{"id":140,"date":"2017-08-23T09:08:02","date_gmt":"2017-08-23T09:08:02","guid":{"rendered":"http:\/\/americanboard.org\/Subjects\/mathematics\/?page_id=140"},"modified":"2017-09-18T18:22:49","modified_gmt":"2017-09-18T18:22:49","slug":"special-triangles","status":"publish","type":"page","link":"https:\/\/americanboard.org\/Subjects\/mathematics\/special-triangles\/","title":{"rendered":"Special Triangles"},"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\/skew-lines\">\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\/area-of-triangles-and-special-quadrilaterals\">Next Lesson \u27a1<\/a><\/div>\n<p><!-- CONTENT BEGINS HERE --><\/p>\n<h1 id=\"title\">Special Triangles<\/h1>\n<h4>Objective<\/h4>\n<p>In this lesson, we will study special triangles and their properties . We will also use special triangles to derive\u00a0the formulas for the areas of special quadrilaterals.<\/p>\n<h4>Previously Covered:<\/h4>\n<ul>\n<li>Two triangles are <strong><em>congruent\u00a0<\/em><\/strong>if there is a correspondence between them such that every pair of\u00a0corresponding sides is congruent and every pair of angles is\u00a0congruent.<\/li>\n<li><strong><em>The Triangle Angle\u00a0Sum Theorem<\/em><\/strong> states that the sum of the interior\u00a0angles of a triangle is 180 \u00b0.<\/li>\n<li>If the sum of the measures of two angles is 180 \u00b0,\u00a0then the angles are called <strong><em>supplementary<\/em><\/strong>, and each angle is called the <em><strong>supplement\u00a0<\/strong><\/em>of the other.<\/li>\n<\/ul>\n<section>\n<h3><strong>How do we classify triangles according to their side lengths?<\/strong><\/h3>\n<p><span style=\"text-decoration: none;\">A <\/span><abbr title=\"a triangle with no two sides congruent\"><span style=\"text-decoration: none;\">scalene\u00a0triangle<\/span><\/abbr> <span style=\"text-decoration: none;\">is\u00a0a triangle in which no two sides are congruent.<\/span><\/p>\n<p><span style=\"text-decoration: none;\">An <\/span><abbr title=\" a triangle with at least two congruent sides\"><span style=\"text-decoration: none;\">isosceles\u00a0triangle<\/span><\/abbr><span style=\"text-decoration: none;\"> in which at least two sides are congruent. <\/span><\/p>\n<p><span style=\"text-decoration: none;\">In an isosceles triangle,\u00a0the two congruent sides are called the <\/span><abbr title=\" nonparallel sides of a trapezoid or the sides that make the right angle in a right triangle\"><span style=\"text-decoration: none;\">legs<\/span><\/abbr> <span style=\"text-decoration: none;\"> and the third side is called the <\/span><abbr title=\"side or face of a geometrical figure from which an altitude can be constructed \"><span style=\"text-decoration: none;\">base<\/span><\/abbr><span style=\"text-decoration: none;\">.<br \/>\n<\/span><\/p>\n<p><span style=\"text-decoration: none;\">An <\/span><abbr title=\" a triangle with three congruent sides \"><span style=\"text-decoration: none;\">equilateral\u00a0triangle<\/span><\/abbr> <span style=\"text-decoration: none;\">in\u00a0which all three sides are congruent. <\/span><\/p>\n<p class=\"lesson_subhead\"><strong>How do we classify triangles according to their angles?<\/strong><\/p>\n<p><span style=\"text-decoration: none;\">An <\/span><abbr title=\"a triangle with three acute angles\"><span style=\"text-decoration: none;\">acute\u00a0triangle<\/span><\/abbr><span style=\"text-decoration: none;\"> is a triangle in which all three angles are acute angles. <\/span><\/p>\n<p><span style=\"text-decoration: none;\">A <\/span><abbr title=\" a triangle with only one right angle \"><span style=\"text-decoration: none;\">right\u00a0triangle<\/span><\/abbr><span style=\"text-decoration: none;\"> is a triangle in which there is only one right angle. <\/span><\/p>\n<p><span style=\"text-decoration: none;\">In a right triangle, the\u00a0side opposite the right angle is the\u00a0<\/span><abbr title=\" the longest side of a right triangle; it is always opposite of the right angle. \"><span style=\"text-decoration: none;\">hypotenuse<\/span><\/abbr><span style=\"text-decoration: none;\">,\u00a0and the other two sides are the <\/span> <abbr title=\"nonparallel sides of a trapezoid or the sides that make the right angle in a right triangle \"><span style=\"text-decoration: none;\">legs<\/span><\/abbr><span style=\"text-decoration: none;\">.<br \/>\n<\/span><\/p>\n<p><span style=\"text-decoration: none;\">An <\/span><abbr title=\"a triangle with only one angle greater than 90 degrees\"><span style=\"text-decoration: none;\">obtuse\u00a0triangle<\/span><\/abbr><span style=\"text-decoration: none;\"> is a triangle in which there is only one obtuse angle. <\/span><\/p>\n<p><strong>For any triangle, we also have the following parts:<\/strong><\/p>\n<p>A\u00a0<abbr title=\"a segment in a triangle from any vertex to the midpoint of the opposite side\">median<\/abbr> of a triangle is the segment from any vertex to the midpoint of\u00a0the opposite side.<\/p>\n<p>An\u00a0<abbr title=\" the segment from any vertex of a triangle that is perpendicular to the line containing the opposite side\">altitude<\/abbr> of a triangle is the segment from any vertex that is perpendicular\u00a0to the line containing the opposite side.<\/p>\n<p>You may recall the notion of an exterior angle from the\u00a0previous section on convex polygons . Every triangle has six\u00a0exterior angles, as shown below.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/MAth%20Mod%204.3%20Art%20021.JPG\" alt=\" exterior angles of a triangle\" width=\"188\" height=\"175\" name=\"graphics3\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>I<span style=\"text-decoration: none;\">n the figure above, the\u00a0angles angle<\/span><em><span style=\"text-decoration: none;\"> B\u00a0<\/span><\/em><span style=\"text-decoration: none;\">and angle<\/span><em><span style=\"text-decoration: none;\"> C <\/span><\/em><span style=\"text-decoration: none;\">of\u00a0the triangle <\/span><em><span style=\"text-decoration: none;\">ABC\u00a0<\/span><\/em><span style=\"text-decoration: none;\">are referred to as\u00a0the <\/span><abbr title=\" two interior angles of a triangle that are not adjacent to the exterior angle, which is drawn by extending one of the sides \"><span style=\"text-decoration: none;\">remote\u00a0interior angles<\/span><\/abbr><span style=\"text-decoration: none;\"> of the exterior angles angle 1 and angle 2.<\/span><\/p>\n<h3><strong>Exterior Angle Theorem <\/strong><\/h3>\n<blockquote><p><em><span style=\"text-decoration: none;\">An exterior\u00a0angle of a triangle is equal to the sum of its remote interior\u00a0angles. <\/span><\/em><\/p><\/blockquote>\n<p>The Exterior Angle Theorem follows from the fact that a fixed\u00a0angle and one of its exterior angles are supplementary, and that\u00a0the sum of the angles of a triangle is 180\u00b0.<\/p>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p><span style=\"text-decoration: none;\">What is the measure of the\u00a0angle marked <\/span><em><span style=\"text-decoration: none;\">y\u00a0<\/span><\/em><span style=\"text-decoration: none;\">in the triangle shown below? <\/span><\/p>\n<p style=\"text-decoration: none;\" align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/MAth%20Mod%204.3%20Art%20022.JPG\" alt=\"Triangle with 160 degree exterior angle\" width=\"254\" height=\"104\" name=\"graphics4\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<ol>\n<li>40 \u00b0<\/li>\n<li>140 \u00b0<\/li>\n<li>160 \u00b0<\/li>\n<li>Cannot be determined<\/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><span style=\"text-decoration: none;\">The correct answer is B.\u00a0The measure of <\/span><em><span style=\"text-decoration: none;\">x\u00a0<\/span><\/em><span style=\"text-decoration: none;\">is 20 \u00b0 because it is the supplement of the angle indicated.<br \/>\nSince the sum of the interior angles is 180 \u00b0, we have 20 +\u00a020 + <\/span><em><span style=\"text-decoration: none;\">y <\/span><\/em><span style=\"text-decoration: none;\">=\u00a0180 . Thus, <\/span><em><span style=\"text-decoration: none;\">y <\/span><\/em><span style=\"text-decoration: none;\">=\u00a0140 \u00b0.<\/span><\/p>\n<\/div>\n<\/section>\n<h3><strong>What is the relationship between congruent angles and\u00a0congruent sides in special triangles?<\/strong><\/h3>\n<p><em><span>\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0Two sides of a\u00a0triangle are congruent if and only if the angles opposite those\u00a0sides are congruent.<\/span><\/em><\/p>\n<p style=\"text-decoration: none;\">This means that an isosceles\u00a0triangle has at least two congruent angles. Conversely, if a\u00a0triangle has at least two congruent angles it is an isosceles\u00a0triangle.<\/p>\n<blockquote><p><em><span style=\"text-decoration: none;\">A triangle has\u00a0three congruent sides if and only if it has three congruent\u00a0angles. <\/span><\/em><\/p><\/blockquote>\n<p style=\"text-decoration: none;\">Any equilateral triangle is\u00a0equiangular and vice versa.<\/p>\n<blockquote><p>&nbsp;<\/p>\n<p>If angle ABC\u00a0is an isosceles triangle with congruent sides\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/s10_p3_html_299c8a9a.gif\" width=\"27\" height=\"21\" name=\"graphics13\" align=\"BOTTOM\" border=\"0\" \/>\u00a0and\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/s10_p3_html_786a0686.gif\" width=\"27\" height=\"23\" name=\"graphics14\" align=\"BOTTOM\" border=\"0\" \/>,\u00a0then the median from B to\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/s10_p3_html_m76b936f1.gif\" width=\"28\" height=\"23\" name=\"graphics15\" align=\"BOTTOM\" border=\"0\" \/>\u00a0is the altitude from B to\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/s10_p3_html_m76b936f1.gif\" width=\"28\" height=\"23\" name=\"graphics16\" align=\"BOTTOM\" border=\"0\" \/>.<\/p>\n<p>&nbsp;<\/p><\/blockquote>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/Math%20Mod%204.3%20Art%20023.JPG\" alt=\"Triangle median \" width=\"101\" height=\"104\" name=\"graphics3\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>What is the formula for the altitude of an equilateral triangle\u00a0if <em>s <\/em>= length of a side?<\/p>\n<ol>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/s10_p3_clip_image002.gif\" width=\"9\" height=\"30\" name=\"graphics4\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/s10_p3_clip_image004.gif\" width=\"25\" height=\"18\" name=\"graphics5\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li><em>s<\/em><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/s10_p3_clip_image006.gif\" width=\"29\" height=\"38\" name=\"graphics6\" align=\"BOTTOM\" border=\"0\" \/><\/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. By the preceding theorem the altitude\u00a0is the median in an equilateral triangle, so the shorter side has\u00a0length\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/s10_p3_clip_image007.gif\" width=\"9\" height=\"30\" name=\"graphics7\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0In fact, the altitude divides the triangle into two congruent\u00a030-60-90 triangles. Therefore, the longer leg has length\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/s10_p3_clip_image008.gif\" width=\"9\" height=\"30\" name=\"graphics8\" align=\"ABSMIDDLE\" border=\"0\" \/><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/s10_p3_clip_image010.gif\" width=\"18\" height=\"18\" name=\"graphics9\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0=\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/s10_p3_clip_image011.gif\" width=\"29\" height=\"38\" name=\"graphics10\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/p>\n<\/div>\n<\/section>\n<div class=\"callout\">\n<h4>Important Tidbit<\/h4>\n<p class=\"notebox_text\" align=\"LEFT\">In an equilateral triangle, the altitudes (which are also the medians)\u00a0and angle\u00a0bisectors all intersect at one point in the interior called the <strong>center of gravity.<\/strong><\/p>\n<\/div>\n<h3><strong>HL Theorem<\/strong><\/h3>\n<blockquote><p><span style=\"text-decoration: none;\">I<span style=\"text-decoration: none;\">f\u00a0the hypotenuse and one leg of a right triangle are congruent to\u00a0corresponding sides of another right triangle, then the triangles\u00a0are congruent. <\/span><\/span><\/p><\/blockquote>\n<h4><strong>What are the properties of special right triangles?<\/strong><\/h4>\n<p>In a 30-60-90 triangle, the length of the hypotenuse is twice\u00a0the length of the shorter leg and the length of the longer leg is\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/s10_p4_clip_image002.gif\" width=\"18\" height=\"18\" name=\"graphics3\" align=\"BOTTOM\" border=\"0\" \/>\u00a0times the shorter leg.<\/p>\n<p>In a 45-45-90 triangle, the length of the hypotenuse is\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/s10_p4_clip_image004.gif\" width=\"19\" height=\"18\" name=\"graphics4\" align=\"BOTTOM\" border=\"0\" \/>\u00a0times the shorter leg.<\/p>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>On a baseball diamond, the distance between consecutive bases\u00a0is 90 feet and the diamond is actually a square. What is the\u00a0distance from second base to home plate?<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/Math%20Mod%204.3%20Art%20024.JPG\" alt=\" Baseball diamond\" width=\"124\" height=\"124\" name=\"graphics5\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<ol>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/s10_p4_clip_image006.gif\" width=\"40\" height=\"18\" name=\"graphics6\" align=\"BOTTOM\" border=\"0\" \/>\u00a0feet<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/s10_p4_clip_image008.gif\" width=\"39\" height=\"18\" name=\"graphics7\" align=\"BOTTOM\" border=\"0\" \/>\u00a0feet<\/li>\n<li>90 feet<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/s10_p4_clip_image010.gif\" width=\"39\" height=\"18\" name=\"graphics8\" align=\"BOTTOM\" border=\"0\" \/>\u00a0feet<\/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. This distance is in fact a diagonal,\u00a0and since a diagonal divides a square into two 45-45-90 triangles\u00a0. A leg has length 90, so the hypotenuse has length\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/s10_p4_clip_image011.gif\" width=\"39\" height=\"18\" name=\"graphics9\" align=\"BOTTOM\" border=\"0\" \/>.<\/p>\n<\/div>\n<\/section>\n<\/section>\n<p><!-- CONTENT ENDS HERE --><\/p>\n<div class=\"advance\"><a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/skew-lines\">\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\/area-of-triangles-and-special-quadrilaterals\">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 Special Triangles Objective In this lesson, we will study special triangles and their properties . We will also use special triangles to derive\u00a0the formulas for the areas of special quadrilaterals. Previously Covered: Two triangles are congruent\u00a0if there is a correspondence between them such that every pair of\u00a0corresponding sides is [&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-140","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/140","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=140"}],"version-history":[{"count":15,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/140\/revisions"}],"predecessor-version":[{"id":758,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/140\/revisions\/758"}],"wp:attachment":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/media?parent=140"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}