{"id":4212,"date":"2010-10-18T21:54:03","date_gmt":"2010-10-18T20:54:03","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=4212"},"modified":"2022-01-18T23:59:48","modified_gmt":"2022-01-18T23:59:48","slug":"angulos-internos-e-externos-de-um-triangulo","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=4212","title":{"rendered":"\u00c2ngulos internos e externos de um tri\u00e2ngulo"},"content":{"rendered":"<p><ul id='GTTabs_ul_4212' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_4212' class='GTTabs_curr'><a  id=\"4212_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_4212' ><a  id=\"4212_1\" onMouseOver=\"GTTabsShowLinks('Resolu\u00e7\u00e3o'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Resolu\u00e7\u00e3o<\/a><\/li>\n<\/ul>\n\n<div class='GTTabs_divs GTTabs_curr_div' id='GTTabs_0_4212'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/pag102-1.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"4215\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=4215\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/pag102-1.jpg\" data-orig-size=\"327,279\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;HP pstc4380&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;}\" data-image-title=\"Tri\u00e2ngulo\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/pag102-1.jpg\" class=\"alignright wp-image-4215 size-full\" title=\"Tri\u00e2ngulo\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/pag102-1.jpg\" alt=\"\" width=\"327\" height=\"279\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/pag102-1.jpg 327w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/pag102-1-300x255.jpg 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/pag102-1-150x127.jpg 150w\" sizes=\"auto, (max-width: 327px) 100vw, 327px\" \/><\/a>Utilizando os dados da figura, calcula:<\/p>\n<ol>\n<li>A medida de cada um dos \u00e2ngulos internos do tri\u00e2ngulo [MNP];<\/li>\n<li>A soma dos \u00e2ngulos externos do tri\u00e2ngulo.<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_4212' onClick='GTTabs_show(1,4212)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_4212'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<ol>\n<li><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"4215\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=4215\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/pag102-1.jpg\" data-orig-size=\"327,279\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;HP pstc4380&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;}\" data-image-title=\"Tri\u00e2ngulo\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/pag102-1.jpg\" class=\"alignright wp-image-4215 size-full\" title=\"Tri\u00e2ngulo\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/pag102-1.jpg\" alt=\"\" width=\"327\" height=\"279\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/pag102-1.jpg 327w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/pag102-1-300x255.jpg 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/pag102-1-150x127.jpg 150w\" sizes=\"auto, (max-width: 327px) 100vw, 327px\" \/>Considerando que os \u00e2ngulos seguintes s\u00e3o suplementares, temos:<br \/>\n\\[\\begin{array}{*{35}{l}}<br \/>\n{\\hat{P}} &amp; = &amp; 180{}^\\text{o}-N\\hat{P}Q\u00a0 \\\\<br \/>\n{} &amp; = &amp; 180{}^\\text{o}-145{}^\\text{o}\u00a0 \\\\<br \/>\n{} &amp; = &amp; 35{}^\\text{o}\u00a0 \\\\<br \/>\n\\end{array}\\]<br \/>\ne<br \/>\n\\[\\begin{array}{*{35}{l}}<br \/>\n{\\hat{M}} &amp; = &amp; 180{}^\\text{o}-N\\hat{M}P\u00a0 \\\\<br \/>\n{} &amp; = &amp; 180{}^\\text{o}-77{}^\\text{o}\u00a0 \\\\<br \/>\n{} &amp; = &amp; 103{}^\\text{o}\u00a0 \\\\<br \/>\n\\end{array}\\]<br \/>\nComo a soma dos \u00e2ngulos internos de um tri\u00e2ngulo \u00e9 um \u00e2ngulo raso, vem: \\[\\begin{array}{*{35}{l}}<br \/>\n{\\hat{N}} &amp; = &amp; 180{}^\\text{o}-(\\hat{M}+\\hat{P})\u00a0 \\\\<br \/>\n{} &amp; = &amp; 180{}^\\text{o}-(103{}^\\text{o}+35{}^\\text{o})\u00a0 \\\\<br \/>\n{} &amp; = &amp; 180{}^\\text{o}-138{}^\\text{o}\u00a0 \\\\<br \/>\n{} &amp; = &amp; 42{}^\\text{o}\u00a0 \\\\<br \/>\n\\end{array}\\]<br \/>\n\u00ad<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"4215\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=4215\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/pag102-1.jpg\" data-orig-size=\"327,279\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;HP pstc4380&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;}\" data-image-title=\"Tri\u00e2ngulo\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/pag102-1.jpg\" class=\"alignright wp-image-4215 size-full\" title=\"Tri\u00e2ngulo\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/pag102-1.jpg\" alt=\"\" width=\"327\" height=\"279\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/pag102-1.jpg 327w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/pag102-1-300x255.jpg 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/pag102-1-150x127.jpg 150w\" sizes=\"auto, (max-width: 327px) 100vw, 327px\" \/>Considerando que os \u00e2ngulos seguintes s\u00e3o suplementares, temos: \\[\\begin{array}{*{35}{l}}<br \/>\nM\\hat{N}S &amp; = &amp; 180{}^\\text{o}-\\hat{N}\u00a0 \\\\<br \/>\n{} &amp; = &amp; 180{}^\\text{o}-42{}^\\text{o}\u00a0 \\\\<br \/>\n{} &amp; = &amp; 138{}^\\text{o}\u00a0 \\\\<br \/>\n\\end{array}\\]<br \/>\nLogo, $N\\hat{P}Q+R\\hat{M}Q+S\\hat{N}M=145{}^\\text{o}+77{}^\\text{o}+138{}^\\text{o}=360{}^\\text{o}$.<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_4212' onClick='GTTabs_show(0,4212)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Utilizando os dados da figura, calcula: A medida de cada um dos \u00e2ngulos internos do tri\u00e2ngulo [MNP]; A soma dos \u00e2ngulos externos do tri\u00e2ngulo. Resolu\u00e7\u00e3o &gt;&gt; Resolu\u00e7\u00e3o &lt;&lt; Enunciado<\/p>\n","protected":false},"author":1,"featured_media":20625,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[100,97,102],"tags":[424,105,67],"series":[],"class_list":["post-4212","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-8--ano","category-aplicando","category-do-espaco-ao-plano","tag-8-o-ano","tag-angulos","tag-geometria"],"views":6495,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/7V2Pag102-1_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/4212","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=4212"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/4212\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/20625"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=4212"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=4212"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=4212"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=4212"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}