><\/p>\n<p style=\"text-align: justify;\">And <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>D = <img decoding=\"async\" style=\"height: 19px; width: 128px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image015.png\" \/><\/p>\n<p style=\"text-align: justify;\">Hence angles of given quadrilateral are <img decoding=\"async\" style=\"height: 21px; width: 89px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image016.png\" \/> and <img decoding=\"async\" style=\"height: 19px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image017.png\" \/><\/p>\n<hr \/>\n<h6 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"2_If_the_diagonals_of_a_parallelogram_are_equal_show_that_it_is_a_rectangle\"><\/span><strong>2. If the diagonals of a parallelogram are equal, show that it is a rectangle.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h6>\n<p style=\"text-align: justify;\"><strong>Ans. Given<\/strong>: ABCD is a parallelogram with diagonal AC = diagonal BD<\/p>\n<p style=\"text-align: justify;\"><strong>To prove<\/strong>: ABCD is a rectangle.<\/p>\n<p style=\"text-align: justify;\"><strong><img decoding=\"async\" id=\"Picture 10\" style=\"height: 132px; width: 176px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image018.jpg\" \/><\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Proo<\/strong>f: In triangles ABC and ABD,<\/p>\n<p style=\"text-align: justify;\">AB = AB[Common]\n<p style=\"text-align: justify;\">AC = BD[Given]\n<p style=\"text-align: justify;\">AD = BC[opp. Sides of a <img decoding=\"async\" style=\"height: 21px; width: 11px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image019.png\" \/>gm]\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image007.png\" \/> <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image020.png\" \/>ABC <img decoding=\"async\" style=\"height: 17px; width: 28px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image021.png\" \/>BAD [By SSS congruency]\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image008.png\" \/> <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>DAB = <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>CBA [By C.P.C.T.] \u2026\u2026\u2026.(i)<\/p>\n<p style=\"text-align: justify;\">But <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>DAB + <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>CBA = <img decoding=\"async\" style=\"height: 19px; width: 33px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image022.png\" \/> \u2026\u2026\u2026.(ii)<\/p>\n<p style=\"text-align: justify;\">[<img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image023.png\" \/> AD<img decoding=\"async\" style=\"height: 21px; width: 11px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image019.png\" \/>BC and AB cuts them, the sum of the interior angles of the same side of transversal is <img decoding=\"async\" style=\"height: 19px; width: 33px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image022.png\" \/>]\n<p style=\"text-align: justify;\">From eq. (i) and (ii),<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>DAB = <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>CBA = <img decoding=\"async\" style=\"height: 21px; width: 24px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image024.png\" \/><\/p>\n<p style=\"text-align: justify;\">Hence ABCD is a rectangle.<\/p>\n<hr \/>\n<h6 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"3_Show_that_is_diagonals_of_a_quadrilateral_bisect_each_other_at_right_angles_then_it_is_a_rhombus\"><\/span><strong>3. Show that is diagonals of a quadrilateral bisect each other at right angles, then it is a rhombus.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h6>\n<p style=\"text-align: justify;\"><strong>Ans. Given<\/strong>: Let ABCD is a quadrilateral.<\/p>\n<p style=\"text-align: justify;\"><strong><img decoding=\"async\" id=\"Picture 12\" style=\"height: 172px; width: 170px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image025.jpg\" \/><\/strong><\/p>\n<p style=\"text-align: justify;\">Let its diagonal AC and BD bisect each other at right angle at point O.<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image007.png\" \/>OA = OC, OB = OD<\/p>\n<p style=\"text-align: justify;\">And <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>AOB = <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>BOC = <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>COD = <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>AOD = <img decoding=\"async\" style=\"height: 21px; width: 24px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image024.png\" \/><\/p>\n<p style=\"text-align: justify;\"><strong>To prove<\/strong>: ABCD is a rhombus.<\/p>\n<p style=\"text-align: justify;\"><strong>Proof<\/strong>: In <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image020.png\" \/>AOD and <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image020.png\" \/>BOC,<\/p>\n<p style=\"text-align: justify;\">OA = OC[Given]\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>AOD = <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>BOC [Given]\n<p style=\"text-align: justify;\">OB = OD[Given]\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image007.png\" \/><img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image020.png\" \/>AOD <img decoding=\"async\" style=\"height: 17px; width: 28px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image021.png\" \/>COB [By SAS congruency]\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image008.png\" \/>AD = CB[By C.P.C.T.] \u2026\u2026\u2026.(i)<\/p>\n<p style=\"text-align: justify;\">Again, In <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image020.png\" \/>AOB and <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image020.png\" \/>COD,<\/p>\n<p style=\"text-align: justify;\">OA = OC[Given]\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>AOB = <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>COD [Given]\n<p style=\"text-align: justify;\">OB = OD[Given]\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image007.png\" \/><img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image020.png\" \/>AOB <img decoding=\"async\" style=\"height: 17px; width: 28px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image021.png\" \/>COD [By SAS congruency]\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image008.png\" \/>AD = CB [By C.P.C.T.] \u2026\u2026\u2026.(ii)<\/p>\n<p style=\"text-align: justify;\">Now In <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image020.png\" \/>AOD and <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image020.png\" \/>BOC,<\/p>\n<p style=\"text-align: justify;\">OA = OC[Given]\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>AOB = <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>BOC [Given]\n<p style=\"text-align: justify;\">OB = OB[Common]\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image007.png\" \/><img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image020.png\" \/>AOB <img decoding=\"async\" style=\"height: 17px; width: 28px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image021.png\" \/>COB [By SAS congruency]\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image008.png\" \/>AB = BC[By C.P.C.T.] \u2026\u2026\u2026.(iii)<\/p>\n<p style=\"text-align: justify;\">From eq. (i), (ii) and (iii),<\/p>\n<p style=\"text-align: justify;\">AD = BC = CD = AB<\/p>\n<p style=\"text-align: justify;\">And the diagonals of quadrilateral ABCD bisect each other at right angle.<\/p>\n<p style=\"text-align: justify;\">Therefore, ABCD is a rhombus.<\/p>\n<hr \/>\n<h6 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"4_Show_that_the_diagonals_of_a_square_are_equal_and_bisect_each_other_at_right_angles\"><\/span><strong>4. Show that the diagonals of a square are equal and bisect each other at right angles.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h6>\n<p style=\"text-align: justify;\"><strong>Ans. Given<\/strong>: ABCD is a square. AC and BD are its diagonals bisect each other at point O.<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 11\" style=\"height: 138px; width: 149px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image026.jpg\" \/><\/p>\n<p style=\"text-align: justify;\"><strong>To prove<\/strong>: AC = BD and AC <img decoding=\"async\" style=\"height: 17px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image027.png\" \/> BD at point O.<\/p>\n<p style=\"text-align: justify;\"><strong>Proof:<\/strong> In triangles ABC and BAD,<\/p>\n<p style=\"text-align: justify;\">AB = AB[Common]\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>ABC = <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>BAD = <img decoding=\"async\" style=\"height: 21px; width: 24px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image024.png\" \/><\/p>\n<p style=\"text-align: justify;\">BC = AD[Sides of a square]\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image007.png\" \/><img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image020.png\" \/>ABC <img decoding=\"async\" style=\"height: 17px; width: 28px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image021.png\" \/>BAD [By SAS congruency]\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image008.png\" \/>AC = BD[By C.P.C.T.]Hence proved.<\/p>\n<p style=\"text-align: justify;\">Now in triangles AOB and AOD,<\/p>\n<p style=\"text-align: justify;\">AO = AO[Common]\n<p style=\"text-align: justify;\">AB = AD[Sides of a square]\n<p style=\"text-align: justify;\">OB = OD[Diagonals of a square bisect each other]\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image007.png\" \/><img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image020.png\" \/>AOB <img decoding=\"async\" style=\"height: 17px; width: 28px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image021.png\" \/>AOD [By SSS congruency]\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>AOB = <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>AOD [By C.P.C.T.]\n<p style=\"text-align: justify;\">But <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>AOB + <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>AOD = <img decoding=\"async\" style=\"height: 19px; width: 33px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image022.png\" \/>[Linear pair]\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image007.png\" \/><img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>AOB = <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>AOD = <img decoding=\"async\" style=\"height: 19px; width: 27px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image028.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image008.png\" \/>OA <img decoding=\"async\" style=\"height: 17px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image027.png\" \/> BD or AC <img decoding=\"async\" style=\"height: 17px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image027.png\" \/> BD Hence proved.<\/p>\n<hr \/>\n<p style=\"text-align: center;\">NCERT Solutions for Class 9 Maths Exercise 8.1<\/p>\n<h6 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"5_Show_that_if_the_diagonals_of_a_quadrilateral_are_equal_and_bisect_each_other_at_right_angles_then_it_is_a_square\"><\/span><strong>5. Show that if the diagonals of a quadrilateral are equal and bisect each other at right angles, then it is a square.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h6>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Let ABCD be a quadrilateral in which equal diagonals AC and BD bisect each other at right angle at point O.<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 13\" style=\"height: 138px; width: 144px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image029.jpg\" \/><\/p>\n<p style=\"text-align: justify;\">We haveAC = BD and OA = OC\u2026\u2026\u2026.(i)<\/p>\n<p style=\"text-align: justify;\">And OB = OD\u2026\u2026\u2026.(ii)<\/p>\n<p style=\"text-align: justify;\">Now OA + OC = OB + OD<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image008.png\" \/>OC + OC = OB + OB [Using (i) &amp; (ii)]\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image008.png\" \/> 2OC = 2OB<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image008.png\" \/> OC = OB\u2026\u2026\u2026.(iii)<\/p>\n<p style=\"text-align: justify;\">From eq. (i), (ii) and (iii), we get, OA = OB = OC = OD \u2026\u2026\u2026.(iv)<\/p>\n<p style=\"text-align: justify;\">Now in <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image020.png\" \/>AOB and <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image020.png\" \/>COD,<\/p>\n<p style=\"text-align: justify;\">OA = OD [proved]\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>AOB = <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>COD[vertically opposite angles]\n<p style=\"text-align: justify;\">OB = OC [proved]\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image007.png\" \/><img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image020.png\" \/>AOB <img decoding=\"async\" style=\"height: 17px; width: 28px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image021.png\" \/>DOC[By SAS congruency]\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image008.png\" \/>AB = DC [By C.P.C.T.]\u2026\u2026\u2026.(v)<\/p>\n<p style=\"text-align: justify;\">Similarly, <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image020.png\" \/>BOC <img decoding=\"async\" style=\"height: 17px; width: 28px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image021.png\" \/>AOD [By SAS congruency]\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image008.png\" \/>BC = AD [By C.P.C.T.]\u2026\u2026\u2026.(vi)<\/p>\n<p style=\"text-align: justify;\">From eq. (v) and (vi), it is concluded that ABCD is a parallelogram because opposite sides of a quadrilateral are equal.<\/p>\n<p style=\"text-align: justify;\">Now in <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image020.png\" \/>ABC and <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image020.png\" \/>BAD,<\/p>\n<p style=\"text-align: justify;\">AB = BA [Common]\n<p style=\"text-align: justify;\">BC = AD [proved above]\n<p style=\"text-align: justify;\">AC = BD [Given]\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image007.png\" \/><img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image020.png\" \/>ABC <img decoding=\"async\" style=\"height: 17px; width: 28px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image021.png\" \/>BAD[By SSS congruency]\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image008.png\" \/><img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>ABC = <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>BAD[By C.P.C.T.]\u2026\u2026\u2026.(vii)<\/p>\n<p style=\"text-align: justify;\">But <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>ABC + <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>BAD = <img decoding=\"async\" style=\"height: 19px; width: 33px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image022.png\" \/>[ABCD is a parallelogram]\u2026\u2026\u2026.(viii)<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image007.png\" \/>AD <img decoding=\"async\" style=\"height: 21px; width: 11px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image019.png\" \/> BC and AB is a transversal.<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image008.png\" \/><img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>ABC + <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>ABC = <img decoding=\"async\" style=\"height: 19px; width: 33px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image022.png\" \/>[Using eq. (vii) and (viii)]\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image008.png\" \/>2<img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>ABC = <img decoding=\"async\" style=\"height: 19px; width: 33px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image022.png\" \/><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image008.png\" \/><img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>ABC = <img decoding=\"async\" style=\"height: 19px; width: 27px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image028.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image007.png\" \/> <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>ABC = <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>BAD =<img decoding=\"async\" style=\"height: 19px; width: 27px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image028.png\" \/>\u2026\u2026\u2026.(ix)<\/p>\n<p style=\"text-align: justify;\">Opposite angles of a parallelogram are equal.<\/p>\n<p style=\"text-align: justify;\">But <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>ABC = <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>BAD =<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image007.png\" \/><img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>ABC = <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>ADC = <img decoding=\"async\" style=\"height: 19px; width: 27px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image028.png\" \/>\u2026\u2026\u2026.(x)<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image007.png\" \/><img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>BAD = <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>BDC =<img decoding=\"async\" style=\"height: 19px; width: 27px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image028.png\" \/>\u2026\u2026\u2026.(xi)<\/p>\n<p style=\"text-align: justify;\">From eq. (x) and (xi), we get<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>ABC = <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>ADC = <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>BAD = <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>BDC =<img decoding=\"async\" style=\"height: 19px; width: 27px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image028.png\" \/>\u2026\u2026\u2026.(xii)<\/p>\n<p style=\"text-align: justify;\">Now in <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image020.png\" \/>AOB and <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image020.png\" \/>BOC,<\/p>\n<p style=\"text-align: justify;\">OA = OC [Given]\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>AOB = <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>BOC = <img decoding=\"async\" style=\"height: 21px; width: 24px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image024.png\" \/>[Given]\n<p style=\"text-align: justify;\">OB = OB [Common]\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image007.png\" \/><img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image020.png\" \/>AOB <img decoding=\"async\" style=\"height: 17px; width: 28px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image021.png\" \/>COB[By SAS congruency]\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image008.png\" \/>AB = BC\u2026\u2026\u2026.(xiii)<\/p>\n<p style=\"text-align: justify;\">From eq. (v), (vi) and (xiii), we get,<\/p>\n<p style=\"text-align: justify;\">AB = BC = CD = AD \u2026\u2026\u2026.(xiv)<\/p>\n<p style=\"text-align: justify;\">Now, from eq. (xii) and (xiv), we have a quadrilateral whose equal diagonals bisect each other at right angle.<\/p>\n<p style=\"text-align: justify;\">Also sides are equal make an angle of <img decoding=\"async\" style=\"height: 21px; width: 24px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image024.png\" \/> with each other.<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image007.png\" \/>ABCD is a square.<\/p>\n<hr \/>\n<p style=\"text-align: center;\">NCERT Solutions for Class 9 Maths Exercise 8.1<\/p>\n<h6 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"6_Diagonal_AC_of_a_parallelogram_ABCD_bisects_A_See_figure_Show_that\"><\/span><strong>6. Diagonal AC of a parallelogram ABCD bisects <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>A (See figure). Show that: <\/strong><span class=\"ez-toc-section-end\"><\/span><\/h6>\n<p style=\"text-align: justify;\"><strong><img decoding=\"async\" id=\"Picture 9\" style=\"height: 138px; width: 157px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image030.png\" \/><\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(i) It bisects <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>C also.<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(ii) ABCD is a rhombus.<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Diagonal AC bisects <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>A of the parallelogram ABCD.<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 14\" style=\"height: 142px; width: 161px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image031.jpg\" \/><\/p>\n<p style=\"text-align: justify;\"><strong>(i)<\/strong> Since AB <img decoding=\"async\" style=\"height: 21px; width: 11px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image019.png\" \/> DC and AC intersects them.<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image007.png\" \/><img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>1 = <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>3 [Alternate angles] \u2026\u2026\u2026.(i)<\/p>\n<p style=\"text-align: justify;\">Similarly <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>2 = <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>4 \u2026\u2026\u2026.(ii)<\/p>\n<p style=\"text-align: justify;\">But <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>1 = <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>2 [Given]\u2026\u2026\u2026.(iii)<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image007.png\" \/><img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>3 = <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>4[Using eq. (i), (ii) and (iii)]\n<p style=\"text-align: justify;\">Thus AC bisects <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>C.<\/p>\n<p style=\"text-align: justify;\"><strong>(ii)<\/strong> <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>2 = <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>3 = <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>4 = <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>1<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image008.png\" \/>AD = CD[Sides opposite to equal angles]\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image007.png\" \/>AB = CD = AD = BC<\/p>\n<p style=\"text-align: justify;\">Hence ABCD is a rhombus.<\/p>\n<hr \/>\n<p style=\"text-align: center;\">NCERT Solutions for Class 9 Maths Exercise 8.1<\/p>\n<h6 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"7_ABCD_is_a_rhombus_Show_that_the_diagonal_AC_bisects_A_as_well_as_C_and_diagonal_BD_bisects_B_as_well_as_D\"><\/span><strong>7. ABCD is a rhombus. Show that the diagonal AC bisects <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>A as well as <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>C and diagonal BD bisects <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>B as well as <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>D.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h6>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>ABCD is a rhombus. Therefore, AB = BC = CD = AD<\/p>\n<p style=\"text-align: justify;\">Let O be the point of bisection of diagonals.<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 15\" style=\"height: 189px; width: 187px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image032.jpg\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image007.png\" \/>OA = OC and OB = OD<\/p>\n<p style=\"text-align: justify;\">In <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image020.png\" \/>AOB and <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image020.png\" \/>AOD,<\/p>\n<p style=\"text-align: justify;\">OA = OA [Common]\n<p style=\"text-align: justify;\">AB = AD [Equal sides of rhombus]\n<p style=\"text-align: justify;\">OB = OD (diagonals of rhombus bisect each other]\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image007.png\" \/><img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image020.png\" \/>AOB <img decoding=\"async\" style=\"height: 17px; width: 28px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image021.png\" \/>AOD [By SSS congruency]\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image008.png\" \/><img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>OAD = <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>OAB [By C.P.C.T.]\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image008.png\" \/> OA bisects <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>A \u2026\u2026\u2026.(i)<\/p>\n<p style=\"text-align: justify;\">Similarly, <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image020.png\" \/>BOC <img decoding=\"async\" style=\"height: 17px; width: 28px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image021.png\" \/>DOC [By SSS congruency]\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image008.png\" \/> <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>OCB = <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>OCD[By C.P.C.T.]\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image008.png\" \/> OC bisects <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>C \u2026\u2026\u2026.(ii)<\/p>\n<p style=\"text-align: justify;\">From eq. (i) and (ii), we can say that diagonal AC bisects <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>A and <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>C.<\/p>\n<p style=\"text-align: justify;\">Now in <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image020.png\" \/>AOB and <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image020.png\" \/>BOC,<\/p>\n<p style=\"text-align: justify;\">OB = OB [Common]\n<p style=\"text-align: justify;\">AB = BC [Equal sides of rhombus]\n<p style=\"text-align: justify;\">OA = OC (diagonals of rhombus bisect each other]\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image007.png\" \/> <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image020.png\" \/>AOB <img decoding=\"async\" style=\"height: 17px; width: 28px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image021.png\" \/>COB [By SSS congruency]\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image008.png\" \/><img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>OBA = <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>OBC [By C.P.C.T.]\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image008.png\" \/> OB bisects <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>B \u2026\u2026\u2026.(iii)<\/p>\n<p style=\"text-align: justify;\">Similarly, <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image020.png\" \/>AOD <img decoding=\"async\" style=\"height: 17px; width: 28px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image021.png\" \/>COD [By SSS congruency]\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image008.png\" \/><img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>ODA = <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>ODC[By C.P.C.T.]\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image008.png\" \/> BD bisects <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>D \u2026\u2026\u2026.(iv)<\/p>\n<p style=\"text-align: justify;\">From eq. (iii) and (iv), we can say that diagonal BD bisects <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>B and <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>D.<\/p>\n<hr \/>\n<p style=\"text-align: center;\">NCERT Solutions for Class 9 Maths Exercise 8.1<\/p>\n<h6 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"8_ABCD_is_a_rectangle_in_which_diagonal_AC_bisects_A_as_well_as_C_Show_that\"><\/span><strong>8. ABCD is a rectangle in which diagonal AC bisects <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>A as well as <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>C. Show that:<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h6>\n<p style=\"text-align: justify;\"><strong>(i) ABCD is a square.<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(ii) Diagonal BD bisects both <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>B as well as <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>D.<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>ABCD is a rectangle. Therefore AB = DC\u2026\u2026\u2026.(i)<\/p>\n<p style=\"text-align: justify;\">And BC = AD<\/p>\n<p style=\"text-align: justify;\">Also <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>A = <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>B = <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>C = <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>D = <img decoding=\"async\" style=\"height: 21px; width: 24px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image024.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 16\" style=\"height: 150px; width: 210px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image033.jpg\" \/><\/p>\n<p style=\"text-align: justify;\"><strong>(i)<\/strong> In <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image020.png\" \/>ABC and <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image020.png\" \/>ADC<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>1 = <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>2 and <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>3 = <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>4<\/p>\n<p style=\"text-align: justify;\">[AC bisects <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>A and <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>C (given)]\n<p style=\"text-align: justify;\">AC = AC [Common]\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image007.png\" \/> <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image020.png\" \/>ABC <img decoding=\"async\" style=\"height: 17px; width: 28px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image021.png\" \/>ADC[By ASA congruency]\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image008.png\" \/> AB = AD \u2026\u2026\u2026.(ii)<\/p>\n<p style=\"text-align: justify;\">From eq. (i) and (ii), AB = BC = CD = AD<\/p>\n<p style=\"text-align: justify;\">Hence ABCD is a square.<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 17\" style=\"height: 125px; width: 122px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image034.jpg\" \/><\/p>\n<p style=\"text-align: justify;\"><strong>(ii)<\/strong> In <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image020.png\" \/>ABC and <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image020.png\" \/>ADC<\/p>\n<p style=\"text-align: justify;\">AB = BA [Since ABCD is a square]\n<p style=\"text-align: justify;\">AD = DC [Since ABCD is a square]\n<p style=\"text-align: justify;\">BD = BD [Common]\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image007.png\" \/> <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image020.png\" \/>ABD <img decoding=\"async\" style=\"height: 17px; width: 28px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image021.png\" \/>CBD [By SSS congruency]\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image008.png\" \/> <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>ABD = <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>CBD [By C.P.C.T.]\u2026\u2026\u2026.(iii)<\/p>\n<p style=\"text-align: justify;\">And <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>ADB = <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>CDB[By C.P.C.T.]\u2026\u2026\u2026.(iv)<\/p>\n<p style=\"text-align: justify;\">From eq. (iii) and (iv), it is clear that diagonal BD bisects both <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>B and <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>D.<\/p>\n<hr \/>\n<p style=\"text-align: center;\">NCERT Solutions for Class 9 Maths Exercise 8.1<\/p>\n<h6 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"9_In_parallelogram_ABCD_two_points_P_and_Q_are_taken_on_diagonal_BD_such_that_DP_BQ_See_figure_Show_that\"><\/span><strong>9. In parallelogram ABCD, two points P and Q are taken on diagonal BD such that DP = BQ (See figure). Show that: <\/strong><span class=\"ez-toc-section-end\"><\/span><\/h6>\n<p style=\"text-align: justify;\"><strong><img decoding=\"async\" id=\"Picture 8\" style=\"height: 153px; width: 202px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image035.jpg\" \/><\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(i) <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image020.png\" \/>APD <img decoding=\"async\" style=\"height: 17px; width: 28px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image021.png\" \/>CQB<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(ii) AP = CQ<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(iii) <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image020.png\" \/>AQB <img decoding=\"async\" style=\"height: 17px; width: 28px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image021.png\" \/>CPD<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(iv) AQ = CP<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(v) APCQ is a parallelogram.<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. (i)<\/strong> In <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image020.png\" \/>APD and <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image020.png\" \/>CQB,<\/p>\n<p style=\"text-align: justify;\">DP = BQ[Given]\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>ADP = <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>QBC [Alternate angles (AD<img decoding=\"async\" style=\"height: 21px; width: 11px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image019.png\" \/>BC and BD is transversal)]\n<p style=\"text-align: justify;\">AD = CB [Opposite sides of parallelogram]\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image007.png\" \/><img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image020.png\" \/>APD <img decoding=\"async\" style=\"height: 17px; width: 28px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image021.png\" \/>CQB [By SAS congruency]\n<p style=\"text-align: justify;\"><strong>(ii)<\/strong> Since <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image020.png\" \/>APD <img decoding=\"async\" style=\"height: 17px; width: 28px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image021.png\" \/>CQB<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image008.png\" \/> AP = CQ[By C.P.C.T.]\n<p style=\"text-align: justify;\"><strong>(iii)<\/strong> In <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image020.png\" \/>AQB and <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image020.png\" \/>CPD,<\/p>\n<p style=\"text-align: justify;\">BQ = DP[Given]\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>ABQ = <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>PDC [Alternate angles (AB<img decoding=\"async\" style=\"height: 21px; width: 11px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image019.png\" \/>CD and BD is transversal)]\n<p style=\"text-align: justify;\">AB = CD[Opposite sides of parallelogram]\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image007.png\" \/><img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image020.png\" \/>AQB <img decoding=\"async\" style=\"height: 17px; width: 28px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image021.png\" \/>CPD [By SAS congruency]\n<p style=\"text-align: justify;\"><strong>(iv)<\/strong> Since <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image020.png\" \/>AQB <img decoding=\"async\" style=\"height: 17px; width: 28px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image021.png\" \/>CPD<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image008.png\" \/>AQ = CP[By C.P.C.T.]\n<p style=\"text-align: justify;\"><strong>(v)<\/strong> In quadrilateral APCQ,<\/p>\n<p style=\"text-align: justify;\">AP = CQ[proved in part (i)]\n<p style=\"text-align: justify;\">AQ = CP[proved in part (iv)]\n<p style=\"text-align: justify;\">Since opposite sides of quadrilateral APCQ are equal.<\/p>\n<p style=\"text-align: justify;\">Hence APCQ is a parallelogram.<\/p>\n<hr \/>\n<p style=\"text-align: center;\">NCERT Solutions for Class 9 Maths Exercise 8.1<\/p>\n<h6 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"10_ABCD_is_a_parallelogram_and_AP_and_CQ_are_the_perpendiculars_from_vertices_A_and_C_on_its_diagonal_BD_See_figure_Show_that\"><\/span><strong>10. ABCD is a parallelogram and AP and CQ are the perpendiculars from vertices A and C on its diagonal BD (See figure). Show that: <\/strong><span class=\"ez-toc-section-end\"><\/span><\/h6>\n<p style=\"text-align: justify;\"><strong><img decoding=\"async\" id=\"Picture 7\" style=\"height: 135px; width: 275px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image036.jpg\" \/><\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(i) <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image020.png\" \/>APB <img decoding=\"async\" style=\"height: 17px; width: 28px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image021.png\" \/>CQD<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(ii) AP = CQ<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. Given<\/strong>: ABCD is a parallelogram. AP <img decoding=\"async\" style=\"height: 17px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image027.png\" \/> BD and CQ <img decoding=\"async\" style=\"height: 17px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image027.png\" \/> BD<\/p>\n<p style=\"text-align: justify;\"><strong>To prove<\/strong>: (i) <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image020.png\" \/>APB <img decoding=\"async\" style=\"height: 17px; width: 28px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image021.png\" \/>CQD (ii) AP = CQ<\/p>\n<p style=\"text-align: justify;\"><strong>Proof<\/strong>: <strong>(i)<\/strong> In <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image020.png\" \/>APB and <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image020.png\" \/>CQD,<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>1 = <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>2[Alternate interior angles]\n<p style=\"text-align: justify;\">AB = CD[Opposite sides of a parallelogram are equal]\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>APB = <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>CQD = <img decoding=\"async\" style=\"height: 21px; width: 24px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image024.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image007.png\" \/><img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image020.png\" \/>APB <img decoding=\"async\" style=\"height: 17px; width: 28px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image021.png\" \/>CQD [By ASA Congruency]\n<p style=\"text-align: justify;\"><strong>(ii)<\/strong> Since <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image020.png\" \/>APB <img decoding=\"async\" style=\"height: 17px; width: 28px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image021.png\" \/>CQD<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image007.png\" \/> AP = CQ [By C.P.C.T.]\n<hr \/>\n<p style=\"text-align: center;\">NCERT Solutions for Class 9 Maths Exercise 8.1<\/p>\n<h6 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"11_An_ABC_and_DEF_AB_DE_AB_DE_BC_EF_and_BC_EF_Vertices_A_B_and_C_are_joined_to_vertices_D_E_and_F_respectively_See_figure_Show_that\"><\/span><strong>11. An <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image020.png\" \/>ABC and <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image020.png\" \/>DEF, AB = DE, AB <img decoding=\"async\" style=\"height: 21px; width: 11px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image019.png\" \/> DE, BC = EF and BC <img decoding=\"async\" style=\"height: 21px; width: 11px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image019.png\" \/> EF. Vertices A, B and C are joined to vertices D, E and F respectively (See figure). Show that: <\/strong><span class=\"ez-toc-section-end\"><\/span><\/h6>\n<p style=\"text-align: justify;\"><strong><img decoding=\"async\" id=\"Picture 6\" style=\"height: 186px; width: 212px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image037.jpg\" \/><\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(i) Quadrilateral ABED is a parallelogram.<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(ii) Quadrilateral BEFC is a parallelogram.<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(iii) AD <img decoding=\"async\" style=\"height: 21px; width: 11px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image019.png\" \/> CF and AD = CF<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(iv) Quadrilateral ACFD is a parallelogram.<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(v) AC = DF<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(vi) <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image020.png\" \/>ABC <img decoding=\"async\" style=\"height: 17px; width: 28px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image021.png\" \/>DEF<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. (i)<\/strong> In <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image020.png\" \/>ABC and <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image020.png\" \/>DEF<\/p>\n<p style=\"text-align: justify;\">AB = DE[Given]\n<p style=\"text-align: justify;\">And AB <img decoding=\"async\" style=\"height: 21px; width: 11px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image019.png\" \/> DE[Given]\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image007.png\" \/>ABED is a parallelogram.<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 18\" style=\"height: 162px; width: 184px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image038.jpg\" \/><\/p>\n<p style=\"text-align: justify;\"><strong>(ii)<\/strong> In <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image020.png\" \/>ABC and <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image020.png\" \/>DEF<\/p>\n<p style=\"text-align: justify;\">BC = EF[Given]\n<p style=\"text-align: justify;\">And BC <img decoding=\"async\" style=\"height: 21px; width: 11px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image019.png\" \/> EF[Given]\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image007.png\" \/>BEFC is a parallelogram.<\/p>\n<p style=\"text-align: justify;\"><strong>(iii)<\/strong> As ABED is a parallelogram.<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image007.png\" \/>AD <img decoding=\"async\" style=\"height: 21px; width: 11px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image019.png\" \/> BE and AD = BE \u2026\u2026\u2026.(i)<\/p>\n<p style=\"text-align: justify;\">Also BEFC is a parallelogram.<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image007.png\" \/>CF <img decoding=\"async\" style=\"height: 21px; width: 11px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image019.png\" \/> BE and CF = BE\u2026\u2026\u2026.(ii)<\/p>\n<p style=\"text-align: justify;\">From (i) and (ii), we get<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image007.png\" \/>AD <img decoding=\"async\" style=\"height: 21px; width: 11px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image019.png\" \/> CF and AD = CF<\/p>\n<p style=\"text-align: justify;\"><strong>(iv)<\/strong> As AD <img decoding=\"async\" style=\"height: 21px; width: 11px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image019.png\" \/> CF and AD = CF<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image008.png\" \/>ACFD is a parallelogram.<\/p>\n<p style=\"text-align: justify;\"><strong>(v)<\/strong> As ACFD is a parallelogram.<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image007.png\" \/>AC = DF<\/p>\n<p style=\"text-align: justify;\"><strong>(vi)<\/strong> In <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image020.png\" \/>ABC and <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image020.png\" \/>DEF,<\/p>\n<p style=\"text-align: justify;\">AB = DE [Given]\n<p style=\"text-align: justify;\">BC = EF [Given]\n<p style=\"text-align: justify;\">AC = DF [Proved]\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image007.png\" \/><img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image020.png\" \/>ABC <img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image039.png\" \/> <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image020.png\" \/>DEF[By SSS congruency]\n<hr \/>\n<p style=\"text-align: center;\">NCERT Solutions for Class 9 Maths Exercise 8.1<\/p>\n<h6 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"12_ABCD_is_a_trapezium_in_which_AB_CD_and_AD_BC_See_figure_Show_that\"><\/span><strong>12. ABCD is a trapezium in which AB <img decoding=\"async\" style=\"height: 21px; width: 11px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image019.png\" \/> CD and AD = BC (See figure). Show that:<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h6>\n<p style=\"text-align: justify;\"><strong><img decoding=\"async\" id=\"Picture 5\" style=\"height: 115px; width: 209px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image040.png\" \/><\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(i) <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>A = <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>B <\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(ii) <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>C = <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>D<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(iii) <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image020.png\" \/>ABC <img decoding=\"async\" style=\"height: 17px; width: 28px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image021.png\" \/>BAD<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(iv) Diagonal AC = Diagonal BD<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. Given<\/strong>: ABCD is a trapezium.<\/p>\n<p style=\"text-align: justify;\">AB <img decoding=\"async\" style=\"height: 21px; width: 11px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image019.png\" \/> CD and AD = BC<\/p>\n<p style=\"text-align: justify;\"><strong><img decoding=\"async\" id=\"Picture 19\" style=\"height: 129px; width: 258px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image041.png\" \/><\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>To prove<\/strong>: (i) <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>A = <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>B<\/p>\n<p style=\"text-align: justify;\">(ii) <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>C = <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>D<\/p>\n<p style=\"text-align: justify;\">(iii) <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image020.png\" \/>ABC <img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image039.png\" \/> <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image020.png\" \/>BAD<\/p>\n<p style=\"text-align: justify;\">(iv) Diag. AC = Diag. BD<\/p>\n<p style=\"text-align: justify;\"><strong>Construction<\/strong>: Draw CE <img decoding=\"async\" style=\"height: 21px; width: 11px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image019.png\" \/> AD and extend<\/p>\n<p style=\"text-align: justify;\">AB to intersect CE at E.<\/p>\n<p style=\"text-align: justify;\"><strong>Proof<\/strong>: <strong>(i)<\/strong> As AECD is a parallelogram.[By construction]\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image007.png\" \/>AD = EC<\/p>\n<p style=\"text-align: justify;\">But AD = BC [Given]\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image007.png\" \/>BC = EC<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image008.png\" \/><img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>3 = <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>4 [Angles opposite to equal sides are equal]\n<p style=\"text-align: justify;\">Now <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>1 + <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>4 = <img decoding=\"async\" style=\"height: 19px; width: 33px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image022.png\" \/> [Interior angles]\n<p style=\"text-align: justify;\">And <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>2 + <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>3 = <img decoding=\"async\" style=\"height: 19px; width: 33px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image022.png\" \/> [Linear pair]\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image008.png\" \/><img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>1 + <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>4 = <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>2 + <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>3<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image008.png\" \/><img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>1 = <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>2 [<img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image023.png\" \/> <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>3 = <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>4 ]\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image008.png\" \/><img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>A = <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>B<\/p>\n<p style=\"text-align: justify;\"><strong>(ii)<\/strong> <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>3 = <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>C[Alternate interior angles]\n<p style=\"text-align: justify;\">And <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>D = <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>4 [Opposite angles of a parallelogram]\n<p style=\"text-align: justify;\">But <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>3 = <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>4 [<img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image020.png\" \/>BCE is an isosceles triangle]\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image007.png\" \/><img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>C = <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>D<\/p>\n<p style=\"text-align: justify;\"><strong>(iii)<\/strong> In <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image020.png\" \/>ABC and <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image020.png\" \/>BAD,<\/p>\n<p style=\"text-align: justify;\">AB = AB [Common]\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>1 = <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image001.png\" \/>2 [Proved]\n<p style=\"text-align: justify;\">AD = BC [Given]\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image007.png\" \/><img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image020.png\" \/>ABC <img decoding=\"async\" style=\"height: 17px; width: 28px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image021.png\" \/>BAD[By SAS congruency]\n<p style=\"text-align: justify;\"><strong>(iv)<\/strong> We had observed that,<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image007.png\" \/><img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image020.png\" \/>ABC <img decoding=\"async\" style=\"height: 17px; width: 28px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image021.png\" \/>BAD<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/09\/maths\/ch08\/Ex8.1\/image008.png\" \/>AC = BD [By C.P.C.T.]\n<h2><span class=\"ez-toc-section\" id=\"NCERT_Solutions_for_Class_9_Maths_Exercise_81\"><\/span>NCERT Solutions for Class 9 Maths Exercise 8.1<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>NCERT Solutions for Class 9 Maths PDF (Download) Free from myCBSEguide app and myCBSEguide website. Ncert solution class 9 Maths includes text book solutions from Mathematics Book. NCERT Solutions for CBSE Class 9 Maths have total 15 chapters. 9 Maths NCERT Solutions in PDF for free Download on our website. Ncert Maths class 9 solutions PDF and Maths ncert class 9 PDF solutions with latest modifications and as per the latest CBSE syllabus are only available in myCBSEguide.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"CBSE_app_for_Class_9\"><\/span>CBSE app for Class 9<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>To download NCERT Solutions for Class 9 Maths, Computer Science, Home Science,Hindi ,English, Social Science do check myCBSEguide app or website. myCBSEguide provides sample papers with solution, test papers for chapter-wise practice, NCERT solutions, NCERT Exemplar solutions, quick revision notes for ready reference, CBSE guess papers and CBSE important question papers. Sample Paper all are made available through\u00a0<a href=\"https:\/\/play.google.com\/store\/apps\/details?id=in.techchefs.MyCBSEGuide&amp;referrer=utm_source%3Dmycbse_bottom%26utm_medium%3Dtext%26utm_campaign%3Dmycbseads\"><strong>the best app for CBSE students<\/strong><\/a>\u00a0and myCBSEguide website.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>NCERT Solutions for Class 9 Maths Exercise 8.1 book solutions are available in PDF format for free download. These ncert book chapter wise questions and answers are very helpful for CBSE board exam. CBSE recommends NCERT books and most of the questions in CBSE exam are asked from NCERT text books. Class 9 Maths chapter &#8230; <a title=\"NCERT Solutions for Class 9 Maths Exercise 8.1\" class=\"read-more\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-for-class-9-maths-exercise-8-1\/\" aria-label=\"More on NCERT Solutions for Class 9 Maths Exercise 8.1\">Read more<\/a><\/p>\n","protected":false},"author":1,"featured_media":-1,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1376,281],"tags":[283,1344,321,1485,282],"class_list":["post-5032","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-mathematics-cbse-class-09","category-ncert-solutions","tag-cbse-study-material","tag-class-9","tag-mathematics","tag-maths","tag-ncert-solution"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.0 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>NCERT Solutions for Class 9 Maths Exercise 8.1 | myCBSEguide<\/title>\n<meta name=\"description\" content=\"NCERT Solutions for Class 9 Maths Exercise 8.1 in PDF format for free download. 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