{"id":139,"date":"2017-08-23T09:07:26","date_gmt":"2017-08-23T09:07:26","guid":{"rendered":"http:\/\/americanboard.org\/Subjects\/mathematics\/?page_id=139"},"modified":"2017-09-18T18:15:38","modified_gmt":"2017-09-18T18:15:38","slug":"skew-lines","status":"publish","type":"page","link":"https:\/\/americanboard.org\/Subjects\/mathematics\/skew-lines\/","title":{"rendered":"Skew Lines"},"content":{"rendered":"<div class=\"twelve columns\" style=\"margin-top: 10%;\">\n<div class=\"advance\"><a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/straightedge-and-compass-constructions\">\u2b05 Previous Lesson<\/a>\u00a0<a class=\"button\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/geometry-spatial-reasoning\">Workshop Index<\/a>\u00a0<a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/special-triangles\">Next Lesson \u27a1<\/a><\/div>\n<p><!-- CONTENT BEGINS HERE --><\/p>\n<h1 id=\"title\">Skew Lines<\/h1>\n<h4>Objective<\/h4>\n<p>In this lesson, we will define skew lines and show how to illustrate them.<\/p>\n<h4>Previously Covered:<\/h4>\n<ul>\n<li>If the sum of the measures of\u00a0two angles is 180\u00b0, then the angles are called <em><strong>supplementary<\/strong><\/em>, and each\u00a0angle is called\u00a0the <em><strong>supplement<\/strong><\/em> of the other.<\/li>\n<li>If the sum of the measures of\u00a0two angles is 90, then the angles are called <strong><em>complementary\u00a0<\/em><\/strong>and each angle is called the <strong><em>complement of the angle<\/em><\/strong>.<\/li>\n<li>The <em><strong><span style=\"text-decoration: none;\">angle\u00a0bisector\u00a0<\/span><\/strong><\/em>of angle\u00a0<em><span style=\"text-decoration: none;\">BAC\u00a0<\/span><\/em><span style=\"text-decoration: none;\">is the ray\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/s9_p1_html_mb09fd5c.gif\" width=\"28\" height=\"21\" name=\"graphics5\" align=\"BOTTOM\" border=\"0\" \/>such\u00a0that\u00a0angle<\/span><em><span style=\"text-decoration: none;\"> BAD\u00a0<\/span><\/em><span style=\"text-decoration: none;\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/s9_p1_html_11aa9d18.gif\" width=\"15\" height=\"13\" name=\"graphics4\" align=\"BOTTOM\" border=\"0\" \/>angle\u00a0<\/span><em><span style=\"text-decoration: none;\">DAC<\/span><\/em><span style=\"text-decoration: none;\">. <\/span><\/li>\n<\/ul>\n<section>\n<h3>Skew Lines<\/h3>\n<p>We already know that in a plane, two lines can either be\u00a0parallel or can intersect at one point. There is one more\u00a0possibility for two lines in a plane: they can be\u00a0<abbr title=\"two or more lines that are actually the same line\">coincident<\/abbr>. This\u00a0means that they are the same line. So there are three\u00a0possibilities for defining a pair of coplanar lines: parallel,\u00a0intersecting, or coincident.<\/p>\n<p class=\"lesson_subhead\"><strong>What about lines in three-dimensional space?<\/strong><\/p>\n<p>There are actually four ways that two lines can be situated in\u00a0three-dimensional space . We have all of the possibilities for\u00a0coplanar lines, plus one more.<\/p>\n<p>Two lines that are not coplanar and do not intersect are called\u00a0<abbr title=\"two lines in three-dimensional space that are not coplanar and do not intersect \">skew<\/abbr> lines.<\/p>\n<p>The fact that this is the only other possibility in three dimensions is guaranteed by the following two theorems.<\/p>\n<p><em><span style=\"text-decoration: none;\">Two parallel\u00a0lines lie in a unique plane. <\/span><\/em><em><span style=\"text-decoration: none;\">Two\u00a0intersecting lines lie in a unique plane. <\/span><\/em>If two lines are actually the same line, then they are\u00a0automatically coplanar.<\/p>\n<p>Therefore, by the theorems above, skew lines are distinct lines\u00a0that do not intersect and are not parallel.<\/p>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>Which choice describes a pair of skew lines in the figure\u00a0below?<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/Math%20Mod%204.3%20Art%20020.JPG\" alt=\"Skew lines\" width=\"239\" height=\"128\" name=\"graphics3\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<ol>\n<li><em><span style=\"text-decoration: none;\">L\u00a0<\/span><\/em><span style=\"text-decoration: none;\">and <\/span><em><span style=\"text-decoration: none;\">M <\/span><\/em><\/li>\n<li><em><span style=\"text-decoration: none;\">M\u00a0<\/span><\/em><span style=\"text-decoration: none;\">and <\/span><em><span style=\"text-decoration: none;\">N<\/span><\/em><\/li>\n<li><em><span style=\"text-decoration: none;\">N\u00a0<\/span><\/em><span style=\"text-decoration: none;\">and <\/span><em><span style=\"text-decoration: none;\">O<\/span><\/em><\/li>\n<li><em><span style=\"text-decoration: none;\">L\u00a0<\/span><\/em><span style=\"text-decoration: none;\">and <\/span><em><span style=\"text-decoration: none;\">O<\/span><\/em><\/li>\n<\/ol>\n<\/div>\n<p><a class=\"button button-primary q-answer\"> Reveal Answer <\/a><\/p>\n<div class=\"q-reveal\">\n<p>The correct answer is D . <span style=\"text-decoration: none;\">The\u00a0pairs (<\/span><em><span style=\"text-decoration: none;\">L<\/span><\/em><span style=\"text-decoration: none;\">,\u00a0<\/span><em><span style=\"text-decoration: none;\">M<\/span><\/em><span style=\"text-decoration: none;\">)\u00a0and (<\/span><em><span style=\"text-decoration: none;\">M<\/span><\/em><span style=\"text-decoration: none;\">,\u00a0<\/span><em><span style=\"text-decoration: none;\">N<\/span><\/em><span style=\"text-decoration: none;\">)\u00a0are parallel. This fact is verified by the faces of the\u00a0rectangular prism shown in the figure . The pair (<\/span><em><span style=\"text-decoration: none;\">N<\/span><\/em><span style=\"text-decoration: none;\">,\u00a0<\/span><em><span style=\"text-decoration: none;\">O<\/span><\/em><span style=\"text-decoration: none;\">)\u00a0intersects . <\/span><em><span style=\"text-decoration: none;\">L <\/span><\/em><span style=\"text-decoration: none;\">and\u00a0<\/span><em><span style=\"text-decoration: none;\">O <\/span><\/em><span style=\"text-decoration: none;\">are\u00a0therefore the only possible skew lines. <\/span><\/p>\n<\/div>\n<\/section>\n<p>The example shows how one can use a rectangular prism to\u00a0illustrate skew lines.<\/p>\n<p>Other polyhedra can also be used to illustrate skew lines.<\/p>\n<\/section>\n<p><!-- CONTENT ENDS HERE --><\/p>\n<div class=\"advance\"><a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/straightedge-and-compass-constructions\">\u2b05 Previous Lesson<\/a>\u00a0<a class=\"button\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/geometry-spatial-reasoning\">Workshop Index<\/a>\u00a0<a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/special-triangles\">Next Lesson \u27a1<\/a><\/div>\n<p><a class=\"backtotop\" href=\"#title\">Back to Top<\/a><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>\u2b05 Previous Lesson\u00a0Workshop Index\u00a0Next Lesson \u27a1 Skew Lines Objective In this lesson, we will define skew lines and show how to illustrate them. Previously Covered: If the sum of the measures of\u00a0two angles is 180\u00b0, then the angles are called supplementary, and each\u00a0angle is called\u00a0the supplement of the other. If the sum of the measures [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-139","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/139","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/comments?post=139"}],"version-history":[{"count":7,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/139\/revisions"}],"predecessor-version":[{"id":756,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/139\/revisions\/756"}],"wp:attachment":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/media?parent=139"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}