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sizes=\"(max-width: 402px) 100vw, 402px\" \/><\/figure><\/div>\n\n\n\n<p>\u0cb8\u0cae\u0cbe\u0ca8 \u0cac\u0cbf\u0c82\u0ca6\u0cc1\u0cb5\u0cbf\u0ca8\u0cb2\u0ccd\u0cb2\u0cbf \u0c8e\u0cb0\u0ca1\u0cc1 \u0cb8\u0cb0\u0cb3 \u0cb0\u0cc7\u0c96\u0cc6\u0c97\u0cb3\u0cc1 \u0caa\u0cb0\u0cb8\u0ccd\u0caa\u0cb0 \u0c9b\u0cc7\u0ca6\u0cbf\u0cb8\u0cc1\u0cb5 \u0cb0\u0cc7\u0c96\u0cc6\u0c97\u0cb3\u0cc1<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">\u0cb2\u0c82\u0cac\u0cb0\u0cc7\u0c96\u0cc6\u0c97\u0cb3\u0cc1<\/h3>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-large is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/themindpalace.in\/wp-content\/uploads\/2020\/10\/lambarekhegalu.jpg\" alt=\"\u0cb2\u0c82\u0cac\u0cb0\u0cc7\u0c96\u0cc6\u0c97\u0cb3\u0cc1\" class=\"wp-image-2834\" width=\"387\" height=\"408\" srcset=\"https:\/\/themindpalace.in\/wp-content\/uploads\/2020\/10\/lambarekhegalu.jpg 774w, 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src=\"https:\/\/themindpalace.in\/wp-content\/uploads\/2020\/10\/parshva-konagalu.jpg\" alt=\"\u0caa\u0cbe\u0cb0\u0ccd\u0cb6\u0ccd\u0cb5 \u0c95\u0ccb\u0ca8\u0c97\u0cb3\u0cc1\" class=\"wp-image-2835\" width=\"407\" height=\"278\" srcset=\"https:\/\/themindpalace.in\/wp-content\/uploads\/2020\/10\/parshva-konagalu.jpg 814w, https:\/\/themindpalace.in\/wp-content\/uploads\/2020\/10\/parshva-konagalu-300x205.jpg 300w, https:\/\/themindpalace.in\/wp-content\/uploads\/2020\/10\/parshva-konagalu-768x524.jpg 768w\" sizes=\"(max-width: 407px) 100vw, 407px\" \/><\/figure><\/div>\n\n\n\n<p>\u0caf\u0cbe\u0cb5\u0cc1\u0ca6\u0cc7 \u0c8e\u0cb0\u0ca1\u0cc1 \u0c95\u0ccb\u0ca8\u0c97\u0cb3\u0cc1 \u0c89\u0cad\u0caf \u0cb8\u0cbe\u0cae\u0cbe\u0ca8\u0ccd\u0caf \u0cac\u0cbe\u0cb9\u0cc1, \u0c92\u0c82\u0ca6\u0cc1 \u0c89\u0cad\u0caf \u0cb8\u0cbe\u0cae\u0cbe\u0ca8\u0ccd\u0caf \u0cb6\u0cc3\u0c82\u0c97 \u0cac\u0cbf\u0c82\u0ca6\u0cc1\u0cb5\u0ca8\u0ccd\u0ca8\u0cc1 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src=\"https:\/\/themindpalace.in\/wp-content\/uploads\/2020\/10\/paripooraka-kona.jpg\" alt=\"\u0caa\u0cb0\u0cbf\u0caa\u0cc2\u0cb0\u0c95 \u0c95\u0ccb\u0ca8\u0c97\u0cb3\u0cc1 \" class=\"wp-image-2864\" width=\"485\" height=\"332\" srcset=\"https:\/\/themindpalace.in\/wp-content\/uploads\/2020\/10\/paripooraka-kona.jpg 969w, https:\/\/themindpalace.in\/wp-content\/uploads\/2020\/10\/paripooraka-kona-300x206.jpg 300w, https:\/\/themindpalace.in\/wp-content\/uploads\/2020\/10\/paripooraka-kona-768x526.jpg 768w\" sizes=\"(max-width: 485px) 100vw, 485px\" \/><\/figure><\/div>\n\n\n\n<p>\u0c8e\u0cb0\u0ca1\u0cc1 \u0c95\u0ccb\u0ca8\u0c97\u0cb3 \u0cae\u0cca\u0ca4\u0ccd\u0ca4 180\u02da \u0c87\u0ca6\u0ccd\u0ca6\u0cb0\u0cc6, \u0c86 \u0c8e\u0cb0\u0ca1\u0cc1 \u0c95\u0ccb\u0ca8\u0c97\u0cb3\u0ca8\u0ccd\u0ca8\u0cc1 \u201c\u0caa\u0cb0\u0cbf\u0caa\u0cc2\u0cb0\u0c95 \u0c95\u0ccb\u0ca8\u0c97\u0cb3\u0cc1\u201d (\u0c85\u0ca5\u0cb5\u0cbe \u0cb8\u0c82\u0caa\u0cc2\u0cb0\u0c95 \u0c95\u0ccb\u0ca8\u0c97\u0cb3\u0cc1)(supplementary) \u0c8e\u0ca8\u0ccd\u0ca8\u0cc1\u0ca4\u0ccd\u0ca4\u0cc7\u0cb5\u0cc6<\/p>\n\n\n\n<p>\u0c87\u0cb2\u0ccd\u0cb2\u0cbf 62\u02da \u0cae\u0ca4\u0ccd\u0ca4\u0cc1 118\u02da \u0c85\u0cb3\u0ca4\u0cc6\u0caf \u0c95\u0ccb\u0ca8\u0c97\u0cb3\u0cc1 \u0cb8\u0c82\u0caa\u0cc2\u0cb0\u0c95 \u0c95\u0ccb\u0ca8\u0c97\u0cb3\u0cbe\u0c97\u0cbf\u0cb5\u0cc6<\/p>\n\n\n\n<p>\u0c8e\u0cb0\u0ca1\u0cc1 \u0cb8\u0c82\u0caa\u0cc2\u0cb0\u0c95 \u0c95\u0ccb\u0ca8\u0c97\u0cb3\u0ca8\u0ccd\u0ca8\u0cc1 \u0caa\u0cb0\u0cb8\u0ccd\u0caa\u0cb0 \u0cb8\u0c82\u0caa\u0cc2\u0cb0\u0c95 \u0c8e\u0c82\u0ca6\u0cc1 \u0c95\u0cb0\u0cc6\u0caf\u0cb2\u0cbe\u0c97\u0cc1\u0ca4\u0ccd\u0ca4\u0ca6\u0cc6<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">\u0caa\u0cbe\u0cb0\u0ccd\u0cb6\u0ccd\u0cb5 \u0c95\u0ccb\u0ca8\u0c97\u0cb3\u0cc1<\/h3>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-large is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/themindpalace.in\/wp-content\/uploads\/2020\/10\/parshva-konagalu-2.jpg\" alt=\"\u0caa\u0cbe\u0cb0\u0ccd\u0cb6\u0ccd\u0cb5 \u0c95\u0ccb\u0ca8\u0c97\u0cb3\u0cc1\" class=\"wp-image-2863\" width=\"421\" height=\"423\" srcset=\"https:\/\/themindpalace.in\/wp-content\/uploads\/2020\/10\/parshva-konagalu-2.jpg 841w, https:\/\/themindpalace.in\/wp-content\/uploads\/2020\/10\/parshva-konagalu-2-298x300.jpg 298w, https:\/\/themindpalace.in\/wp-content\/uploads\/2020\/10\/parshva-konagalu-2-150x150.jpg 150w, https:\/\/themindpalace.in\/wp-content\/uploads\/2020\/10\/parshva-konagalu-2-768x773.jpg 768w, https:\/\/themindpalace.in\/wp-content\/uploads\/2020\/10\/parshva-konagalu-2-80x80.jpg 80w\" sizes=\"(max-width: 421px) 100vw, 421px\" \/><\/figure><\/div>\n\n\n\n<p>\u0c92\u0c82\u0ca6\u0cc1 \u0cb8\u0cbe\u0cae\u0cbe\u0ca8\u0ccd\u0caf \u0cb6\u0cc3\u0c82\u0c97, \u0c92\u0c82\u0ca6\u0cc1 \u0cb8\u0cbe\u0cae\u0cbe\u0ca8\u0ccd\u0caf \u0cac\u0cbe\u0cb9\u0cc1 \u0cae\u0ca4\u0ccd\u0ca4\u0cc1 \u0cb8\u0cbe\u0cae\u0cbe\u0ca8\u0ccd\u0caf\u0cb5\u0cb2\u0ccd\u0cb2\u0ca6 \u0c8e\u0cb0\u0ca1\u0cc1 \u0cac\u0cbe\u0cb9\u0cc1\u0c97\u0cb3\u0cc1 \u0cb8\u0cbe\u0cae\u0cbe\u0ca8\u0ccd\u0caf \u0cac\u0cbe\u0cb9\u0cc1\u0cb5\u0cbf\u0ca8 \u0cb5\u0cbf\u0cb5\u0cbf\u0ca7 \u0c95\u0ca1\u0cc6 \u0c87\u0ca6\u0ccd\u0ca6\u0cbe\u0c97 \u0c89\u0c82\u0c9f\u0cbe\u0c97\u0cc1\u0cb5 \u0c95\u0ccb\u0ca8\u0c97\u0cb3\u0ca8\u0ccd\u0ca8\u0cc1&nbsp;&nbsp; \u0caa\u0cbe\u0cb0\u0ccd\u0cb6\u0ccd\u0cb5\u0c95\u0ccb\u0ca8\u0c97\u0cb3\u0cc1 \u0c8e\u0ca8\u0ccd\u0ca8\u0cc1\u0ca4\u0ccd\u0ca4\u0cc7\u0cb5\u0cc6 (adjacent angles)<\/p>\n\n\n\n<p>\u0c85\u0cb5\u0cc1\u0c97\u0cb3 \u0cb8\u0cbe\u0cae\u0cbe\u0ca8\u0ccd\u0caf\u0cb5\u0cb2\u0ccd\u0cb2\u0ca6 \u0cac\u0cbe\u0cb9\u0cc1\u0c97\u0cb3\u0cc1 \u0cb8\u0cbe\u0cae\u0cbe\u0ca8\u0ccd\u0caf \u0cac\u0cbe\u0cb9\u0cc1\u0cb5\u0cbf\u0ca8 \u0c8e\u0cb0\u0ca1\u0cc2 \u0cac\u0ca6\u0cbf\u0caf\u0cb2\u0ccd\u0cb2\u0cbf\u0cb5\u0cc6<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">\u0cb8\u0cb0\u0cb3\u0caf\u0cc1\u0c97\u0ccd\u0cae \u0c95\u0ccb\u0ca8\u0c97\u0cb3\u0cc1<\/h3>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-large is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/themindpalace.in\/wp-content\/uploads\/2020\/10\/saralayugma-kona-1024x601.jpg\" alt=\"\u0cb8\u0cb0\u0cb3\u0caf\u0cc1\u0c97\u0ccd\u0cae \u0c95\u0ccb\u0ca8\u0c97\u0cb3\u0cc1 \" class=\"wp-image-2862\" width=\"512\" height=\"301\" srcset=\"https:\/\/themindpalace.in\/wp-content\/uploads\/2020\/10\/saralayugma-kona-1024x601.jpg 1024w, https:\/\/themindpalace.in\/wp-content\/uploads\/2020\/10\/saralayugma-kona-300x176.jpg 300w, https:\/\/themindpalace.in\/wp-content\/uploads\/2020\/10\/saralayugma-kona-768x451.jpg 768w, https:\/\/themindpalace.in\/wp-content\/uploads\/2020\/10\/saralayugma-kona.jpg 1091w\" sizes=\"(max-width: 512px) 100vw, 512px\" \/><\/figure><\/div>\n\n\n\n<p>\u0c92\u0c82\u0ca6\u0cc1 \u0cb8\u0cb0\u0cb3\u0cb0\u0cc7\u0c96\u0cc6\u0caf \u0cae\u0cc7\u0cb2\u0cc6 \u0c92\u0c82\u0ca6\u0cc1 \u0cb0\u0cc7\u0c96\u0cbe\u0c95\u0cbf\u0cb0\u0ca3 \u0ca8\u0cbf\u0c82\u0ca4\u0cbe\u0c97 \u0c86 \u0cb8\u0cbe\u0cae\u0cbe\u0ca8\u0ccd\u0caf \u0cac\u0cbf\u0c82\u0ca6\u0cc1\u0cb5\u0cbf\u0ca8\u0cb2\u0ccd\u0cb2\u0cbf \u0c89\u0c82\u0c9f\u0cbe\u0c97\u0cc1\u0cb5 \u0c95\u0ccb\u0ca8\u0c97\u0cb3 \u0cae\u0cca\u0ca4\u0ccd\u0ca4 180\u02da&nbsp; \u0c87\u0cb0\u0cc1\u0ca4\u0ccd\u0ca4\u0ca6\u0cc6. \u0c86 \u0c95\u0ccb\u0ca8\u0c97\u0cb3\u0ca8\u0ccd\u0ca8\u0cc1 \u0cb8\u0cb0\u0cb3\u0caf\u0cc1\u0c97\u0ccd\u0cae \u0c95\u0ccb\u0ca8\u0c97\u0cb3\u0cc1 \u0c8e\u0ca8\u0ccd\u0ca8\u0cc1\u0ca4\u0ccd\u0ca4\u0cc7\u0cb5\u0cc6.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">\u0cb6\u0cc3\u0c82\u0c97\u0cbe\u0cad\u0cbf\u0cae\u0cc1\u0c96 \u0c95\u0ccb\u0ca8\u0c97\u0cb3\u0cc1<\/h3>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-large is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/themindpalace.in\/wp-content\/uploads\/2020\/10\/shringabhimukha-kona-2.jpg\" alt=\"\u0cb6\u0cc3\u0c82\u0c97\u0cbe\u0cad\u0cbf\u0cae\u0cc1\u0c96 \u0c95\u0ccb\u0ca8\u0c97\u0cb3\u0cc1\" class=\"wp-image-2861\" width=\"482\" height=\"376\" srcset=\"https:\/\/themindpalace.in\/wp-content\/uploads\/2020\/10\/shringabhimukha-kona-2.jpg 963w, https:\/\/themindpalace.in\/wp-content\/uploads\/2020\/10\/shringabhimukha-kona-2-300x234.jpg 300w, https:\/\/themindpalace.in\/wp-content\/uploads\/2020\/10\/shringabhimukha-kona-2-768x600.jpg 768w\" sizes=\"(max-width: 482px) 100vw, 482px\" \/><\/figure><\/div>\n\n\n\n<p>\u0c8e\u0cb0\u0ca1\u0cc1 \u0cb8\u0cb0\u0cb3\u0cb0\u0cc7\u0c96\u0cc6\u0c97\u0cb3\u0cc1 \u0caa\u0cb0\u0cb8\u0ccd\u0caa\u0cb0 \u0c9b\u0cc7\u0ca6\u0cbf\u0cb8\u0cbf\u0ca6\u0cbe\u0c97 \u0c89\u0c82\u0c9f\u0cbe\u0c97\u0cc1\u0cb5 \u0cb6\u0cc3\u0c82\u0c97\u0cbe\u0cad\u0cbf\u0cae\u0cc1\u0c96 \u0c95\u0ccb\u0ca8\u0c97\u0cb3\u0cc1<\/p>\n\n\n\n<p>\u0cb6\u0cc3\u0c82\u0c97\u0cbe\u0cad\u0cbf\u0cae\u0cc1\u0c96 \u0c95\u0ccb\u0ca8\u0c97\u0cb3\u0cc1 \u0caa\u0cb0\u0cb8\u0ccd\u0caa\u0cb0 \u0cb8\u0cae\u0cb5\u0cbe\u0c97\u0cbf\u0cb0\u0cc1\u0ca4\u0ccd\u0ca4\u0cb5\u0cc6<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">\u0c9b\u0cc7\u0ca6\u0c95 \u0cb0\u0cc7\u0c96\u0cc6<\/h3>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-large is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/themindpalace.in\/wp-content\/uploads\/2020\/10\/chedaka-rekhe.jpg\" alt=\"\u0c9b\u0cc7\u0ca6\u0c95 \u0cb0\u0cc7\u0c96\u0cc6\" class=\"wp-image-2860\" width=\"401\" height=\"409\" srcset=\"https:\/\/themindpalace.in\/wp-content\/uploads\/2020\/10\/chedaka-rekhe.jpg 802w, https:\/\/themindpalace.in\/wp-content\/uploads\/2020\/10\/chedaka-rekhe-294x300.jpg 294w, https:\/\/themindpalace.in\/wp-content\/uploads\/2020\/10\/chedaka-rekhe-768x782.jpg 768w\" sizes=\"(max-width: 401px) 100vw, 401px\" \/><\/figure><\/div>\n\n\n\n<p>\u0c92\u0c82\u0ca6\u0cc7 \u0cb8\u0cae\u0ca4\u0cb2\u0ca6\u0cb2\u0ccd\u0cb2\u0cbf\u0cb0\u0cc1\u0cb5 \u0c8e\u0cb0\u0ca1\u0cc1 \u0cb8\u0cb0\u0cb3 \u0cb0\u0cc7\u0c96\u0cc6\u0c97\u0cb3\u0ca8\u0ccd\u0ca8\u0cc1 \u0cac\u0cc7\u0cb0\u0cc6 \u0cac\u0cc7\u0cb0\u0cc6 \u0cac\u0cbf\u0c82\u0ca6\u0cc1\u0c97\u0cb3\u0cb2\u0ccd\u0cb2\u0cbf \u0c9b\u0cc7\u0ca6\u0cbf\u0cb8\u0cc1\u0cb5 \u0cb0\u0cc7\u0c96\u0cc6\u0caf\u0ca8\u0ccd\u0ca8\u0cc1 \u2018\u0c9b\u0cc7\u0ca6\u0c95 \u0cb0\u0cc7\u0c96\u0cc6\u2019 (transversal) \u0c8e\u0ca8\u0ccd\u0ca8\u0cc1\u0cb5\u0cb0\u0cc1<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">\u0c85\u0c82\u0ca4\u0cb0\u0ccd \u0c95\u0ccb\u0ca8\u0c97\u0cb3\u0cc1 &amp; \u0cac\u0cb9\u0cbf\u0cb0\u0ccd \u0c95\u0ccb\u0ca8\u0c97\u0cb3\u0cc1<\/h3>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-large is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/themindpalace.in\/wp-content\/uploads\/2020\/10\/antar-mattu-bahir-kona.jpg\" alt=\"\u0c85\u0c82\u0ca4\u0cb0\u0ccd \u0c95\u0ccb\u0ca8\u0c97\u0cb3\u0cc1 &amp;  \u0cac\u0cb9\u0cbf\u0cb0\u0ccd \u0c95\u0ccb\u0ca8\u0c97\u0cb3\u0cc1\" class=\"wp-image-2859\" width=\"397\" height=\"402\" srcset=\"https:\/\/themindpalace.in\/wp-content\/uploads\/2020\/10\/antar-mattu-bahir-kona.jpg 793w, https:\/\/themindpalace.in\/wp-content\/uploads\/2020\/10\/antar-mattu-bahir-kona-296x300.jpg 296w, https:\/\/themindpalace.in\/wp-content\/uploads\/2020\/10\/antar-mattu-bahir-kona-768x779.jpg 768w, https:\/\/themindpalace.in\/wp-content\/uploads\/2020\/10\/antar-mattu-bahir-kona-80x80.jpg 80w\" sizes=\"(max-width: 397px) 100vw, 397px\" \/><\/figure><\/div>\n\n\n\n<p class=\"has-text-align-center\"><strong>3, 4, 5 and 6 \u0c85\u0c82\u0ca4\u0cb0\u0ccd \u0c95\u0ccb\u0ca8\u0c97\u0cb3\u0cc1<\/strong><\/p>\n\n\n\n<p class=\"has-text-align-center\"><strong>1, 2, 7 and 8 \u0cac\u0cb9\u0cbf\u0cb0\u0ccd \u0c95\u0ccb\u0ca8\u0c97\u0cb3\u0cc1<\/strong><\/p>\n\n\n\n<p>\u0c92\u0c82\u0ca6\u0cc1 \u0c9b\u0cc7\u0ca6\u0c95\u0cb5\u0cc1 \u0c92\u0c82\u0ca6\u0cc1 \u0c9c\u0cca\u0ca4\u0cc6 \u0c85\u0c82\u0ca4\u0cb0\u0ccd \u0c95\u0ccb\u0ca8\u0c97\u0cb3\u0cc1 \u0cb9\u0cbe\u0c97\u0cc1 \u0c92\u0c82\u0ca6\u0cc1 \u0c9c\u0cca\u0ca4\u0cc6 \u0cac\u0cb9\u0cbf\u0cb0\u0ccd \u0c95\u0ccb\u0ca8\u0c97\u0cb3\u0ca8\u0ccd\u0ca8\u0cc1&nbsp; \u0c89\u0c82\u0c9f\u0cc1 \u0cae\u0cbe\u0ca1\u0cc1\u0ca4\u0ccd\u0ca4\u0ca6\u0cc6.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">\u0c9b\u0cc7\u0ca6\u0c95\u0ca6\u0cbf\u0c82\u0ca6 \u0c89\u0c82\u0c9f\u0cbe\u0c97\u0cc1\u0cb5 \u0c95\u0ccb\u0ca8\u0c97\u0cb3\u0cc1<\/h2>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-large is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/themindpalace.in\/wp-content\/uploads\/2020\/10\/anuroopa-konagalu.jpg\" alt=\"\u0c85\u0ca8\u0cc1\u0cb0\u0cc2\u0caa\u0c95\u0ccb\u0ca8\u0c97\u0cb3\u0cc1\" class=\"wp-image-2858\" width=\"448\" height=\"392\" srcset=\"https:\/\/themindpalace.in\/wp-content\/uploads\/2020\/10\/anuroopa-konagalu.jpg 895w, https:\/\/themindpalace.in\/wp-content\/uploads\/2020\/10\/anuroopa-konagalu-300x262.jpg 300w, https:\/\/themindpalace.in\/wp-content\/uploads\/2020\/10\/anuroopa-konagalu-768x672.jpg 768w\" sizes=\"(max-width: 448px) 100vw, 448px\" \/><\/figure><\/div>\n\n\n\n<p>\u0c92\u0c82\u0ca6\u0cc1 \u0c9b\u0cc7\u0ca6\u0c95 \u0cb0\u0cc7\u0c96\u0cc6\u0caf\u0cc1 \u0c8e\u0cb0\u0ca1\u0cc1 \u0cb8\u0cae\u0cbe\u0c82\u0ca4\u0cb0 \u0cb8\u0cb0\u0cb3\u0cb0\u0cc7\u0c96\u0cc6\u0c97\u0cb3\u0ca8\u0ccd\u0ca8\u0cc1 \u0c9b\u0cc7\u0ca6\u0cbf\u0cb8\u0cbf\u0ca6\u0cbe\u0c97 \u0c89\u0c82\u0c9f\u0cbe\u0c97\u0cc1\u0cb5 \u0c85\u0ca8\u0cc1\u0cb0\u0cc2\u0caa\u0c95\u0ccb\u0ca8\u0c97\u0cb3\u0cc1 \u0cb8\u0cae\u0cb5\u0cbe\u0c97\u0cbf\u0cb0\u0cc1\u0ca4\u0ccd\u0ca4\u0ca6\u0cc6<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">\u0caa\u0cb0\u0ccd\u0caf\u0cbe\u0caf \u0c85\u0c82\u0ca4\u0cb0\u0ccd \u0c95\u0ccb\u0ca8\u0c97\u0cb3\u0cc1 &amp; 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\u0c9b\u0cc7\u0ca6\u0cbf\u0cb8\u0cbf\u0ca6\u0cbe\u0c97 \u0c89\u0c82\u0c9f\u0cbe\u0c97\u0cc1\u0cb5 \u0c85\u0ca8\u0cc1\u0cb0\u0cc2\u0caa\u0c95\u0ccb\u0ca8\u0c97\u0cb3\u0cc1 \u0cb8\u0cae\u0cb5\u0cbe\u0c97\u0cbf\u0cb0\u0cc1\u0ca4\u0ccd\u0ca4\u0ca6\u0cc6<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-large is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/themindpalace.in\/wp-content\/uploads\/2020\/10\/k-lines-axiom-3.jpg\" alt=\"\" class=\"wp-image-2849\" width=\"420\" height=\"382\" srcset=\"https:\/\/themindpalace.in\/wp-content\/uploads\/2020\/10\/k-lines-axiom-3.jpg 839w, https:\/\/themindpalace.in\/wp-content\/uploads\/2020\/10\/k-lines-axiom-3-300x273.jpg 300w, https:\/\/themindpalace.in\/wp-content\/uploads\/2020\/10\/k-lines-axiom-3-768x699.jpg 768w\" sizes=\"(max-width: 420px) 100vw, 420px\" \/><\/figure><\/div>\n\n\n\n<p>\u0c87\u0ca6\u0ca8\u0ccd\u0ca8\u0cc1 \u0c85\u0ca8\u0cc1\u0cb0\u0cc2\u0caa \u0c95\u0ccb\u0ca8\u0c97\u0cb3\u0cc1 \u0c86\u0ca7\u0cbe\u0cb0&nbsp; \u0caa\u0ccd\u0cb0\u0ca4\u0cbf\u0c9c\u0ccd\u0c9e\u0cc6 \u0c8e\u0c82\u0ca6\u0cc2 \u0cb8\u0cb9 \u0c95\u0cb0\u0cc6\u0caf\u0cc1\u0cb5\u0cb0\u0cc1<\/p>\n\n\n\n<hr class=\"wp-block-separator\"\/>\n\n\n\n<p>\u0c86\u0ca7\u0cbe\u0cb0 \u0caa\u0ccd\u0cb0\u0ca4\u0cbf\u0c9c\u0ccd\u0c9e\u0cc6 4 \u0c92\u0c82\u0ca6\u0cc1 \u0c9b\u0cc7\u0ca6\u0c95\u0cb5\u0cc1 \u0c8e\u0cb0\u0ca1\u0cc1 \u0cb0\u0cc7\u0c96\u0cc6\u0c97\u0cb3\u0ca8\u0ccd\u0ca8\u0cc1 \u0c9b\u0cc7\u0ca6\u0cbf\u0cb8\u0cbf\u0ca6\u0cbe\u0c97 \u0c89\u0c82\u0c9f\u0cbe\u0c97\u0cc1\u0cb5 \u0c92\u0c82\u0ca6\u0cc1 \u0c9c\u0cca\u0ca4\u0cc6 \u0c85\u0ca8\u0cc1\u0cb0\u0cc2\u0caa \u0c95\u0ccb\u0ca8\u0c97\u0cb3\u0cc1 \u0cb8\u0cae\u0cb5\u0cbf\u0ca6\u0ccd\u0ca6\u0cbe\u0c97 \u0c86 \u0c8e\u0cb0\u0ca1\u0cc1 \u0cb0\u0cc7\u0c96\u0cc6\u0c97\u0cb3\u0cc1 \u0caa\u0cb0\u0cb8\u0ccd\u0caa\u0cb0 \u0cb8\u0cae\u0cbe\u0c82\u0ca4\u0cb0\u0cb5\u0cbe\u0c97\u0cbf\u0cb0\u0cc1\u0ca4\u0ccd\u0ca4\u0cb5\u0cc6<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-large is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/themindpalace.in\/wp-content\/uploads\/2020\/10\/k-lines-axiom-4.jpg\" alt=\"\" class=\"wp-image-2848\" width=\"431\" height=\"377\" srcset=\"https:\/\/themindpalace.in\/wp-content\/uploads\/2020\/10\/k-lines-axiom-4.jpg 861w, https:\/\/themindpalace.in\/wp-content\/uploads\/2020\/10\/k-lines-axiom-4-300x263.jpg 300w, https:\/\/themindpalace.in\/wp-content\/uploads\/2020\/10\/k-lines-axiom-4-768x673.jpg 768w\" sizes=\"(max-width: 431px) 100vw, 431px\" \/><\/figure><\/div>\n\n\n\n<hr class=\"wp-block-separator\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">\u0caa\u0ccd\u0cb0\u0cae\u0cc7\u0caf\u0c97\u0cb3\u0cc1<\/h2>\n\n\n\n<p>\u0caa\u0ccd\u0cb0\u0cae\u0cc7\u0caf 1 \u0c8e\u0cb0\u0ca1\u0cc1 \u0cb8\u0cb0\u0cb3\u0cb0\u0cc7\u0c96\u0cc6\u0c97\u0cb3\u0cc1 \u0caa\u0cb0\u0cb8\u0ccd\u0caa\u0cb0 \u0c9b\u0cc7\u0ca6\u0cbf\u0cb8\u0cbf\u0ca6\u0cbe\u0c97 \u0c89\u0c82\u0c9f\u0cbe\u0c97\u0cc1\u0cb5 \u0cb6\u0cc3\u0c82\u0c97\u0cbe\u0cad\u0cbf\u0cae\u0cc1\u0c96 \u0c95\u0ccb\u0ca8\u0c97\u0cb3\u0cc1 \u0caa\u0cb0\u0cb8\u0ccd\u0caa\u0cb0 \u0cb8\u0cae\u0cb5\u0cbe\u0c97\u0cbf\u0cb0\u0cc1\u0ca4\u0ccd\u0ca4\u0cb5\u0cc6<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-large is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/themindpalace.in\/wp-content\/uploads\/2020\/10\/k-lines-theorem-1.jpg\" alt=\"\" class=\"wp-image-2847\" width=\"353\" height=\"223\" srcset=\"https:\/\/themindpalace.in\/wp-content\/uploads\/2020\/10\/k-lines-theorem-1.jpg 706w, https:\/\/themindpalace.in\/wp-content\/uploads\/2020\/10\/k-lines-theorem-1-300x189.jpg 300w\" sizes=\"(max-width: 353px) 100vw, 353px\" \/><\/figure><\/div>\n\n\n\n<hr class=\"wp-block-separator\"\/>\n\n\n\n<p>\u0caa\u0ccd\u0cb0\u0cae\u0cc7\u0caf 2 &nbsp;\u0c8e\u0cb0\u0ca1\u0cc1 \u0cb8\u0cae\u0cbe\u0c82\u0ca4\u0cb0 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https:\/\/themindpalace.in\/wp-content\/uploads\/2020\/10\/k-lines-theorem-2-768x578.jpg 768w\" sizes=\"(max-width: 402px) 100vw, 402px\" \/><\/figure><\/div>\n\n\n\n<hr class=\"wp-block-separator\"\/>\n\n\n\n<p>\u0caa\u0ccd\u0cb0\u0cae\u0cc7\u0caf 3 \u0c92\u0c82\u0ca6\u0cc1 \u0c9b\u0cc7\u0ca6\u0c95\u0cb5\u0cc1 \u0c8e\u0cb0\u0ca1\u0cc1 \u0cb0\u0cc7\u0c96\u0cc6\u0c97\u0cb3\u0ca8\u0ccd\u0ca8\u0cc1 \u0c9b\u0cc7\u0ca6\u0cbf\u0cb8\u0cbf\u0ca6\u0cbe\u0c97 \u0c89\u0c82\u0c9f\u0cbe\u0c97\u0cc1\u0cb5 \u0c92\u0c82\u0ca6\u0cc1 \u0c9c\u0cca\u0ca4\u0cc6 \u0caa\u0cb0\u0ccd\u0caf\u0cbe\u0caf \u0c85\u0c82\u0ca4\u0cb0\u0ccd \u0c95\u0ccb\u0ca8\u0c97\u0cb3\u0cc1 \u0cb8\u0cae\u0cb5\u0cbf\u0ca6\u0ccd\u0ca6\u0cbe\u0c97 \u0c86 \u0c8e\u0cb0\u0ca1\u0cc1 \u0cb0\u0cc7\u0c96\u0cc6\u0c97\u0cb3\u0cc1 \u0caa\u0cb0\u0cb8\u0ccd\u0caa\u0cb0 \u0cb8\u0cae\u0cbe\u0c82\u0ca4\u0cb0\u0cb5\u0cbe\u0c97\u0cbf\u0cb0\u0cc1\u0ca4\u0ccd\u0ca4\u0cb5\u0cc6<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-large is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/themindpalace.in\/wp-content\/uploads\/2020\/10\/k-lines-theorem-3.jpg\" alt=\"\" class=\"wp-image-2845\" width=\"373\" height=\"272\" srcset=\"https:\/\/themindpalace.in\/wp-content\/uploads\/2020\/10\/k-lines-theorem-3.jpg 497w, https:\/\/themindpalace.in\/wp-content\/uploads\/2020\/10\/k-lines-theorem-3-300x219.jpg 300w\" sizes=\"(max-width: 373px) 100vw, 373px\" \/><\/figure><\/div>\n\n\n\n<hr class=\"wp-block-separator\"\/>\n\n\n\n<p>\u0caa\u0ccd\u0cb0\u0cae\u0cc7\u0caf 4 \u0c92\u0c82\u0ca6\u0cc1 \u0c9b\u0cc7\u0ca6\u0c95\u0cb5\u0cc1 \u0c8e\u0cb0\u0ca1\u0cc1 \u0cb8\u0cae\u0cbe\u0c82\u0ca4\u0cb0 \u0cb0\u0cc7\u0c96\u0cc6\u0c97\u0cb3\u0ca8\u0ccd\u0ca8\u0cc1 \u0c9b\u0cc7\u0ca6\u0cbf\u0cb8\u0cbf\u0ca6\u0cbe\u0c97, \u0c9b\u0cc7\u0ca6\u0c95\u0ca6 \u0c92\u0c82\u0ca6\u0cc7 \u0caa\u0cbe\u0cb0\u0ccd\u0cb6\u0ccd\u0cb5\u0ca6\u0cb2\u0ccd\u0cb2\u0cbf\u0cb0\u0cc1\u0cb5 \u0caa\u0ccd\u0cb0\u0ca4\u0cbf\u0caf\u0cca\u0c82\u0ca6\u0cc1 \u0c9c\u0cca\u0ca4\u0cc6 \u0c85\u0c82\u0ca4\u0cb0\u0ccd \u0c95\u0ccb\u0ca8\u0c97\u0cb3\u0cc1 \u0cb8\u0cb0\u0cb3\u0c95\u0ccb\u0ca8 \u0caa\u0cc2\u0cb0\u0c95\u0c97\u0cb3\u0cbe\u0c97\u0cbf\u0cb0\u0cc1\u0ca4\u0ccd\u0ca4\u0cb5\u0cc6<\/p>\n\n\n\n<div class=\"wp-block-jetpack-tiled-gallery aligncenter is-style-rectangular\"><div class=\"tiled-gallery__gallery\"><div class=\"tiled-gallery__row\"><div class=\"tiled-gallery__col\" style=\"flex-basis:45.1969038493197%\"><figure class=\"tiled-gallery__item\"><img decoding=\"async\" srcset=\"https:\/\/i2.wp.com\/themindpalace.in\/wp-content\/uploads\/2020\/10\/k-lines-thorem-4a.jpg?strip=info&#038;w=600&#038;ssl=1 600w,https:\/\/i2.wp.com\/themindpalace.in\/wp-content\/uploads\/2020\/10\/k-lines-thorem-4a.jpg?strip=info&#038;w=781&#038;ssl=1 781w\" alt=\"\" data-height=\"632\" data-id=\"2843\" data-link=\"https:\/\/themindpalace.in\/index.php\/2020\/10\/10\/lines-ad-angles-kannada\/k-lines-thorem-4a\/\" data-url=\"https:\/\/themindpalace.in\/wp-content\/uploads\/2020\/10\/k-lines-thorem-4a.jpg\" data-width=\"781\" src=\"https:\/\/i2.wp.com\/themindpalace.in\/wp-content\/uploads\/2020\/10\/k-lines-thorem-4a.jpg?ssl=1\" data-amp-layout=\"responsive\"\/><\/figure><\/div><div class=\"tiled-gallery__col\" style=\"flex-basis:54.80309615068031%\"><figure class=\"tiled-gallery__item\"><img decoding=\"async\" srcset=\"https:\/\/i1.wp.com\/themindpalace.in\/wp-content\/uploads\/2020\/10\/k-lines-thorem-4b.jpg?strip=info&#038;w=600&#038;ssl=1 600w,https:\/\/i1.wp.com\/themindpalace.in\/wp-content\/uploads\/2020\/10\/k-lines-thorem-4b.jpg?strip=info&#038;w=900&#038;ssl=1 900w,https:\/\/i1.wp.com\/themindpalace.in\/wp-content\/uploads\/2020\/10\/k-lines-thorem-4b.jpg?strip=info&#038;w=1200&#038;ssl=1 1200w,https:\/\/i1.wp.com\/themindpalace.in\/wp-content\/uploads\/2020\/10\/k-lines-thorem-4b.jpg?strip=info&#038;w=1471&#038;ssl=1 1471w\" alt=\"\" data-height=\"981\" data-id=\"2844\" data-link=\"https:\/\/themindpalace.in\/index.php\/2020\/10\/10\/lines-ad-angles-kannada\/k-lines-thorem-4b\/\" data-url=\"https:\/\/themindpalace.in\/wp-content\/uploads\/2020\/10\/k-lines-thorem-4b.jpg\" data-width=\"1471\" src=\"https:\/\/i1.wp.com\/themindpalace.in\/wp-content\/uploads\/2020\/10\/k-lines-thorem-4b.jpg?ssl=1\" data-amp-layout=\"responsive\"\/><\/figure><\/div><\/div><\/div><\/div>\n\n\n\n<hr class=\"wp-block-separator\"\/>\n\n\n\n<p>\u0caa\u0ccd\u0cb0\u0cae\u0cc7\u0caf 5 \u0c92\u0c82\u0ca6\u0cc1 \u0c9b\u0cc7\u0ca6\u0c95\u0cb5\u0cc1 \u0c8e\u0cb0\u0ca1\u0cc1 \u0cb0\u0cc7\u0c96\u0cc6\u0c97\u0cb3\u0ca8\u0ccd\u0ca8\u0cc1 \u0c9b\u0cc7\u0ca6\u0cbf\u0cb8\u0cbf\u0ca6\u0cbe\u0c97, \u0c9b\u0cc7\u0ca6\u0c95\u0ca6 \u0c92\u0c82\u0ca6\u0cc7 \u0caa\u0cbe\u0cb0\u0ccd\u0cb6\u0ccd\u0cb5\u0ca6\u0cb2\u0ccd\u0cb2\u0cbf\u0cb0\u0cc1\u0cb5 \u0caa\u0ccd\u0cb0\u0ca4\u0cbf\u0caf\u0cca\u0c82\u0ca6\u0cc1 \u0c9c\u0cca\u0ca4\u0cc6 \u0c85\u0c82\u0ca4\u0cb0\u0ccd \u0c95\u0ccb\u0ca8\u0c97\u0cb3\u0cc1 \u0cb8\u0cb0\u0cb3\u0c95\u0ccb\u0ca8 \u0caa\u0cc2\u0cb0\u0c95\u0c97\u0cb3\u0cbe\u0c97\u0cbf\u0ca6\u0ccd\u0ca6\u0cb0\u0cc6, \u0c86 \u0c8e\u0cb0\u0ca1\u0cc1 \u0cb0\u0cc7\u0c96\u0cc6\u0c97\u0cb3\u0cc1 \u0cb8\u0cbe\u0cae\u0c82\u0ca4\u0cb0\u0cb5\u0cbe\u0c97\u0cbf\u0cb0\u0cc1\u0ca4\u0ccd\u0ca4\u0cb5\u0cc6<\/p>\n\n\n\n<hr class=\"wp-block-separator\"\/>\n\n\n\n<p>\u0caa\u0ccd\u0cb0\u0cae\u0cc7\u0caf 6 &nbsp;: \u0c92\u0c82\u0ca6\u0cc7 \u0cb0\u0cc7\u0c96\u0cc6\u0c97\u0cc6 \u0cb8\u0cae\u0cbe\u0c82\u0ca4\u0cb0\u0cb5\u0cbe\u0c97\u0cbf\u0cb0\u0cc1\u0cb5 \u0c8e\u0cb0\u0ca1\u0cc1 \u0cb0\u0cc7\u0c96\u0cc6\u0c97\u0cb3\u0cc1 \u0caa\u0cb0\u0cb8\u0ccd\u0caa\u0cb0 \u0cb8\u0cae\u0cbe\u0c82\u0ca4\u0cb0\u0cb5\u0cbe\u0c97\u0cbf\u0cb0\u0cc1\u0ca4\u0ccd\u0ca4\u0cb5\u0cc6<\/p>\n\n\n\n<figure class=\"wp-block-pullquote\"><blockquote><p>\u0caa\u0ca0\u0ccd\u0caf\u0ca6 \u0c85\u0cad\u0ccd\u0caf\u0cbe\u0cb8\u0c97\u0cb3\u0cc1<\/p><\/blockquote><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\">\u0caa\u0ca0\u0ccd\u0caf\u0ca6 \u0c85\u0cad\u0ccd\u0caf\u0cbe\u0cb8\u0c97\u0cb3\u0cc1 <\/h2>\n\n\n\n<p><strong><span class=\"has-inline-color has-vivid-purple-color\">1.\u0c9a\u0cbf\u0ca4\u0ccd\u0cb0\u0ca6\u0cb2\u0ccd\u0cb2\u0cbf AB \u0cae\u0ca4\u0ccd\u0ca4\u0cc1 CD \u0cb8\u0cb0\u0cb3\u0cb0\u0cc7\u0c96\u0cc6\u0c97\u0cb3\u0cc1 O \u0cac\u0cbf\u0c82\u0ca6\u0cc1\u0cb5\u0cbf\u0ca8\u0cb2\u0ccd\u0cb2\u0cbf \u0c9b\u0cc7\u0ca6\u0cbf\u0cb8\u0cc1\u0ca4\u0ccd\u0ca4\u0cbf\u0cb5\u0cc6. \u2220AOC + \u2220BOE = 70\u00b0 \u0cae\u0ca4\u0ccd\u0ca4\u0cc1 \u2220BOD = 40\u00b0 \u0c86\u0ca6\u0cb0\u0cc6 \u2220BOE \u0cae\u0ca4\u0ccd\u0ca4\u0cc1 \u0cb8\u0cb0\u0cb3\u0cbe\u0ca7\u0cbf\u0c95 \u2220COE \u0c95\u0c82\u0ca1\u0cc1\u0cb9\u0cbf\u0ca1\u0cbf\u0caf\u0cbf\u0cb0\u0cbf.<\/span><\/strong><\/p>\n\n\n\n<p><strong><span class=\"has-inline-color has-vivid-red-color\">\u0c95\u0cca\u0c9f\u0ccd\u0c9f\u0cbf\u0cb0\u0cc1\u0cb5\u0c82\u0ca4\u0cc6 <\/span>:<\/strong> \u2220AOC + \u2220BOE = 70\u00b0<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"376\" height=\"257\" src=\"https:\/\/themindpalace.in\/wp-content\/uploads\/2020\/09\/1-pic.jpg\" alt=\"\" class=\"wp-image-925\" srcset=\"https:\/\/themindpalace.in\/wp-content\/uploads\/2020\/09\/1-pic.jpg 376w, https:\/\/themindpalace.in\/wp-content\/uploads\/2020\/09\/1-pic-300x205.jpg 300w\" sizes=\"(max-width: 376px) 100vw, 376px\" \/><\/figure><\/div>\n\n\n\n<p>&nbsp;&nbsp; \u2220BOD = 40\u00b0<\/p>\n\n\n\n<p><strong>\u0c95\u0c82\u0ca1\u0cc1\u0cb9\u0cbf\u0ca1\u0cbf\u0caf\u0cb2\u0cc1 : <\/strong>\u2220BOE \u0cae\u0ca4\u0ccd\u0ca4\u0cc1 \u0cb8\u0cb0\u0cb3\u0cbe\u0ca7\u0cbf\u0c95 \u2220COE &nbsp;<\/p>\n\n\n\n<p><span class=\"has-inline-color has-vivid-purple-color\">\u0caa\u0cb0\u0cbf\u0cb9\u0cbe\u0cb0 :<\/span><br>AB \u0c92\u0c82\u0ca6\u0cc1 \u0cb8\u0cb0\u0cb3\u0cb0\u0cc7\u0c96\u0cc6\u0caf\u0cbe\u0c97\u0cbf\u0ca6\u0cc6. \u0c86\u0ca6\u0ccd\u0ca6\u0cb0\u0cbf\u0c82\u0ca6 ,<\/p>\n\n\n\n<p>\u2234 \u2220AOC + \u2220COE + \u2220EOB = 180\u00b0<\/p>\n\n\n\n<p>(\u2220AOC + \u2220BOE) + \u2220COE = 180<\/p>\n\n\n\n<p>70\u00b0 + \u2220COE = 180\u00b0&nbsp;<\/p>\n\n\n\n<p>(\u0c95\u0cca\u0c9f\u0ccd\u0c9f\u0cbf\u0cb0\u0cc1\u0cb5\u0c82\u0ca4\u0cc6)<\/p>\n\n\n\n<p>\u2220COE = 180\u00b0 \u2013 70\u00b0<\/p>\n\n\n\n<p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; = 110\u00b0<\/p>\n\n\n\n<p>\u2234 \u0cb8\u0cb0\u0cb3\u0cbe\u0ca7\u0cbf\u0c95\u2220COE = 360\u00b0 \u2013 110\u00b0<\/p>\n\n\n\n<p>&nbsp; &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; = 250\u00b0<\/p>\n\n\n\n<p>AB \u0cae\u0ca4\u0ccd\u0ca4\u0cc1 CD &#8216;O&#8217; \u0cac\u0cbf\u0c82\u0ca6\u0cc1\u0cb5\u0cbf\u0ca8\u0cb2\u0ccd\u0cb2\u0cbf \u0c9b\u0cc7\u0ca6\u0cbf\u0cb8\u0cc1\u0ca4\u0ccd\u0ca4\u0cbf\u0cb5\u0cc6<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"349\" height=\"249\" src=\"https:\/\/themindpalace.in\/wp-content\/uploads\/2020\/09\/2-pic.jpg\" alt=\"\" class=\"wp-image-926\" srcset=\"https:\/\/themindpalace.in\/wp-content\/uploads\/2020\/09\/2-pic.jpg 349w, https:\/\/themindpalace.in\/wp-content\/uploads\/2020\/09\/2-pic-300x214.jpg 300w\" sizes=\"(max-width: 349px) 100vw, 349px\" \/><\/figure><\/div>\n\n\n\n<p>\u2234\u2220COA = \u2220BOD \u0c85\u0cb5\u0cc1 \u0cb6\u0cc3\u0c82\u0c97\u0cbe\u0cad\u0cbf\u0cae\u0cc1\u0c96 \u0c95\u0ccb\u0ca8\u0c97\u0cb3\u0cc1 \u0c86\u0c97\u0cbf\u0cb5\u0cc6.<\/p>\n\n\n\n<p>\u0c86\u0ca6\u0cb0\u0cc6 \u2220BOD = 40\u00b0 [\u0c95\u0cca\u0c9f\u0ccd\u0c9f\u0cbf\u0cb0\u0cc1\u0cb5\u0c82\u0ca4\u0cc6]<\/p>\n\n\n\n<p>\u2234 \u2220COA = 40\u00b0<\/p>\n\n\n\n<p>\u0cb9\u0cbe\u0c97\u0cc2, \u2220AOC + \u2220BOE = 70\u00b0<\/p>\n\n\n\n<p>\u2234 40\u00b0 + \u2220 BOE = 70\u00b0<\/p>\n\n\n\n<p>&nbsp;&nbsp;&nbsp;&nbsp; \u2220BOE = 70\u00b0 -40\u00b0 = 30\u00b0<\/p>\n\n\n\n<p><strong>\u0cb9\u0cc0\u0c97\u0cc6 , \u2220BOE = 30\u00b0 \u0cae\u0ca4\u0ccd\u0ca4\u0cc1 <strong>\u0cb8\u0cb0\u0cb3\u0cbe\u0ca7\u0cbf\u0c95<\/strong><\/strong> <strong>\u2220COE = 250\u00b0.<\/strong><\/p>\n\n\n\n<hr class=\"wp-block-separator\"\/>\n\n\n\n<p><strong>2.<\/strong> \u0c95\u0cc6\u0cb3\u0c97\u0cbf\u0ca8 \u0c9a\u0cbf\u0ca4\u0ccd\u0cb0\u0ca6\u0cb2\u0ccd\u0cb2\u0cbf <strong>XY<\/strong> \u0cae\u0ca4\u0ccd\u0ca4\u0cc1 <strong>MN<\/strong> \u0cb8\u0cb0\u0cb3\u0cb0\u0cc7\u0c96\u0cc6\u0c97\u0cb3\u0cc1 O \u0cac\u0cbf\u0c82\u0ca6\u0cc1\u0cb5\u0cbf\u0ca8\u0cb2\u0ccd\u0cb2\u0cbf \u0c9b\u0cc7\u0ca6\u0cbf\u0cb8\u0cc1\u0ca4\u0ccd\u0ca4\u0cbf\u0cb5\u0cc6. <strong>\u2220POY = 90\u00b0<\/strong> \u0cae\u0ca4\u0ccd\u0ca4\u0cc1 <strong>: b = 2 : 3<\/strong> \u0c86\u0ca6\u0cb0\u0cc6 <strong>c<\/strong> \u0c95\u0c82\u0ca1\u0cc1\u0cb9\u0cbf\u0ca1\u0cbf\u0caf\u0cbf\u0cb0\u0cbf.<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"242\" height=\"217\" src=\"https:\/\/themindpalace.in\/wp-content\/uploads\/2020\/09\/2-prb-1.jpg\" alt=\"\" class=\"wp-image-928\"\/><\/figure><\/div>\n\n\n\n<p><strong>\u0c95\u0cca\u0c9f\u0ccd\u0c9f\u0cbf\u0cb0\u0cc1\u0cb5\u0c82\u0ca4\u0cc6:&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <\/strong>a : b = 2 : 3 , \u2220POY = 90\u00b0<\/p>\n\n\n\n<p><strong><strong>\u0c95\u0c82\u0ca1\u0cc1\u0cb9\u0cbf\u0ca1\u0cbf\u0caf\u0cb2\u0cc1<\/strong>:&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <\/strong>a, b, \u0cae\u0ca4\u0ccd\u0ca4\u0cc1 c<\/p>\n\n\n\n<p><strong><span class=\"has-inline-color has-vivid-purple-color\">\u0caa\u0cb0\u0cbf\u0cb9\u0cbe\u0cb0 <\/span><\/strong><\/p>\n\n\n\n<p> XOY \u0c92\u0c82\u0ca6\u0cc1 \u0cb8\u0cb0\u0cb3\u0cb0\u0cc7\u0c96\u0cc6\u0caf\u0cbe\u0c97\u0cbf\u0ca6\u0cc6<\/p>\n\n\n\n<p>\u2234 b+a+\u2220POY= 180\u00b0 [\u0cb8\u0cb0\u0cb3\u0caf\u0cc1\u0c97\u0ccd\u0cae \u0c95\u0ccb\u0ca8\u0c97\u0cb3\u0cc1]<\/p>\n\n\n\n<p>\u0c86\u0ca6\u0cb0\u0cc6 \u2220POY = 90\u00b0 [\u0c95\u0cca\u0c9f\u0ccd\u0c9f\u0cbf\u0cb0\u0cc1\u0cb5\u0c82\u0ca4\u0cc6]<\/p>\n\n\n\n<p>\u2234 b + a = 180\u00b0 \u2013 90\u00b0<\/p>\n\n\n\n<p>&nbsp;&nbsp; b + a = 90\u00b0<\/p>\n\n\n\n<p> a : b = 2 : 3 =&gt; a=2x  \u0cae\u0ca4\u0ccd\u0ca4\u0cc1 b= 3x \u0c8e\u0c82\u0ca6\u0cc1 \u0ca8\u0cbe\u0cb5\u0cc1 \u0cac\u0cb0\u0cc6\u0caf\u0cac\u0cb9\u0cc1\u0ca6\u0cc1.<\/p>\n\n\n\n<p>\u0c85\u0c82\u0ca4\u0cc6\u0caf\u0cc7 <\/p>\n\n\n\n<p>\u21d2 2x + 3x = 90<sup>0<\/sup> \u0c86\u0c97\u0cc1\u0ca4\u0ccd\u0ca4\u0ca6\u0cc6.<\/p>\n\n\n\n<p>\u21d2 5x= 90<sup>0<\/sup><\/p>\n\n\n\n<p>X=90\/5<\/p>\n\n\n\n<p><strong>X= 18<sup>0<\/sup><\/strong><\/p>\n\n\n\n<p>\u0c85\u0ca6\u0cc7 \u0cb0\u0cc0\u0ca4\u0cbf a= 2x<\/p>\n\n\n\n<p>b= 3x \u0c88\u0c97 c= a + 90<sup>0<\/sup> [V O A]<\/p>\n\n\n\n<p>=2 X 18<sup>0<\/sup>= 36<sup>0<\/sup><\/p>\n\n\n\n<p>= 3 x 18<sup>0 <\/sup>=54<sup>0<\/sup><\/p>\n\n\n\n<p>C= 36<sup>0<\/sup> + 90<sup>0<\/sup> =126<sup>0<\/sup><\/p>\n\n\n\n<p><strong>\u2234 a = 36<sup>0<\/sup> , b =&nbsp; 54<sup>0<\/sup> and C=126<sup>0<\/sup><\/strong><\/p>\n\n\n\n<hr class=\"wp-block-separator\"\/>\n\n\n\n<p>3. \u0c95\u0cc6\u0cb3\u0c97\u0cbf\u0ca8 \u0c9a\u0cbf\u0ca4\u0ccd\u0cb0\u0ca6\u0cb2\u0ccd\u0cb2\u0cbf, \u2220PQR = \u2220PRQ  \u0c86\u0ca6\u0cb0\u0cc6 \u2220PQS = \u2220PRT \u0c8e\u0c82\u0ca6\u0cc1 \u0cb8\u0cbe\u0ca7\u0cbf\u0cb8\u0cbf..<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"268\" height=\"213\" src=\"https:\/\/themindpalace.in\/wp-content\/uploads\/2020\/09\/3-prb.jpg\" alt=\"\" class=\"wp-image-930\"\/><\/figure><\/div>\n\n\n\n<p><strong><strong>\u0c95\u0cca\u0c9f\u0ccd\u0c9f\u0cbf\u0cb0\u0cc1\u0cb5\u0c82\u0ca4\u0cc6<\/strong>:<\/strong> \u2220PQR = \u2220PRQ<\/p>\n\n\n\n<p><strong>\u0cb8\u0cbe\u0ca7\u0cbf\u0cb8\u0cac\u0cc7\u0c95\u0cbe\u0c97\u0cbf\u0cb0\u0cc1\u0cb5\u0cc1\u0ca6\u0cc1: <\/strong>\u2220PQS = \u2220PRT<\/p>\n\n\n\n<p><strong>\u0caa\u0cc1\u0cb0\u0cbe\u0cb5\u0cc6<\/strong>: ST \u0c92\u0c82\u0ca6\u0cc1 \u0cb8\u0cb0\u0cb3\u0cb0\u0cc7\u0c96\u0cc6\u0caf\u0cbe\u0c97\u0cbf\u0ca6\u0cc6.<\/p>\n\n\n\n<p>\u2234 \u2220PQR + \u2220PQS = 180\u00b0 \u2026(1) [\u0cb8\u0cb0\u0cb3\u0caf\u0cc1\u0c97\u0ccd\u0cae \u0c95\u0ccb\u0ca8\u0c97\u0cb3\u0cc1]<\/p>\n\n\n\n<p>\u0c85\u0ca6\u0cc7 \u0cb0\u0cc0\u0ca4\u0cbf\u0caf\u0cbe\u0c97\u0cbf , \u2220PRT + \u2220PRQ = 180\u00b0 \u2026(2) [\u0cb8\u0cb0\u0cb3\u0caf\u0cc1\u0c97\u0ccd\u0cae \u0c95\u0ccb\u0ca8\u0c97\u0cb3\u0cc1]<\/p>\n\n\n\n<p> (1) \u0cae\u0ca4\u0ccd\u0ca4\u0cc1 (2) \u0cb0\u0cbf\u0c82\u0ca6 \u0ca8\u0cbe\u0cb5\u0cc1 <\/p>\n\n\n\n<p>\u2220PQS + \u2220PQR = \u2220PRT + \u2220PRQ \u0c8e\u0c82\u0ca6\u0cc1 \u0cb8\u0cbe\u0ca7\u0cbf\u0cb8\u0cac\u0cb9\u0cc1\u0ca6\u0cc1 <\/p>\n\n\n\n<p>\u0c86\u0ca6\u0cb0\u0cc6 \u2220PQR = \u2220PRQ [\u0c95\u0cca\u0c9f\u0ccd\u0c9f\u0cbf\u0cb0\u0cc1\u0cb5\u0c82\u0ca4\u0cc6]<\/p>\n\n\n\n<p>\u2234 \u2220PQS = \u2220PRT<\/p>\n\n\n\n<hr class=\"wp-block-separator\"\/>\n\n\n\n<p>4. \u0c95\u0cc6\u0cb3\u0c97\u0cbf\u0ca8 \u0c9a\u0cbf\u0ca4\u0ccd\u0cb0\u0ca6\u0cb2\u0ccd\u0cb2\u0cbf   x + y = w + z  \u0c86\u0ca6\u0cb0\u0cc6  AOB \u0c92\u0c82\u0ca6\u0cc1 \u0cb8\u0cb0\u0cb3\u0cb0\u0cc7\u0c96\u0cc6 \u0c8e\u0c82\u0ca6\u0cc1 \u0cb8\u0cbe\u0ca7\u0cbf\u0cb8\u0cbf.<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"205\" height=\"233\" src=\"https:\/\/themindpalace.in\/wp-content\/uploads\/2020\/09\/prb-4.jpg\" alt=\"\" class=\"wp-image-932\"\/><\/figure><\/div>\n\n\n\n<p><strong><strong><strong>\u0c95\u0cca\u0c9f\u0ccd\u0c9f\u0cbf\u0cb0\u0cc1\u0cb5\u0c82\u0ca4\u0cc6<\/strong><\/strong><\/strong> <strong>: <\/strong>x + y = z + w<\/p>\n\n\n\n<p><strong>\u0cb8\u0cbe\u0ca7\u0cbf\u0cb8\u0cac\u0cc7\u0c95\u0cbe\u0c97\u0cbf\u0cb0\u0cc1\u0cb5\u0cc1\u0ca6\u0cc1:<\/strong> \u2234 AOB \u0c92\u0c82\u0ca6\u0cc1 \u0cb8\u0cb0\u0cb3\u0cb0\u0cc7\u0c96\u0cc6 \u0c8e\u0c82\u0ca6\u0cc1<\/p>\n\n\n\n<p><strong><strong>\u0caa\u0cc1\u0cb0\u0cbe\u0cb5\u0cc6<\/strong>:<\/strong> &#8216;O&#8217; \u0cac\u0cbf\u0c82\u0ca6\u0cc1\u0cb5\u0cbf\u0ca8 \u0cb8\u0cc1\u0ca4\u0ccd\u0ca4\u0cb2\u0cbf\u0ca8 \u0c8e\u0cb2\u0ccd\u0cb2 \u0c95\u0ccb\u0ca8\u0c97\u0cb3 \u0c92\u0c9f\u0ccd\u0c9f\u0cc1 \u0cae\u0cca\u0ca4\u0ccd\u0ca4 = 360\u00b0<\/p>\n\n\n\n<p>x+ y+ z + w = 360<sup>0<\/sup>  [\u0caa\u0cc2\u0cb0\u0ccd\u0ca3 \u0c95\u0ccb\u0ca8]<\/p>\n\n\n\n<p>x + y + x + y = 360<sup>0<\/sup>  [x + y = z + w]<\/p>\n\n\n\n<p>2x + 2y = 360<sup>0<\/sup><\/p>\n\n\n\n<p>2(x + y) = 360<sup>0<\/sup><\/p>\n\n\n\n<p>(x + y) =360<sup>0<\/sup>\/2<\/p>\n\n\n\n<p>(x + y) = 180<sup>0<\/sup><\/p>\n\n\n\n<p>\u0cb8\u0cb0\u0cb3\u0caf\u0cc1\u0c97\u0ccd\u0cae \u0c86\u0ca7\u0cbe\u0cb0 \u0caa\u0ccd\u0cb0\u0ca4\u0cbf\u0c9c\u0ccd\u0c9e\u0cc6\u0c97\u0cb3\u0cc1 \u0cb8\u0cbe\u0ca7\u0cbf\u0cb8\u0cc1\u0cb5\u0c82\u0ca4\u0cc6  AOB&nbsp; \u0c92\u0c82\u0ca6\u0cc1 \u0cb8\u0cb0\u0cb3\u0cb0\u0cc7\u0c96\u0cc6\u0caf\u0cbe\u0c97\u0cbf\u0ca6\u0cc6.<\/p>\n\n\n\n<hr class=\"wp-block-separator\"\/>\n\n\n\n<p>5. \u0c95\u0cca\u0c9f\u0ccd\u0c9f\u0cbf\u0cb0\u0cc1\u0cb5 \u0c9a\u0cbf\u0ca4\u0ccd\u0cb0\u0ca6\u0cb2\u0ccd\u0cb2\u0cbf POQ \u0c92\u0c82\u0ca6\u0cc1 \u0cb8\u0cb0\u0cb3\u0cb0\u0cc7\u0c96\u0cc6, OR \u0c95\u0cbf\u0cb0\u0ca3\u0cb5\u0cc1 PQ \u0cb8\u0cb0\u0cb3\u0cb0\u0cc7\u0c96\u0cc6\u0c97\u0cc6 \u0cb2\u0c82\u0cac\u0cb5\u0cbe\u0c97\u0cbf\u0ca6\u0cc6.  OP \u0cae\u0ca4\u0ccd\u0ca4\u0cc1 OR \u0c95\u0cbf\u0cb0\u0ca3\u0c97\u0cb3 \u0ca8\u0ca1\u0cc1\u0cb5\u0cc6 OS \u0c95\u0cbf\u0cb0\u0ca3 \u0c87\u0ca6\u0cc6. <\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"257\" height=\"58\" src=\"https:\/\/themindpalace.in\/wp-content\/uploads\/2020\/09\/image-4.png\" alt=\"\" class=\"wp-image-934\"\/><\/figure>\n\n\n\n<p>\u0c8e\u0c82\u0ca6\u0cc1 \u0cb8\u0cbe\u0ca7\u0cbf\u0cb8\u0cbf.<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"235\" height=\"152\" src=\"https:\/\/themindpalace.in\/wp-content\/uploads\/2020\/09\/5-prb.jpg\" alt=\"\" class=\"wp-image-933\"\/><\/figure><\/div>\n\n\n\n<p><strong><strong><strong>\u0c95\u0cca\u0c9f\u0ccd\u0c9f\u0cbf\u0cb0\u0cc1\u0cb5\u0c82\u0ca4\u0cc6<\/strong><\/strong><\/strong> : POQ \u0c92\u0c82\u0ca6\u0cc1 \u0cb8\u0cb0\u0cb3\u0cb0\u0cc7\u0c96\u0cc6\u0caf\u0cbe\u0c97\u0cbf\u0ca6\u0cc6.<\/p>\n\n\n\n<p>\u0c86\u0ca6\u0cb0\u0cc6 OR \u22a5 PQ   \u2234 \u2220ROQ = 90\u00b0<\/p>\n\n\n\n<p>\u21d2 \u2220POS + \u2220ROS + 90\u00b0 = 180\u00b0<\/p>\n\n\n\n<p>\u21d2 \u2220POS + \u2220ROS = 90\u00b0<\/p>\n\n\n\n<p>\u21d2 \u2220ROS = 90\u00b0 \u2013 \u2220POS \u2026 (1)<\/p>\n\n\n\n<p>\u0c88\u0c97,  \u2220ROS + \u2220ROQ = \u2220QOS <\/p>\n\n\n\n<p>\u21d2 \u2220ROS + 90\u00b0 = \u2220QOS<\/p>\n\n\n\n<p>\u21d2 \u2220ROS = \u2220QOS \u2013 90\u00b0 \u2026\u2026(2)<\/p>\n\n\n\n<p>(1) \u0cae\u0ca4\u0ccd\u0ca4\u0cc1 (2)\u0ca8\u0ccd\u0ca8\u0cc1 \u0c95\u0cc2\u0ca1\u0cbf\u0cb8\u0cbf\u0ca6\u0cbe\u0c97 <\/p>\n\n\n\n<p>2 \u2220ROS = (\u2220QOS \u2013 \u2220POS)<\/p>\n\n\n\n<p>2 \u2220ROS = (\u2220QOS \u2013 \u2220POS)<\/p>\n\n\n\n<p><strong>\u2234 \u2220ROS =1\/2 (\u2220QOS &#8211; \u2220POS)<\/strong><\/p>\n\n\n\n<hr class=\"wp-block-separator\"\/>\n\n\n\n<p>&nbsp;6. \u2220XYZ = 64\u00b0  \u0cae\u0ca4\u0ccd\u0ca4\u0cc1  XY \u0caf\u0ca8\u0ccd\u0ca8\u0cc1  P \u0cb5\u0cb0\u0cc6\u0c97\u0cc6 \u0cb5\u0cc3\u0ca6\u0ccd\u0ca7\u0cbf\u0cb8\u0cbf\u0ca6\u0cc6. \u0c88 \u0ca6\u0ca4\u0ccd\u0ca4\u0cb5\u0ca8\u0ccd\u0ca8\u0cc1 \u0c89\u0caa\u0caf\u0ccb\u0c97\u0cbf\u0cb8\u0cbf \u0c92\u0c82\u0ca6\u0cc1 \u0c9a\u0cbf\u0ca4\u0ccd\u0cb0 \u0cb0\u0c9a\u0cbf\u0cb8\u0cbf.  \u2220ZYP \u0caf\u0ca8\u0ccd\u0ca8\u0cc1  YQ  \u0ca6\u0ccd\u0cb5\u0cbf\u0cad\u0cbe\u0c97\u0cbf\u0cb8\u0cbf\u0ca6\u0cb0\u0cc6,  \u2220XYQ  \u0cae\u0ca4\u0ccd\u0ca4\u0cc1 \u0cb8\u0cb0\u0cb3\u0cbe\u0ca7\u0cbf\u0c95  \u2220QYP  \u0c95\u0c82\u0ca1\u0cc1\u0cb9\u0cbf\u0ca1\u0cbf\u0caf\u0cbf\u0cb0\u0cbf.  <\/p>\n\n\n\n<p><strong><strong><strong>\u0c95\u0cca\u0c9f\u0ccd\u0c9f\u0cbf\u0cb0\u0cc1\u0cb5\u0c82\u0ca4\u0cc6<\/strong><\/strong><\/strong> <strong>: <\/strong>\u2220XYZ = 64\u00b0 , \u2220ZYP \u0caf\u0ca8\u0ccd\u0ca8\u0cc1 YQ \u0ca6\u0ccd\u0cb5\u0cbf\u0cad\u0cbe\u0c97\u0cbf\u0cb8\u0cc1\u0ca4\u0ccd\u0ca4\u0ca6\u0cc6.<\/p>\n\n\n\n<p><strong><strong>\u0cb8\u0cbe\u0ca7\u0cbf\u0cb8\u0cac\u0cc7\u0c95\u0cbe\u0c97\u0cbf\u0cb0\u0cc1\u0cb5\u0cc1\u0ca6\u0cc1<\/strong><\/strong> <strong>: <\/strong> \u2220XYQ  \u0cae\u0ca4\u0ccd\u0ca4\u0cc1 \u0cb8\u0cb0\u0cb3\u0cbe\u0ca7\u0cbf\u0c95   \u2220QYP<\/p>\n\n\n\n<p><strong>\u0caa\u0cb0\u0cbf\u0cb9\u0cbe\u0cb0 :<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-image size-large is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/themindpalace.in\/wp-content\/uploads\/2020\/10\/example-6-lines-and-angles.png\" alt=\"\" class=\"wp-image-3826\" width=\"331\" height=\"233\" srcset=\"https:\/\/themindpalace.in\/wp-content\/uploads\/2020\/10\/example-6-lines-and-angles.png 662w, https:\/\/themindpalace.in\/wp-content\/uploads\/2020\/10\/example-6-lines-and-angles-300x211.png 300w\" sizes=\"(max-width: 331px) 100vw, 331px\" \/><\/figure>\n\n\n\n<p>XYP \u0c92\u0c82\u0ca6\u0cc1 \u0cb8\u0cb0\u0cb3\u0cb0\u0cc7\u0c96\u0cc6 \u0c86\u0c97\u0cbf\u0ca6\u0cc6, \u0c86 \u0cb8\u0cb0\u0cb3\u0cb0\u0cc7\u0c96\u0cc6\u0caf \u0cae\u0cc7\u0cb2\u0cc6 YQ \u0cae\u0ca4\u0ccd\u0ca4\u0cc1 YZ \u0c95\u0cbf\u0cb0\u0ca3\u0c97\u0cb3\u0cc1 \u0ca8\u0cbf\u0c82\u0ca4\u0cbf\u0cb5\u0cc6.<\/p>\n\n\n\n<p>\u2234 \u2220XYZ + \u2220ZYQ + \u2220QYP = 180\u00b0<\/p>\n\n\n\n<p>\u21d2 64\u00b0 + \u2220ZYQ + \u2220QYP = 180\u00b0  [\u2220ZYP \u0caf\u0ca8\u0ccd\u0ca8\u0cc1 YQ \u0ca6\u0ccd\u0cb5\u0cbf\u0cad\u0cbe\u0c97\u0cbf\u0cb8\u0cc1\u0ca4\u0ccd\u0ca4\u0ca6\u0cc6, \u0c86\u0ca6\u0ccd\u0ca6\u0cb0\u0cbf\u0c82\u0ca6, \u2220QYP = \u2220ZYQ]<\/p>\n\n\n\n<p>\u21d2 2\u2220QYP = 180\u00b0 \u2013 64\u00b0<\/p>\n\n\n\n<p>\u21d2 2\u2220QYP = 116\u00b0<\/p>\n\n\n\n<p>\u21d2 \u2220QYP = &#8220;116\u00b0 &#8221; <strong>\/<\/strong><strong>2<\/strong><\/p>\n\n\n\n<p>\u21d2 \u2220QYP = 58\u00b0<\/p>\n\n\n\n<p>\u2234 \u0cb8\u0cb0\u0cb3\u0cbe\u0ca7\u0cbf\u0c95 \u2220QYP = 360\u00b0 \u2013 58\u00b0 = 302\u00b0<\/p>\n\n\n\n<p>\u2220XYQ = \u2220XYZ + \u2220ZYQ  \u0c86\u0c97\u0cbf\u0cb0\u0cc1\u0cb5\u0cc1\u0ca6\u0cb0\u0cbf\u0c82\u0ca6 <\/p>\n\n\n\n<p>\u21d2 \u2220XYQ = 64\u00b0 + \u2220QYP [\u2235\u2220XYZ = 64\u00b0 (\u0c95\u0cca\u0c9f\u0ccd\u0c9f\u0cbf\u0cb0\u0cc1\u0cb5\u0c82\u0ca4\u0cc6)  \u0cae\u0ca4\u0ccd\u0ca4\u0cc1 \u2220ZYQ = \u2220QYP]<\/p>\n\n\n\n<p>\u21d2 \u2220XYQ = 64\u00b0 + 58\u00b0<\/p>\n\n\n\n<p>&nbsp; &nbsp;= 122\u00b0 [\u2220QYP = 58\u00b0]<\/p>\n\n\n\n<p><strong>\u0c86\u0ca6\u0cc1\u0ca6\u0cb0\u0cbf\u0c82\u0ca6 , \u2220XYQ = 122\u00b0 \u0cae\u0ca4\u0ccd\u0ca4\u0cc1 \u0cb8\u0cb0\u0cb3\u0cbe\u0ca7\u0cbf\u0c95<\/strong> <strong>\u2220QYP = 302\u00b0.<\/strong><\/p>\n\n\n\n<hr class=\"wp-block-separator\"\/>\n\n\n\n<h4 class=\"wp-block-heading\">Exercise 3.2<\/h4>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"243\" height=\"379\" src=\"https:\/\/themindpalace.in\/wp-content\/uploads\/2020\/09\/1-3.jpg\" alt=\"\" class=\"wp-image-979\" srcset=\"https:\/\/themindpalace.in\/wp-content\/uploads\/2020\/09\/1-3.jpg 243w, https:\/\/themindpalace.in\/wp-content\/uploads\/2020\/09\/1-3-192x300.jpg 192w\" sizes=\"(max-width: 243px) 100vw, 243px\" \/><\/figure><\/div>\n\n\n\n<p>1. \u0c9a\u0cbf\u0ca4\u0ccd\u0cb0\u0ca6\u0cb2\u0ccd\u0cb2\u0cbf x \u0cae\u0ca4\u0ccd\u0ca4\u0cc1 y \u0cac\u0cc6\u0cb2\u0cc6 \u0c95\u0c82\u0ca1\u0cc1\u0cb9\u0cbf\u0ca1\u0cbf\u0ca6\u0cc1 AB || CD \u0c8e\u0c82\u0ca6\u0cc1 \u0ca4\u0ccb\u0cb0\u0cbf\u0cb8\u0cbf.<br><br><strong>\u0caa\u0cb0\u0cbf\u0cb9\u0cbe\u0cb0 <\/strong>:<\/p>\n\n\n\n<p>\u0c9a\u0cbf\u0ca4\u0ccd\u0cb0\u0ca6\u0cb2\u0ccd\u0cb2\u0cbf CD \u0cae\u0ca4\u0ccd\u0ca4\u0cc1 PQ \u0cb8\u0cb0\u0cb3\u0cb0\u0cc7\u0c96\u0cc6\u0c97\u0cb3\u0cc1  &#8216;F&#8217; \u0cac\u0cbf\u0c82\u0ca6\u0cc1\u0cb5\u0cbf\u0ca8\u0cb2\u0ccd\u0cb2\u0cbf \u0c9b\u0cc7\u0ca6\u0cbf\u0cb8\u0cc1\u0ca4\u0ccd\u0ca4\u0cbf\u0cb5\u0cc6 .<\/p>\n\n\n\n<p>\u2234 y = 130\u00b0 \u2026(1) [\u0cb6\u0cc3\u0c82\u0c97\u0cbe\u0cad\u0cbf\u0cae\u0cc1\u0c96 \u0c95\u0ccb\u0ca8\u0c97\u0cb3\u0cc1]<\/p>\n\n\n\n<p>PQ \u0c92\u0c82\u0ca6\u0cc1 \u0cb8\u0cb0\u0cb3\u0cb0\u0cc7\u0c96\u0cc6\u0caf\u0cbe\u0c97\u0cbf\u0ca6\u0cc6  \u0cb9\u0cbe\u0c97\u0cc1 EA \u0c95\u0cbf\u0cb0\u0ca3\u0cb5\u0cc1 \u0c85\u0ca6\u0cb0 \u0cae\u0cc7\u0cb2\u0cc6 \u0ca8\u0cbf\u0c82\u0ca4\u0cbf\u0ca6\u0cc6.<\/p>\n\n\n\n<p>\u2220AEP + \u2220AEQ = 180\u00b0 [\u0cb8\u0cb0\u0cb3\u0caf\u0cc1\u0c97\u0ccd\u0cae \u0c95\u0ccb\u0ca8\u0c97\u0cb3\u0cc1]<\/p>\n\n\n\n<p>50\u00b0 + x = 180\u00b0<\/p>\n\n\n\n<p>\u21d2 x = 180\u00b0 \u2013 50\u00b0<\/p>\n\n\n\n<p>\u21d2 x = 130\u00b0 \u2026(2)<\/p>\n\n\n\n<p>(1) \u0cae\u0ca4\u0ccd\u0ca4\u0cc1 (2) \u0cb0\u0cbf\u0c82\u0ca6  x = y = 130\u00b0<\/p>\n\n\n\n<p>\u0c85\u0cb5\u0cc1 &nbsp;\u0caa\u0cb0\u0ccd\u0caf\u0cbe\u0caf \u0c85\u0c82\u0ca4\u0cb0\u0ccd \u0c95\u0ccb\u0ca8\u0c97\u0cb3\u0cc1 \u0c86\u0c97\u0cbf\u0cb0\u0cc1\u0cb5\u0cc1\u0ca6\u0cb0\u0cbf\u0c82\u0ca6 <\/p>\n\n\n\n<p><strong>\u2234 AB || CD<\/strong><\/p>\n\n\n\n<hr class=\"wp-block-separator\"\/>\n\n\n\n<p>2. \u0c9a\u0cbf\u0ca4\u0ccd\u0cb0\u0ca6\u0cb2\u0ccd\u0cb2\u0cbf  AB || CD,  CD || EF  \u0cae\u0ca4\u0ccd\u0ca4\u0cc1 y : z =3:7  \u0c86\u0ca6\u0cb0\u0cc6 x \u0cac\u0cc6\u0cb2\u0cc6 \u0c95\u0c82\u0ca1\u0cc1\u0cb9\u0cbf\u0ca1\u0cbf\u0caf\u0cbf\u0cb0\u0cbf.<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"233\" height=\"203\" src=\"https:\/\/themindpalace.in\/wp-content\/uploads\/2020\/09\/2-2.jpg\" alt=\"\" class=\"wp-image-980\"\/><\/figure><\/div>\n\n\n\n<p>\u0c95\u0cca\u0c9f\u0ccd\u0c9f\u0cbf\u0cb0\u0cc1\u0cb5\u0c82\u0ca4\u0cc6 : <strong>AB || CD, CD || EF <\/strong> \u0cae\u0ca4\u0ccd\u0ca4\u0cc1  <strong>y : z = 3 : 7<\/strong><\/p>\n\n\n\n<p>\u0c95\u0c82\u0ca1\u0cc1\u0cb9\u0cbf\u0ca1\u0cbf\u0caf\u0cac\u0cc7\u0c95\u0cbe\u0c97\u0cbf\u0cb0\u0cc1\u0cb5\u0cc1\u0ca6\u0cc1 :  <strong>x<\/strong><\/p>\n\n\n\n<p><strong>\u0caa\u0cb0\u0cbf\u0cb9\u0cbe\u0cb0 <\/strong>: AB || CD, \u0cae\u0ca4\u0ccd\u0ca4\u0cc1 CD || EF [\u0c95\u0cca\u0c9f\u0ccd\u0c9f\u0cbf\u0cb0\u0cc1\u0cb5\u0c82\u0ca4\u0cc6]<\/p>\n\n\n\n<p>\u2234 AB || CD || EF<\/p>\n\n\n\n<p>\u2234 x = z [\u0caa\u0cb0\u0ccd\u0caf\u0cbe\u0caf \u0c85\u0c82\u0ca4\u0cb0\u0ccd \u0c95\u0ccb\u0ca8\u0c97\u0cb3\u0cc1] \u2026.(1)<\/p>\n\n\n\n<p>\u0ca6\u0ca4\u0ccd\u0ca4\u0cbe\u0c82\u0cb6\u0ca6\u0cbf\u0c82\u0ca6  y : z =3 : 7<\/p>\n\n\n\n<p>y \u0cae\u0ca4\u0ccd\u0ca4\u0cc1  z  \u0ca8\u0ca1\u0cc1\u0cb5\u0cbf\u0ca8 \u0cb8\u0cbe\u0cae\u0cbe\u0ca8\u0ccd\u0caf \u0c85\u0ca8\u0cc1\u0caa\u0cbe\u0ca4 &#8216;a&#8217; \u0c86\u0c97\u0cbf\u0cb0\u0cb2\u0cbf <br>\u2234 y = 3a  \u0cae\u0ca4\u0ccd\u0ca4\u0cc1   z=7a  \u0c86\u0c97\u0cc1\u0ca4\u0ccd\u0ca4\u0ca6\u0cc6.<\/p>\n\n\n\n<p>\u0cb9\u0cbe\u0c97\u0cc6\u0caf\u0cc7 <br>x + y = 180\u00b0&nbsp; [\u0c9b\u0cc7\u0ca6\u0c95\u0ca6 \u0c92\u0c82\u0ca6\u0cc7 \u0caa\u0cbe\u0cb0\u0ccd\u0cb6\u0ccd\u0cb5\u0ca6\u0cb2\u0ccd\u0cb2\u0cbf\u0cb0\u0cc1\u0cb5 \u0caa\u0ccd\u0cb0\u0ca4\u0cbf\u0caf\u0cca\u0c82\u0ca6\u0cc1 \u0c9c\u0cca\u0ca4\u0cc6 \u0c85\u0c82\u0ca4\u0cb0\u0ccd \u0c95\u0ccb\u0ca8\u0c97\u0cb3\u0cc1 \u0cb8\u0cb0\u0cb3\u0c95\u0ccb\u0ca8 \u0caa\u0cc2\u0cb0\u0c95\u0cb5\u0cbe\u0c97\u0cbf\u0cb0\u0cc1\u0ca4\u0ccd\u0ca4\u0cb5\u0cc6.]<\/p>\n\n\n\n<p>z + y = 180\u00b0 [\u0cb8\u0cae\u0cc0\u0c95\u0cb0\u0ca3 (1) \u0cb0\u0cbf\u0c82\u0ca6]<br>7a + 3a = 180\u00b0<br>10a = 180\u00b0<\/p>\n\n\n\n<p>a = 180\u00b0 \/ 10a = 18\u00b0<br><br>\u0c88\u0c97 <strong>x<\/strong> = 7a = 7&nbsp;x 18\u00b0 = 126\u00b0<\/p>\n\n\n\n<p><strong>\u2234 x = 126\u00b0<\/strong><\/p>\n\n\n\n<hr class=\"wp-block-separator\"\/>\n\n\n\n<p>3.  \u0c9a\u0cbf\u0ca4\u0ccd\u0cb0\u0ca6\u0cb2\u0ccd\u0cb2\u0cbf AB || CD, EF \u22a5 CD \u0cae\u0ca4\u0ccd\u0ca4\u0cc1 \u2220GED = 126\u00b0 \u0c86\u0ca6\u0cb0\u0cc6 \u2220AGE, \u2220GEF \u0cae\u0ca4\u0ccd\u0ca4\u0cc1 \u2220FGE \u0c95\u0c82\u0ca1\u0cc1\u0cb9\u0cbf\u0ca1\u0cbf\u0caf\u0cbf\u0cb0\u0cbf.<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"274\" height=\"207\" src=\"https:\/\/themindpalace.in\/wp-content\/uploads\/2020\/09\/3-1.jpg\" alt=\"\" class=\"wp-image-981\"\/><\/figure><\/div>\n\n\n\n<p><strong>\u0c95\u0cca\u0c9f\u0ccd\u0c9f\u0cbf\u0cb0\u0cc1\u0cb5\u0c82\u0ca4\u0cc6 <\/strong>: AB || CD, EF \u22a5 CD \u0cae\u0ca4\u0ccd\u0ca4\u0cc1 \u2220GED = 126\u00b0<\/p>\n\n\n\n<p><strong>\u0c95\u0c82\u0ca1\u0cc1\u0cb9\u0cbf\u0ca1\u0cbf\u0caf\u0cac\u0cc7\u0c95\u0cbe\u0c97\u0cbf\u0cb0\u0cc1\u0cb5\u0cc1\u0ca6\u0cc1 <\/strong>: \u2220AGE, \u2220GEF  \u0cae\u0ca4\u0ccd\u0ca4\u0cc1  \u2220FGE.<\/p>\n\n\n\n<p><strong>\u0caa\u0cb0\u0cbf\u0cb9\u0cbe\u0cb0 <\/strong>: AB || CD \u0cae\u0ca4\u0ccd\u0ca4\u0cc1 GE \u0c92\u0c82\u0ca6\u0cc1 \u0c9b\u0cc7\u0ca6\u0c95\u0cb5\u0cbe\u0c97\u0cbf\u0ca6\u0cc6 <\/p>\n\n\n\n<p>\u2234 \u2220AGE = \u2220GED [&nbsp;\u0caa\u0cb0\u0ccd\u0caf\u0cbe\u0caf \u0c85\u0c82\u0ca4\u0cb0\u0ccd \u0c95\u0ccb\u0ca8\u0c97\u0cb3\u0cc1]<\/p>\n\n\n\n<p>\u2220AGE = \u2220GED = 126\u00b0 [\u0c95\u0cca\u0c9f\u0ccd\u0c9f\u0cbf\u0cb0\u0cc1\u0cb5\u0c82\u0ca4\u0cc6 ]<\/p>\n\n\n\n<p>\u0c86\u0ca6\u0cb0\u0cc6 \u2220AGE + \u2220FGE = 180\u00b0 [\u0cb8\u0cb0\u0cb3\u0caf\u0cc1\u0c97\u0ccd\u0cae \u0c95\u0ccb\u0ca8\u0c97\u0cb3\u0cc1]<\/p>\n\n\n\n<p>126\u00b0+ \u2220FGE = 180\u00b0<\/p>\n\n\n\n<p> \u2220FGE = 180\u00b0 &#8211; 126\u00b0 = 54\u00b0<\/p>\n\n\n\n<p>\u0c86\u0ca6\u0ccd\u0ca6\u0cb0\u0cbf\u0c82\u0ca6 , <strong>\u2220AGE = 126\u00b0<\/strong>, <strong>\u2220GEF= 36\u00b0 and \u2220FGE = 54\u00b0.<\/strong><\/p>\n\n\n\n<hr class=\"wp-block-separator\"\/>\n\n\n\n<p>4. \u0c9a\u0cbf\u0ca4\u0ccd\u0cb0\u0ca6\u0cb2\u0ccd\u0cb2\u0cbf  PQ || ST,  \u2220 PQR = 110\u00b0  \u0cae\u0ca4\u0ccd\u0ca4\u0cc1  \u2220 RST = 130\u00b0 \u0c86\u0ca6\u0cb0\u0cc6  \u2220QRS \u0c95\u0c82\u0ca1\u0cc1\u0cb9\u0cbf\u0ca1\u0cbf\u0caf\u0cbf\u0cb0\u0cbf. ( \u0cb8\u0cc1\u0cb3\u0cbf\u0cb5\u0cc1:  R \u0cac\u0cbf\u0c82\u0ca6\u0cc1\u0cb5\u0cbf\u0ca8 \u0cae\u0cc2\u0cb2\u0c95 ST \u0c97\u0cc6 \u0c92\u0c82\u0ca6\u0cc1 \u0cb8\u0cae\u0cbe\u0c82\u0ca4\u0cb0 \u0cb0\u0cc7\u0c96\u0cc6\u0caf\u0ca8\u0ccd\u0ca8\u0cc1 \u0c8e\u0cb3\u0cc6\u0caf\u0cbf\u0cb0\u0cbf).<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"268\" height=\"282\" src=\"https:\/\/themindpalace.in\/wp-content\/uploads\/2020\/09\/4-2.jpg\" alt=\"\" class=\"wp-image-982\"\/><\/figure><\/div>\n\n\n\n<p><strong>\u0c95\u0cca\u0c9f\u0ccd\u0c9f\u0cbf\u0cb0\u0cc1\u0cb5\u0c82\u0ca4\u0cc6<\/strong> : PQ || ST, \u2220 PQR = 110\u00b0 and \u2220 RST = 130\u00b0<\/p>\n\n\n\n<p><strong>\u0c95\u0c82\u0ca1\u0cc1\u0cb9\u0cbf\u0ca1\u0cbf\u0caf\u0cac\u0cc7\u0c95\u0cbe\u0c97\u0cbf\u0cb0\u0cc1\u0cb5\u0cc1\u0ca6\u0cc1<\/strong> : \u2220QRS.<\/p>\n\n\n\n<p><strong><strong>\u0c95\u0c82\u0ca1\u0cc1\u0cb9\u0cbf\u0ca1\u0cbf\u0caf\u0cac\u0cc7\u0c95\u0cbe\u0c97\u0cbf\u0cb0\u0cc1\u0cb5\u0cc1\u0ca6\u0cc1<\/strong><\/strong>:  R \u0cac\u0cbf\u0c82\u0ca6\u0cc1\u0cb5\u0cbf\u0ca8 \u0cae\u0cc2\u0cb2\u0c95 ST \u0c97\u0cc6 \u0c92\u0c82\u0ca6\u0cc1 \u0cb8\u0cae\u0cbe\u0c82\u0ca4\u0cb0 \u0cb0\u0cc7\u0c96\u0cc6 EF \u0c85\u0ca8\u0ccd\u0ca8\u0cc1 \u0c8e\u0cb3\u0cc6\u0caf\u0cbf\u0cb0\u0cbf<\/p>\n\n\n\n<p>\u0c95\u0cca\u0c9f\u0ccd\u0c9f\u0cbf\u0cb0\u0cc1\u0cb5\u0c82\u0ca4\u0cc6 PQ || ST  \u0cae\u0ca4\u0ccd\u0ca4\u0cc1 EF || ST <br><br>\u2234 PQ || EF \u0cae\u0ca4\u0ccd\u0ca4\u0cc1 QR \u0c92\u0c82\u0ca6\u0cc1 \u0c9b\u0cc7\u0ca6\u0c95 \u0cb0\u0cc7\u0c96\u0cc6\u0caf\u0cbe\u0c97\u0cbf\u0ca6\u0cc6<\/p>\n\n\n\n<p>\u21d2 \u2220PQR = \u2220QRF [\u0caa\u0cb0\u0ccd\u0caf\u0cbe\u0caf \u0c85\u0c82\u0ca4\u0cb0\u0ccd \u0c95\u0ccb\u0ca8\u0c97\u0cb3\u0cc1]<\/p>\n\n\n\n<p>\u0c86\u0ca6\u0cb0\u0cc6 \u2220PQR = 110\u00b0 [\u0ca6\u0ca4\u0ccd\u0ca4 ]<br>\u2234\u2220QRF = \u2220QRS + \u2220SRF = 110\u00b0 \u2026(1)<br>\u0cb9\u0cbe\u0c97\u0cc6\u0caf\u0cc7  ST || EF \u0cae\u0ca4\u0ccd\u0ca4\u0cc1 RS \u0c92\u0c82\u0ca6\u0cc1 \u0c9b\u0cc7\u0ca6\u0c95 \u0cb0\u0cc7\u0c96\u0cc6\u0caf\u0cbe\u0c97\u0cbf\u0ca6\u0cc6<\/p>\n\n\n\n<p>\u2234 \u2220RST + \u2220SRF = 180\u00b0 [\u0cb8\u0cb0\u0cb3\u0caf\u0cc1\u0c97\u0ccd\u0cae \u0c95\u0ccb\u0ca8\u0c97\u0cb3\u0cc1] &nbsp;<\/p>\n\n\n\n<p>&nbsp; 130\u00b0 + \u2220SRF = 180\u00b0\u21d2 \u2220SRF = 180\u00b0 \u2013 130\u00b0 = 50\u00b0                  <\/p>\n\n\n\n<p>\u21d2 \u2220QRS = 110\u00b0 \u2013 50\u00b0 = 60\u00b0<\/p>\n\n\n\n<p>\u0cb9\u0cc0\u0c97\u0cbe\u0c97\u0cbf , <strong>\u2220QRS = 60\u00b0<\/strong>.<\/p>\n\n\n\n<hr class=\"wp-block-separator\"\/>\n\n\n\n<p>5. \u0c9a\u0cbf\u0ca4\u0ccd\u0cb0\u0ca6\u0cb2\u0ccd\u0cb2\u0cbf  AB || CD,  \u2220APQ = 50\u00b0  \u0cae\u0ca4\u0ccd\u0ca4\u0cc1  \u2220PRD = 127\u00b0  \u0c86\u0ca6\u0cb0\u0cc6  x \u0cae\u0ca4\u0ccd\u0ca4\u0cc1 y \u0c97\u0cb3  \u0cac\u0cc6\u0cb2\u0cc6\u0caf\u0ca8\u0ccd\u0ca8\u0cc1 \u0c95\u0c82\u0ca1\u0cc1\u0cb9\u0cbf\u0ca1\u0cbf\u0caf\u0cbf\u0cb0\u0cbf.<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"291\" height=\"257\" src=\"https:\/\/themindpalace.in\/wp-content\/uploads\/2020\/09\/5-1.jpg\" alt=\"\" class=\"wp-image-983\"\/><\/figure><\/div>\n\n\n\n<p><strong>\u0c95\u0cca\u0c9f\u0ccd\u0c9f\u0cbf\u0cb0\u0cc1\u0cb5\u0cc1\u0ca6\u0cc1  <\/strong>: AB || CD, \u2220APQ = 50\u00b0 \u0cae\u0ca4\u0ccd\u0ca4\u0cc1 \u2220PRD = 127\u00b0,<\/p>\n\n\n\n<p><strong>\u0c95\u0c82\u0ca1\u0cc1\u0cb9\u0cbf\u0ca1\u0cbf\u0cac\u0cc7\u0c95\u0cbe\u0ca6\u0cc1\u0ca6\u0cc1 <\/strong>: x \u0cae\u0ca4\u0ccd\u0ca4\u0cc1 y.<\/p>\n\n\n\n<p><strong>\u0caa\u0cb0\u0cbf\u0cb9\u0cbe\u0cb0 :<\/strong>  AB || CD \u0cae\u0ca4\u0ccd\u0ca4\u0cc1 PQ \u0c92\u0c82\u0ca6\u0cc1 \u0c9b\u0cc7\u0ca6\u0c95 \u0cb0\u0cc7\u0c96\u0cc6\u0caf\u0cbe\u0c97\u0cbf\u0ca6\u0cc6<\/p>\n\n\n\n<p>\u2234 \u2220APQ = \u2220PQR&nbsp;&nbsp;&nbsp; [\u0caa\u0cb0\u0ccd\u0caf\u0cbe\u0caf \u0c85\u0c82\u0ca4\u0cb0\u0ccd \u0c95\u0ccb\u0ca8\u0c97\u0cb3\u0cc1]<\/p>\n\n\n\n<p>\u21d2 50\u00b0 = x [ \u2235 \u2220APQ = 50\u00b0 (\u0ca6\u0ca4\u0ccd\u0ca4 \/ \u0c95\u0cca\u0c9f\u0ccd\u0c9f\u0cbf\u0cb0\u0cc1\u0cb5\u0c82\u0ca4\u0cc6)]<\/p>\n\n\n\n<p>AB || CD \u0cae\u0ca4\u0ccd\u0ca4\u0cc1 PR \u0c92\u0c82\u0ca6\u0cc1 \u0c9b\u0cc7\u0ca6\u0c95 \u0cb0\u0cc7\u0c96\u0cc6\u0caf\u0cbe\u0c97\u0cbf\u0ca6\u0cc6<br>\u2234 \u2220APR = \u2220PRD [\u0caa\u0cb0\u0ccd\u0caf\u0cbe\u0caf \u0c85\u0c82\u0ca4\u0cb0\u0ccd \u0c95\u0ccb\u0ca8\u0c97\u0cb3\u0cc1]<\/p>\n\n\n\n<p>\u21d2 \u2220APR = 127\u00b0 [ \u2235 \u2220PRD = 127\u00b0 (\u0ca6\u0ca4\u0ccd\u0ca4)]<\/p>\n\n\n\n<p>\u21d2 \u2220APQ + \u2220QPR = 127\u00b0<br>\u21d2 50\u00b0 + y = 127\u00b0 [ \u2235 \u2220APQ = 50\u00b0 (\u0ca6\u0ca4\u0ccd\u0ca4)]<br>\u21d2 y = 127\u00b0- 50\u00b0 = 77\u00b0<br><strong>\u0cb9\u0cc0\u0c97\u0cc6  x = 50\u00b0 \u0cae\u0ca4\u0ccd\u0ca4\u0cc1 y = 77\u00b0<\/strong><\/p>\n\n\n\n<hr class=\"wp-block-separator\"\/>\n\n\n\n<p>6.  \u0c9a\u0cbf\u0ca4\u0ccd\u0cb0\u0ca6\u0cb2\u0ccd\u0cb2\u0cbf  PQ \u0cae\u0ca4\u0ccd\u0ca4\u0cc1 RS \u0caa\u0cb0\u0cb8\u0ccd\u0caa\u0cb0 \u0cb8\u0cae\u0cbe\u0c82\u0ca4\u0cb0\u0cb5\u0cbe\u0c97\u0cbf\u0cb0\u0cc1\u0cb5  \u0c8e\u0cb0\u0ca1\u0cc1 \u0ca6\u0cb0\u0ccd\u0caa\u0ca3\u0c97\u0cb3\u0ca8\u0ccd\u0ca8\u0cc1 \u0c87\u0ca1\u0cb2\u0cbe\u0c97\u0cbf\u0ca6\u0cc6.  AB \u0caa\u0ca4\u0ca8 \u0c95\u0cbf\u0cb0\u0ca3\u0cb5\u0cc1  PQ \u0ca6\u0cb0\u0ccd\u0caa\u0ca3\u0cb5\u0ca8\u0ccd\u0ca8\u0cc1 B \u0cac\u0cbf\u0c82\u0ca6\u0cc1\u0cb5\u0cbf\u0ca8\u0cb2\u0ccd\u0cb2\u0cbf \u0ca4\u0cbe\u0c97\u0cbf \u0caa\u0ccd\u0cb0\u0ca4\u0cbf\u0cab\u0cb2\u0cbf\u0ca4 \u0c95\u0cbf\u0cb0\u0ca3\u0cb5\u0cc1,  BC \u0cb9\u0cbe\u0ca6\u0cbf\u0caf\u0cb2\u0ccd\u0cb2\u0cbf \u0c9a\u0cb2\u0cbf\u0cb8\u0cbf  RS  \u0ca6\u0cb0\u0ccd\u0caa\u0ca3\u0cb5\u0ca8\u0ccd\u0ca8\u0cc1 C \u0cac\u0cbf\u0c82\u0ca6\u0cc1\u0cb5\u0cbf\u0ca8\u0cb2\u0ccd\u0cb2\u0cbf  \u0ca4\u0cbe\u0c97\u0cbf \u0caa\u0cc1\u0ca8\u0c83  CD \u0cb9\u0cbe\u0ca6\u0cbf\u0caf\u0cb2\u0ccd\u0cb2\u0cbf \u0caa\u0ccd\u0cb0\u0ca4\u0cbf\u0cab\u0cb2\u0cbf\u0cb8\u0cc1\u0ca4\u0ccd\u0ca4\u0ca6\u0cc6. AB || CD \u0c8e\u0c82\u0ca6\u0cc1 \u0cb8\u0cbe\u0ca7\u0cbf\u0cb8\u0cbf.<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"267\" height=\"332\" src=\"https:\/\/themindpalace.in\/wp-content\/uploads\/2020\/09\/6.jpg\" alt=\"\" class=\"wp-image-985\" srcset=\"https:\/\/themindpalace.in\/wp-content\/uploads\/2020\/09\/6.jpg 267w, https:\/\/themindpalace.in\/wp-content\/uploads\/2020\/09\/6-241x300.jpg 241w\" sizes=\"(max-width: 267px) 100vw, 267px\" \/><\/figure><\/div>\n\n\n\n<p><strong>\u0cb0\u0c9a\u0ca8\u0cc6<\/strong> : BL \u22a5PQ \u0cae\u0ca4\u0ccd\u0ca4\u0cc1 CM \u22a5 RS \u0cb0\u0c9a\u0cbf\u0cb8\u0cbf <br><strong>\u0caa\u0cb0\u0cbf\u0cb9\u0cbe\u0cb0  <\/strong>:  \u2235 PQ || RS \u21d2 BL || CM&nbsp; [\u2235 BL \u22a5 PQ \u0cae\u0ca4\u0ccd\u0ca4\u0cc1 CM \u22a5 RS]<br>\u0c88\u0c97 , BL || CM \u0cb9\u0cbe\u0c97\u0cc2 BC \u0c92\u0c82\u0ca6\u0cc1 \u0c9b\u0cc7\u0ca6\u0c95 \u0cb0\u0cc7\u0c96\u0cc6\u0caf\u0cbe\u0c97\u0cbf\u0ca6\u0cc6<\/p>\n\n\n\n<p>\u2234 \u2220LBC = \u2220MCB \u2026(1) [\u0caa\u0cb0\u0ccd\u0caf\u0cbe\u0caf \u0c85\u0c82\u0ca4\u0cb0\u0ccd \u0c95\u0ccb\u0ca8\u0c97\u0cb3\u0cc1]<br>\u0cac\u0cc6\u0cb3\u0c95\u0cbf\u0ca8 \u0caa\u0ccd\u0cb0\u0ca4\u0cbf\u0cab\u0cb2\u0ca8 \u0ca8\u0cbf\u0caf\u0cae\u0ca6\u0c82\u0ca4\u0cc6 , \u0caa\u0ca4\u0ca8 \u0c95\u0cbf\u0cb0\u0ca3 \u0c95\u0ccb\u0ca8  = \u0caa\u0ccd\u0cb0\u0ca4\u0cbf\u0cab\u0cb2\u0cbf\u0ca4 \u0c95\u0cbf\u0cb0\u0ca3 \u0c95\u0ccb\u0ca8 <br>\u2220ABL = \u2220LBC \u0cae\u0ca4\u0ccd\u0ca4\u0cc1 \u2220MCB = \u2220MCD<br>\u21d2 \u2220ABL = \u2220MCD \u2026(2) [\u0cb8\u0cae\u0cc0\u0c95\u0cb0\u0ca3 1 \u0cb0\u0cbf\u0c82\u0ca6 ]<\/p>\n\n\n\n<p>(1) \u0cae\u0ca4\u0ccd\u0ca4\u0cc1 (2) \u0ca8\u0ccd\u0ca8\u0cc1 \u0c95\u0cc2\u0ca1\u0cbf\u0cb8\u0cbf\u0ca6\u0cbe\u0c97  <br> <br>\u2220LBC + \u2220ABL = \u2220MCB + \u2220MCD<br>\u21d2 \u2220ABC = \u2220BCD<\/p>\n\n\n\n<p>\u0c86\u0ca6\u0cb0\u0cc6, \u0caa\u0ccd\u0cb0\u0ca4\u0cbf\u0caf\u0cca\u0c82\u0ca6\u0cc1 \u0c9c\u0cca\u0ca4\u0cc6 \u0caa\u0cb0\u0ccd\u0caf\u0cbe\u0caf \u0c85\u0c82\u0ca4\u0cb0\u0ccd \u0c95\u0ccb\u0ca8\u0c97\u0cb3\u0cc1 \u0cb8\u0cae\u0cb5\u0cbe\u0c97\u0cbf\u0cb0\u0cc1\u0ca4\u0ccd\u0ca4\u0cb5\u0cc6<\/p>\n\n\n\n<p>\u2234 AB || CD.<br><\/p>\n\n\n\n<hr class=\"wp-block-separator\"\/>\n\n\n\n<h4 class=\"wp-block-heading\">Exercise 3.2<\/h4>\n\n\n\n<p><strong>\u0ca4\u0ccd\u0cb0\u0cbf\u0cad\u0cc1\u0c9c\u0ca6 \u0c95\u0ccb\u0ca8\u0c97\u0cb3 \u0cae\u0cca\u0ca4\u0ccd\u0ca4\u0ca6 \u0c97\u0cc1\u0ca3\u0cb2\u0c95\u0ccd\u0cb7\u0ca3 :<\/strong><\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"261\" height=\"243\" src=\"https:\/\/themindpalace.in\/wp-content\/uploads\/2020\/09\/11.jpg\" alt=\"\" class=\"wp-image-986\"\/><\/figure><\/div>\n\n\n\n<p>STATEMENT<strong>:<\/strong>&nbsp;The sum of the angles of a triangle is 180\u00ba.<\/p>\n\n\n\n<p><strong>Given:<\/strong>&nbsp;A&nbsp; \u0394 ABC.<\/p>\n\n\n\n<p><strong>To Prove:&nbsp; <\/strong>\u22201 + \u22202 + \u22203 = 180\u00b0<\/p>\n\n\n\n<p><strong>Construction:&nbsp; <\/strong>Let us draw a line DAE&nbsp; parallel to BC.<\/p>\n\n\n\n<p><strong>Proof:<\/strong> Statement                                                                 Reason<\/p>\n\n\n\n<p>DAE ||BC and PQ and AB Is the transversal                   <\/p>\n\n\n\n<p>Hence, \u22204 = \u22202                                                                         (alternate interior angles)<\/p>\n\n\n\n<p>&nbsp; DAE ||BC and AC Is the transversal                                 (alternate interior angles<\/p>\n\n\n\n<p>&nbsp;&nbsp;\u22205 = \u22203<\/p>\n\n\n\n<p>Now, DAE is a straight line<\/p>\n\n\n\n<p>\u22204 + \u22201 +\u22205 = 180\u00b0<\/p>\n\n\n\n<p>\u21d2 \u22201 + \u22204 +\u22205 = 180\u00b0                                                          <\/p>\n\n\n\n<p>Angles on the same side<\/p>\n\n\n\n<p>of DAE at the point A<\/p>\n\n\n\n<p>\u22204 = \u22202 &nbsp;and&nbsp; \u22205 = \u22203<\/p>\n\n\n\n<p><strong>Hence, the sum of the angles of a triangles 180\u00b0<\/strong><\/p>\n\n\n\n<hr class=\"wp-block-separator\"\/>\n\n\n\n<p><strong>Exterior angle property of a triangle theorem<\/strong><\/p>\n\n\n\n<p>STATEMENT<strong>:<\/strong>&nbsp;If a side of a triangle is produced, then the exterior angle so formed is equal to the sum of the two interior opposite angles.<\/p>\n\n\n\n<p><strong>Given<\/strong><strong>: <\/strong>A&nbsp; \u0394 &nbsp;ABC. whose side BC has been produced to D.<\/p>\n\n\n\n<p>&nbsp; Forming exterior angle \u2220ACD<\/p>\n\n\n\n<p><strong>To Prove:&nbsp; <\/strong>\u2220ACD = \u2220BAC + \u2220CBA&nbsp; [\u22204 = \u22201 + \u22202<\/p>\n\n\n\n<p><strong>Proof:<\/strong>. Statement                                                               Reason<\/p>\n\n\n\n<p>We know that \u0394 &nbsp;ABC                                              from the angle sum property<\/p>\n\n\n\n<p>&nbsp;\u22201 + \u22202 + \u22203 = 180\u00b0\u2026..(i)<\/p>\n\n\n\n<p>Also&nbsp;&nbsp; \u22203 +\u22204 = 180\u00b0\u2026..(ii)                                       form a linear pair<\/p>\n\n\n\n<p>From equation (i) and (ii) it follows that:<\/p>\n\n\n\n<p>\u22203 +\u22204 = \u22201 + \u22202 + \u22203<\/p>\n\n\n\n<p>\u2234 \u22204 = \u22201 + \u22202<\/p>\n\n\n\n<p><strong>Hence, proved<\/strong><\/p>\n\n\n\n<hr class=\"wp-block-separator\"\/>\n\n\n\n<p>1. In figure, sides QP and RQ of \u2206PQR are produced to points S and T, &nbsp;&nbsp;respectively. If \u2220SPR = 135\u00b0 and \u2220PQT = 110\u00b0, find \u2220PRQ.<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"182\" height=\"174\" src=\"https:\/\/themindpalace.in\/wp-content\/uploads\/2020\/09\/111.jpg\" alt=\"\" class=\"wp-image-987\"\/><\/figure><\/div>\n\n\n\n<p><strong>Given:<\/strong>&nbsp; \u2220SPR = 135\u00b0 and \u2220PQT = 110\u00b0,<\/p>\n\n\n\n<p><strong>To find: <\/strong>\u2220PRQ.<\/p>\n\n\n\n<p><strong>Solution<\/strong>:We have, \u2220TQP + \u2220PQR = 180\u00b0&nbsp; [Linear pair]<\/p>\n\n\n\n<p>\u21d2 110\u00b0 + \u2220PQR = 180\u00b0<br>\u21d2 \u2220PQR = 180\u00b0 \u2013 110\u00b0 = 70\u00b0<\/p>\n\n\n\n<p>Since, the side QP of \u2206PQR is produced to S.<\/p>\n\n\n\n<p>\u21d2 \u2220PQR + \u2220PRQ = 135\u00b0<br>[Exterior angle property of a triangle]<\/p>\n\n\n\n<p>\u21d2 70\u00b0 + \u2220PRQ = 135\u00b0 [\u2220PQR = 70\u00b0]<br>\u21d2 \u2220PRQ = 135\u00b0 \u2013 70\u00b0 \u21d2 \u2220PRQ = 65\u00b0<\/p>\n\n\n\n<hr class=\"wp-block-separator\"\/>\n\n\n\n<p>2. In figure, \u2220X = 62\u00b0, \u2220XYZ = 54\u00b0, if YO and ZO are the bisectors of \u2220XYZ and \u2220XZY respectively of \u2206XYZ, find \u2220OZY and \u2220YOZ.<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"262\" height=\"206\" src=\"https:\/\/themindpalace.in\/wp-content\/uploads\/2020\/09\/222.jpg\" alt=\"\" class=\"wp-image-988\"\/><\/figure><\/div>\n\n\n\n<p><strong>Given:<\/strong> \u2220X = 62\u00b0, \u2220XYZ = 54\u00b0, YO and ZO are the bisectors of \u2220XYZ and \u2220XZY<\/p>\n\n\n\n<p><strong>To find:<\/strong>\u2220OZY and \u2220YOZ.<\/p>\n\n\n\n<p><strong>Solution:<\/strong>In \u2206XYZ, we have \u2220XYZ + \u2220YZX + \u2220ZXY = 180\u00b0<br>[Angle sum property of a triangle]<\/p>\n\n\n\n<p>But \u2220XYZ = 54\u00b0 and \u2220ZXY = 62\u00b0<\/p>\n\n\n\n<p>\u2234 54\u00b0 + \u2220YZX + 62\u00b0 = 180\u00b0<\/p>\n\n\n\n<p>\u21d2 \u2220YZX = 180\u00b0 \u2013 54\u00b0 \u2013 62\u00b0 = 64\u00b0<\/p>\n\n\n\n<p>YO and ZO are the bisectors of \u2220XYZ and \u2220XZY respectively.<\/p>\n\n\n\n<p>\u2234 \u2220OYZ = \u2220XYZ\/ 2 = 54\u00b0\/ 2 = 27\u00b0<br>and \u2220OZY = \u2220YZX \/ 2 = 64\u00b0\/ 2 = 32\u00b0<\/p>\n\n\n\n<p>in \u2206OYZ, we have<br>\u2220YOZ + \u2220OYZ + \u2220OZY = 180\u00b0 [Angle sum property of a triangle<\/p>\n\n\n\n<p>\u21d2 \u2220YOZ + 27\u00b0 + 32\u00b0 = 180\u00b0<br>\u21d2 \u2220YOZ = 180\u00b0 -27\u00b0 \u2013 32\u00b0 = 121\u00b0<\/p>\n\n\n\n<p><strong>Thus, \u2220OZY = 32\u00b0 and \u2220YOZ = 121\u00b0<\/strong><\/p>\n\n\n\n<hr class=\"wp-block-separator\"\/>\n\n\n\n<p>3. In figure, if AB || DE, \u2220BAC = 35\u00b0 and \u2220CDE = 53\u00b0&nbsp; , find \u2220DCE.<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"211\" height=\"204\" src=\"https:\/\/themindpalace.in\/wp-content\/uploads\/2020\/09\/333.jpg\" alt=\"\" class=\"wp-image-989\"\/><\/figure><\/div>\n\n\n\n<p><strong>Given:<\/strong> AB || DE, \u2220BAC = 35\u00b0 and \u2220CDE = 53\u00b0<\/p>\n\n\n\n<p><strong>To find: <\/strong>\u2220DCE.<\/p>\n\n\n\n<p><strong>Solution:<\/strong>AB || DE and AE is a transversal.<br>So, \u2220BAC = \u2220AED&nbsp; [Alternate interior angles]<\/p>\n\n\n\n<p>and \u2220BAC = 35\u00b0 [Given]<br>&nbsp;&nbsp;&nbsp;&nbsp; \u2234 \u2220AED = 35\u00b0<\/p>\n\n\n\n<p>&nbsp;in \u2206CDE, we have<\/p>\n\n\n\n<p>&nbsp;\u2220CDE + \u2220DEC + \u2220DCE = 180\u00b0&nbsp;&nbsp;&nbsp;&nbsp; [Angle sum property of a triangle]<\/p>\n\n\n\n<p>\u2234 53\u00b0 + 35\u00b0 + \u2220DCE =180\u00b0[\u2235 \u2220DEC = \u2220AED = 35\u00b0 and\u2220CDE = 53\u00b0 (Given)]<\/p>\n\n\n\n<p>\u21d2 \u2220DCE = 180\u00b0 \u2013 53\u00b0 \u2013 35\u00b0 = 92\u00b0<br>Thus, <strong>\u2220DCE = 92\u00b0<\/strong><\/p>\n\n\n\n<hr class=\"wp-block-separator\"\/>\n\n\n\n<p>4. In figure, if lines PQ and RS intersect at point T, such that \u2220 PRT = 40\u00b0, \u2220 RPT = 95\u00b0 and<\/p>\n\n\n\n<p>\u2220TSQ = 75\u00b0, find \u2220 SQT.<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"256\" height=\"207\" src=\"https:\/\/themindpalace.in\/wp-content\/uploads\/2020\/09\/444.jpg\" alt=\"\" class=\"wp-image-990\"\/><\/figure><\/div>\n\n\n\n<p><strong>Given:<\/strong> \u2220PRT = 40\u00b0, \u2220 RPT = 95\u00b0 and \u2220TSQ = 75\u00b0 <strong>&nbsp;<\/strong><\/p>\n\n\n\n<p><strong>To find: <\/strong>\u2220SQT.<\/p>\n\n\n\n<p><strong>Solution<\/strong>:<\/p>\n\n\n\n<p>In \u2206PRT, we have \u2220P + \u2220R + \u2220PTR = 180\u00b0[Angle sum property of a triangle]<\/p>\n\n\n\n<p>\u21d2 95\u00b0 + 40\u00b0 + \u2220PTR =180\u00b0 [ \u2235 \u2220P = 95\u00b0, \u2220R = 40\u00b0 (given)]<\/p>\n\n\n\n<p>\u21d2 \u2220PTR = 180\u00b0 \u2013 95\u00b0 \u2013 40\u00b0 = 45\u00b0<br>But PQ and RS intersect at T.<\/p>\n\n\n\n<p>\u2234 \u2220PTR = \u2220QTS [Vertically opposite angles<\/p>\n\n\n\n<p>\u2234 \u2220QTS = 45\u00b0 [ \u2235 \u2220PTR = 45\u00b0]<\/p>\n\n\n\n<p>in \u2206 TQS, we have \u2220TSQ + \u2220STQ + \u2220SQT = 180\u00b0 [Angle sum property of a triangle]<\/p>\n\n\n\n<p>\u2234 75\u00b0 + 45\u00b0 + \u2220SQT = 180\u00b0 [ \u2235 \u2220TSQ = 75\u00b0 and \u2220STQ = 45\u00b0]<\/p>\n\n\n\n<p>\u21d2 \u2220SQT = 180\u00b0 \u2013 75\u00b0 \u2013 45\u00b0 = 60\u00b0<br>Thus, <strong>\u2220SQT = 60\u00b0<\/strong><\/p>\n\n\n\n<hr class=\"wp-block-separator\"\/>\n\n\n\n<p>5. In figure, if PQ \u22a5 PS, PQ||SR, \u2220SQR = 2S\u00b0 and \u2220QRT = 65\u00b0, then find the values of x and y.<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"270\" height=\"231\" src=\"https:\/\/themindpalace.in\/wp-content\/uploads\/2020\/09\/555.jpg\" alt=\"\" class=\"wp-image-991\"\/><\/figure><\/div>\n\n\n\n<p><strong>Given<\/strong><\/p>\n\n\n\n<p><strong>To find:<\/strong><\/p>\n\n\n\n<p><strong>Solution<\/strong>: In \u2206 QRS, the side SR is produced to T.<\/p>\n\n\n\n<p>\u2234 \u2220QRT = \u2220RQS + \u2220RSQ [Exterior angle property of a triangle]<\/p>\n\n\n\n<p>But \u2220RQS = 28\u00b0 and \u2220QRT = 65\u00b0<\/p>\n\n\n\n<p>So, 28\u00b0 + \u2220RSQ = 65\u00b0<br>\u21d2 \u2220RSQ = 65\u00b0 \u2013 28\u00b0 = 37\u00b0<\/p>\n\n\n\n<p>Since, PQ || SR and QS is a transversal.<\/p>\n\n\n\n<p>\u2234 \u2220PQS = \u2220RSQ = 37\u00b0 [Alternate interior angles]<\/p>\n\n\n\n<p>\u21d2 x = 37\u00b0<\/p>\n\n\n\n<p>PQ \u22a5 PS \u21d2 AP = 90\u00b0<br>&nbsp;in \u2206PQS, \u2220P + \u2220PQS + \u2220PSQ = 180\u00b0 [Angle sum property of a triangle]<\/p>\n\n\n\n<p>\u21d2 90\u00b0 + 37\u00b0 + y = 180\u00b0<\/p>\n\n\n\n<p>\u21d2 y = 180\u00b0 \u2013 90\u00b0 \u2013 37\u00b0 = 53\u00b0<\/p>\n\n\n\n<p><strong>Thus, x = 37\u00b0 and y = 53\u00b0<\/strong><\/p>\n\n\n\n<hr class=\"wp-block-separator\"\/>\n\n\n\n<p>6. In figure, the side QR of \u2206PQR is produced to a point S. If the bisectors of \u2220PQR and \u2220PRS meet at point T, then prove that \u2220QTR =&nbsp;1\/2\u2220QPR<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"278\" height=\"222\" src=\"https:\/\/themindpalace.in\/wp-content\/uploads\/2020\/09\/666.jpg\" alt=\"\" class=\"wp-image-992\"\/><\/figure><\/div>\n\n\n\n<p>Solution:In \u2206PQR, side QR is produced to S, so by exterior angle property, \u2220PRS = \u2220P + \u2220PQR<\/p>\n\n\n\n<p>\u21d2&nbsp;1\/2\u2220PRS =&nbsp;1\/2\u2220P +&nbsp;12\u2220PQR<\/p>\n\n\n\n<p>Now\u21d2 \u2220TRS =&nbsp;1\/2\u2220P + \u2220TQR \u2026(1)[\u2235 QT and RT are bisectors of \u2220PQR and &nbsp; \u2220PRS respectively.]<\/p>\n\n\n\n<p>\u2220TRS = \u2220TQR + \u2220T \u2026(2)[Exterior angle property of a triangle]<\/p>\n\n\n\n<p>From (1) and (2),<\/p>\n\n\n\n<p>we have \u2220TQR +&nbsp;1\/2\u2220P = \u2220TQR + \u2220T<\/p>\n\n\n\n<p>\u21d2&nbsp;1\/2\u2220P = \u2220T<\/p>\n\n\n\n<p>\u21d2&nbsp;1\/2\u2220QPR = \u2220QTR or <strong>\u2220QTR =&nbsp;<\/strong>1\/2<strong>\u2220QPR<\/strong><\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u0cac\u0cbf\u0c82\u0ca6\u0cc1\u0c97\u0cb3\u0cc1 \u0cac\u0cbf\u0c82\u0ca6\u0cc1\u0cb5\u0ca8\u0ccd\u0ca8\u0cc1 \u0c9a\u0cc1\u0c95\u0ccd\u0c95\u0cbf\u0caf\u0cbf\u0c82\u0ca6 \u0cb8\u0cc2\u0c9a\u0cbf\u0cb8\u0cb2\u0cbe\u0c97\u0cc1\u0ca4\u0ccd\u0ca4\u0ca6\u0cc6. \u0c92\u0c82\u0ca6\u0cc1 \u0cac\u0cbf\u0c82\u0ca6\u0cc1\u0cb5\u0cc1 \u0c92\u0c82\u0ca6\u0cc1 \u0cb8\u0ccd\u0ca5\u0cb3\u0cb5\u0ca8\u0ccd\u0ca8\u0cc1 \u0ca8\u0cbf\u0cb0\u0ccd\u0ca7\u0cb0\u0cbf\u0cb8\u0cc1\u0ca4\u0ccd\u0ca4\u0ca6\u0cc6. \u0cb8\u0cbe\u0cae\u0cbe\u0ca8\u0ccd\u0caf \u0cac\u0cbf\u0c82\u0ca6\u0cc1\u0cb5\u0ca8\u0ccd\u0ca8\u0cc1 \u0cb6\u0cc3\u0c82\u0c97\u0cac\u0cbf\u0c82\u0ca6\u0cc1 \u0c8e\u0ca8\u0ccd\u0ca8\u0cc1\u0ca4\u0ccd\u0ca4\u0cc7\u0cb5\u0cc6 \u0c92\u0c82\u0ca6\u0cc1 \u0cb0\u0cc7\u0c96\u0cc6\u0caf\u0cc1 \u0cb9\u0cb2\u0cb5\u0cbe\u0cb0\u0cc1 \u0cac\u0cbf\u0c82\u0ca6\u0cc1\u0c97\u0cb3\u0cbf\u0c82\u0ca6\u0cbe\u0c97\u0cbf\u0cb0\u0cc1\u0ca4\u0ccd\u0ca4\u0ca6\u0cc6. \u0cb8\u0cb9\u0cb0\u0cc7\u0c96\u0cbf \u0cac\u0cbf\u0c82\u0ca6\u0cc1\u0c97\u0cb3\u0cc1 \u0c92\u0c82\u0ca6\u0cc1 \u0cb8\u0cb0\u0cb3 \u0cb0\u0cc7\u0c96\u0cc6\u0caf \u0cae\u0cc7\u0cb2\u0cc6 \u0cb8\u0cae\u0cbe\u0ca8\u0cb5\u0cbe\u0c97\u0cbf \u0cb9\u0cb0\u0ca1\u0cbf\u0c95\u0cca\u0c82\u0ca1\u0cbf\u0cb0\u0cc1\u0ca4\u0ccd\u0ca4\u0cb5\u0cc6. \u0cb8\u0cb9\u0cb0\u0cc7\u0c96\u0cbf \u0c85\u0cb2\u0ccd\u0cb2\u0ca6 \u0cac\u0cbf\u0c82\u0ca6\u0cc1\u0c97\u0cb3\u0cc1 \u0c92\u0c82\u0ca6\u0cc7 \u0cb8\u0cb0\u0cb3\u0cb0\u0cc7\u0c96\u0cc6\u0caf \u0cae\u0cc7\u0cb2\u0cc6 \u0c87\u0cb0\u0cc1\u0cb5\u0cc1\u0ca6\u0cbf\u0cb2\u0ccd\u0cb2. \u0cb0\u0cc7\u0c96\u0cbe\u0c96\u0c82\u0ca1 \u0c8e\u0cb0\u0ca1\u0cc1&#46;&#46;&#46;<\/p>\n","protected":false},"author":2,"featured_media":1129,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[111,14,4],"tags":[],"cp_meta_data":{"_edit_lock":["1629955765:2"],"_edit_last":["2"],"_layout":["inherit"],"_heateor_sss_meta":["a:2:{s:7:\"sharing\";i:0;s:16:\"vertical_sharing\";i:0;}"],"_thumbnail_id":["1129"],"_oembed_95287caaddeb112cd4edfcbd8e525566":["<iframe title=\"Introduction of Computers  Part1\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/SzIGR3gp_F4?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; 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