{"id":134,"date":"2017-08-23T09:04:48","date_gmt":"2017-08-23T09:04:48","guid":{"rendered":"http:\/\/americanboard.org\/Subjects\/mathematics\/?page_id=134"},"modified":"2017-09-18T17:02:39","modified_gmt":"2017-09-18T17:02:39","slug":"triangles","status":"publish","type":"page","link":"https:\/\/americanboard.org\/Subjects\/mathematics\/triangles\/","title":{"rendered":"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\/two-dimensional-representations-of-three-dimensional-objects\">\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\/similarity-ratio\">Next Lesson \u27a1<\/a><\/div>\n<p><!-- CONTENT BEGINS HERE --><\/p>\n<h1 id=\"title\">Triangles<\/h1>\n<h4>Objective<\/h4>\n<p>In this lesson, you will study some of these basic properties of transversals and triangles in this section and give\u00a0sufficient conditions for the congruence of triangles. You will also prove some of these theorems as simple\u00a0corollaries of others.<\/p>\n<h4>Previously Covered:<\/h4>\n<p>\u2022 A <em><strong>polygon<\/strong><\/em> is the union of the <em>n\u00a0<\/em>segments <em>P<sub>1,\u00a0<\/sub>P<sub>2<\/sub><\/em>,\u00a0<em>P<sub>2,\u00a0<\/sub>P<sub>3<\/sub><\/em>, . . . <em>P<sub>3,\u00a0<\/sub>P<sub>1<\/sub><\/em> such that <em>P<sub>1<\/sub>,\u00a0P<sub>2<\/sub> . . . P<sub>n<\/sub><\/em> is a sequence of <em>n\u00a0<\/em>distinct points\u00a0in the plane, with\u00a0<em><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/s4_p1_clip_image002.gif\" width=\"31\" height=\"11\" name=\"graphics2\" align=\"BOTTOM\" border=\"0\" \/><\/em> satisfying the following properties:<\/p>\n<ol>\n<li><em>No two segments\u00a0intersect except at their endpoints.<\/em><\/li>\n<li><em> No two segments\u00a0with a common endpoint are collinear. <\/em><\/li>\n<\/ol>\n<p>\u2022 A <strong>triangle\u00a0<\/strong>is a polygon in which <em>n\u00a0<\/em>= 3.<\/p>\n<section>\n<h3><strong>What is a transversal and how does it help us in the\u00a0case of parallel lines?<\/strong><\/h3>\n<p>A <abbr title=\"a line that intersects two lines in two different points \">transversal<\/abbr> to two lines in the plane is a line that intersects them in two\u00a0different points.<\/p>\n<p align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/MathRedo_TriScreen2_Img1.jpg\" alt=\"transversals\" width=\"200\" height=\"200\" name=\"graphics3\" align=\"BOTTOM\" border=\"0\" \/><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/MathRedo_TriScreen2_Img2.jpg\" alt=\"transversals\" width=\"200\" height=\"200\" name=\"graphics4\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>The lines cut by the transversal are not necessarily parallel,\u00a0as shown in the first picture above. In both cases, the angles\u00a0labeled by 1 and 2 are referred to as <abbr title=\" a line that intersects two lines in two different points \">alternate\u00a0interior angles<\/abbr>. Note that there are two pairs of\u00a0alternate interior angles.<\/p>\n<p>The most important theorems involving transversals are in the\u00a0case of parallel lines:<\/p>\n<ul>\n<li style=\"list-style-type: none;\">\n<ul>\n<li><em>If two lines\u00a0are cut by a transversal and any pair of alternate interior angles\u00a0are congruent, then the lines are parallel.<\/em><\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p><center><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/MathRedo_TriScreen2_Img3.jpg\" alt=\"Alternate interior angles \" width=\"200\" height=\"200\" name=\"graphics5\" align=\"BOTTOM\" border=\"0\" \/><\/center><\/p>\n<ul>\n<li><em>If two\u00a0parallel lines are cut by a transversal, then any pair of\u00a0alternate interior angles is congruent.<\/em><\/li>\n<\/ul>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>Which is the measure of the angle denoted by <em>x\u00a0<\/em>in the\u00a0figure below?<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/MathRedo_TriScreen2_Img4.jpg\" alt=\"Alternate interior angles x, y, y, and 140\" width=\"200\" height=\"200\" name=\"graphics6\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<ol>\n<li>40<\/li>\n<li>100<\/li>\n<li><em>y <\/em>+ 40<\/li>\n<li>140<\/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.\u00a0The existence of the pair of congruent alternate interior angles\u00a0(the pair with measure <em>y-<\/em>span\u00a0&gt;)\u00a0tells us that the lines <em>L\u00a01<\/em> and <em>L\u00a02<\/em> are parallel.\u00a0The fact that the lines are parallel allows us to use the second\u00a0theorem to deduce that the other pair of alternate interior\u00a0angles is congruent, so <em>x\u00a0<\/em>= 140.<\/p>\n<\/div>\n<\/section>\n<p>In fact, the example above is a\u00a0special case of a corollary of the two theorems on transversals.<\/p>\n<ul>\n<li><em>If two lines\u00a0are cut by a transversal, and one pair of alternate interior\u00a0angles is congruent, then the other pair of alternate interior\u00a0angles is also congruent.<\/em><\/li>\n<\/ul>\n<p>The proof of this is\u00a0sketched in the explanation of the correct answer to the example\u00a0question above. If we replace the number 140 with an arbitrary\u00a0angle measure of <em>x\u00a0<\/em>and follow the\u00a0same reasoning, then we have a proof.<\/p>\n<h3><strong>When do parallel lines exist?<\/strong><\/h3>\n<p>The answer to this question is actually a postulate created by\u00a0Euclid, the famous geometer.<\/p>\n<p><em>Existence and Uniqueness of Parallel Lines:\u00a0<\/em>Let L\u00a0be any line and P be a point not on L. Then there is only one line\u00a0containing P parallel to L.<\/p>\n<h4><strong>What is the angle sum of a triangle?<\/strong><\/h4>\n<p>We can label any triangle by labeling its vertices. The\u00a0triangle <em>ABC <\/em>is\u00a0shown below.<\/p>\n<p>The most basic property concerning the angles of the triangle\u00a0is given by the following theorem.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/MathRedo_TriScreen3_Img1.jpg\" width=\"200\" height=\"200\" name=\"graphics3\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p><em>The Triangle Angle Sum Theorem:<\/em>The\u00a0sum of the interior angles of a triangle is 180\u00b0.<\/p>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>What is the measure of angle A in the picture below?<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/MathRedo_TriScreen3_Img2.jpg\" width=\"200\" height=\"115\" name=\"graphics4\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<ol>\n<li>90\u00b0<\/li>\n<li>80\u00b0<\/li>\n<li>100\u00b0<\/li>\n<li>95\u00b0<\/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 B. We find the measure of angle A,\u00a0which we denote by the variable <em>x\u00a0<span style=\"font-style: normal;\">and solving the equation\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/s4_p3_html_m225d46e0.gif\" width=\"113\" height=\"19\" name=\"graphics7\" align=\"absmiddle\" border=\"0\" \/>.<\/span><\/em><\/p>\n<\/div>\n<\/section>\n<h3><strong>How can the Triangle Angle Sum Theorem be proven using\u00a0these theorems about transversals and parallel lines?<\/strong><\/h3>\n<p>As is often the case in geometric proofs, it boils down to\u00a0thinking about the right picture. If we consider the first picture\u00a0below, then we are in an ideal position to use the theorem on\u00a0transversals. But, as in the case with any mathematical proof, we\u00a0need to justify each step and our picture.<\/p>\n<p><strong>Step 1<\/strong> For any triangle <em>span\u00a0&gt;ABC<\/em>,\u00a0we can label the measure of the three interior angles by the\u00a0variables <em>x<\/em>,\u00a0<em>y<\/em>,\u00a0and <em>z<\/em>.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/Math%20Mod%204.1%20Art%20048.JPG\" alt=\"triangle ABC with angles x, y, z \" width=\"160\" height=\"92\" name=\"graphics3\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p><strong>Step 2 <\/strong>By the theorem on the existence of\u00a0parallel lines in the previous subsection, we know that there\u00a0exists a line parallel to\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/s4_p4_html_299c8a9a.gif\" width=\"27\" height=\"21\" name=\"graphics4\" align=\"BOTTOM\" border=\"0\" \/>\u00a0containing <em>C<\/em>.\u00a0This justifies the picture drawn below.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/Math%20Mod%204.1%20Art%20049.JPG\" alt=\"Line parallel to AB through C\" width=\"176\" height=\"92\" name=\"graphics5\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p><strong>Step 3 <\/strong>Apply the transversal theorem involving\u00a0alternate interior angles to the angles <em>y<\/em>span\u00a0&gt;\u00a0and <em>z<\/em>.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/Math%20Mod%204.1%20Art%20050.JPG\" alt=\"Transversal theorem for angles y and z \" width=\"176\" height=\"92\" name=\"graphics6\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p><strong>Step 4<\/strong> Thus, the angles <em>x<\/em>,\u00a0<em>y<\/em>,\u00a0and <em>z\u00a0<\/em>are supplementary,\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/s4_p4_clip_image004.gif\" width=\"98\" height=\"14\" name=\"graphics7\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/p>\n<p>Just by drawing the right picture and using one other theorem,\u00a0we were able to show that the angle sum of the triangle is 180\u00b0.<\/p>\n<p>Now use the Triangle Angle Sum Formula to prove the following\u00a0theorem:<\/p>\n<p style=\"text-align: center;\"><em>If two angles\u00a0of one triangle are congruent to two angles of another triangle,\u00a0then the remaining angles are congruent. <\/em><\/p>\n<p>Suppose we denote the\u00a0measure of the first pair of angles by <em>x\u00a0<\/em>and the second by <em>y<\/em>.\u00a0Then\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/s4_p4_clip_image006.gif\" width=\"93\" height=\"14\" name=\"graphics8\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0and\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/s4_p4_clip_image008.gif\" width=\"91\" height=\"14\" name=\"graphics9\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0by the Triangle Angle Sum Theorem. So <em>w\u00a0<\/em>= <em>span\u00a0&gt; z<\/em>, which shows that the remaining angles are congruent.<\/p>\n<h3><strong>When are two triangles congruent?<\/strong><\/h3>\n<p>We compare triangles by comparing their sides and angles.\u00a0Angles are <abbr title=\"equal in shape and size\">congruent<\/abbr> if they have the same angular measure. Sides are congruent if they\u00a0have the same length. We can extend the idea of congruence to\u00a0triangles as follows.<\/p>\n<p>A <abbr title=\"a particular similarity or congruence(i.e. a match up in the angles or sides of triangles)\">correspondence<\/abbr> between two triangles\u00a0is a match-up between its vertices. For example, in the two\u00a0triangles shown below, if the vertex <em>A\u00a0<\/em>corresponds to <em>D\u00a0<\/em>and the vertex<em> B\u00a0<\/em>corresponds to <em>E\u00a0<\/em>and the vertex <em>C\u00a0<\/em>corresponds to <em>F<\/em>,\u00a0then the triangle <em>ABC\u00a0<\/em>corresponds to the triangle <em>DEF<\/em>.\u00a0We write <em>ABC \u00a0<\/em><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/double%20arrow.gif\" width=\"21\" height=\"14\" name=\"graphics3\" align=\"ABSMIDDLE\" border=\"0\" \/><em>DEF<\/em>.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/Math%20Mod%204.1%20Art%20060.jpg\" width=\"273\" height=\"107\" name=\"graphics4\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>The ordering is very important. Any correspondence between\u00a0triangles automatically gives us a correspondence between its\u00a0sides and its angles. This will allow us to define congruent\u00a0triangles.<\/p>\n<p>A correspondence <em>ABC\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/double%20arrow.gif\" width=\"21\" height=\"14\" name=\"graphics5\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0DEF<\/em> between two triangles is called <em><strong>congruence\u00a0<\/strong><\/em>if every pair of corresponding sides is congruent and every pair\u00a0of angles is congruent. If such congruence exists, we denote this\u00a0relationship by <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/triangle.gif\" width=\"13\" height=\"12\" name=\"graphics6\" align=\"ABSMIDDLE\" border=\"0\" \/><em>ABC\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/cong.gif\" width=\"14\" height=\"13\" name=\"graphics7\" align=\"ABSMIDDLE\" border=\"0\" \/><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/triangle.gif\" width=\"13\" height=\"12\" name=\"graphics8\" align=\"ABSMIDDLE\" border=\"0\" \/>DEF<\/em>.\u00a0In fact, congruence between two triangles is actually six\u00a0relations: three angle congruences and three side congruences.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/s4_p5_clip_image006.gif\" width=\"201\" height=\"69\" name=\"graphics10\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>When comparing triangles, indicate congruence of sides and\u00a0angles by marking the figures as shown below.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/Math%20Mod%204.1%20Art%20051.JPG\" alt=\"Corresponding parts of triangles\" width=\"253\" height=\"95\" name=\"graphics11\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>In the example above, we showed that if two pairs of\u00a0corresponding angles are congruent then the third pair of\u00a0corresponding angles is congruent. If we relax some of these six\u00a0conditions, however, is the congruence of the triangles still\u00a0implied? There are three well-known situations in which this is\u00a0the case. These are known as the SSS postulate, the SAS postulate,\u00a0and the ASA postulate.<\/p>\n<div class=\"callout\">\n<h4>Important Tidbit<\/h4>\n<p>A side of a triangle is said to be <em><strong>included<\/strong><\/em> by the angles\u00a0the vertices of which are the endpoints of the side . In the picture below the side <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/s4_p5_html_m6c718ed6.gif\" width=\"28\" height=\"23\" name=\"graphics15\" align=\"BOTTOM\" border=\"0\" \/> is\u00a0included by angle <em>F <\/em>and angle <em>G<\/em>.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/Math%20Mod%204.1%20Art%20052.JPG\" alt=\"Triangle FGH\" width=\"135\" height=\"104\" name=\"graphics12\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>An angle of a triangle is said to be included by the sides of the triangle which\u00a0make up that angle . In the picture above, angle <em>H<\/em> is included by the sides <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/s4_p5_html_m6a56d609.gif\" width=\"29\" height=\"21\" name=\"graphics16\" align=\"BOTTOM\" border=\"0\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/s4_p5_html_m43c43aa.gif\" width=\"29\" height=\"23\" name=\"graphics17\" align=\"BOTTOM\" border=\"0\" \/>.<\/p>\n<\/div>\n<h4><strong>Side-Side-Side (SSS) postulate<\/strong><\/h4>\n<blockquote><p>If three sides\u00a0of one triangle are congruent to three sides of another triangle,\u00a0then the two triangles are congruent.<\/p><\/blockquote>\n<p align=\"CENTER\"><strong><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/Math%20Mod%204.1%20Art%20053.JPG\" alt=\"SSS postulate\" width=\"253\" height=\"95\" name=\"graphics3\" align=\"BOTTOM\" border=\"0\" \/><\/strong><\/p>\n<h4>Side-Angle-Side (SAS) postulate<\/h4>\n<p style=\"margin-left: 0.52in; margin-right: 0.52in; border: none; padding: 0in;\">If two sides and the\u00a0included angle of one triangle are congruent to two sides and the\u00a0included angle of another triangle, then the two triangles are\u00a0congruent.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/Math%20Mod%204.1%20Art%20054.JPG\" alt=\"SAS postulate\" width=\"253\" height=\"95\" name=\"graphics4\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<h4><strong>Angle-Side-Angle (ASA) postulate<\/strong><\/h4>\n<blockquote><p>If two angles\u00a0and the included side of one triangle are congruent to two angles\u00a0and the included side of another triangle, then the two triangles\u00a0are congruent.<\/p><\/blockquote>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/Math%20Mod%204.1%20Art%20055.JPG\" alt=\"ASA postulate \" width=\"253\" height=\"95\" name=\"graphics5\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>In the figures below, which pair of triangles is NOT\u00a0necessarily congruent?<\/p>\n<ol>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/MAth%20Mod%204.1%20Art%20056.JPG\" alt=\"ASA example\" width=\"204\" height=\"125\" name=\"graphics6\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/Math%20Mod%204.1%20Art%20057.JPG\" alt=\"SSS example\" width=\"204\" height=\"125\" name=\"graphics7\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li><a name=\"one\"><\/a><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/MAth%20Mod%204.1%20Art%20058.JPG\" alt=\" SSA example\" width=\"204\" height=\"125\" name=\"graphics8\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/Math%20Mod%204.1%20Art%20059.JPG\" alt=\"SAS example\" width=\"204\" height=\"125\" name=\"graphics9\" 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 C . There is no SSA Postulate as shown\u00a0by the pair of triangles below.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/MAth%20Mod%204.1%20Art%20058.JPG\" alt=\" SSA example\" width=\"204\" height=\"125\" name=\"graphics10\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<\/div>\n<\/section>\n<p>Although it is not as well known as its siblings, there is one\u00a0more postulate we can use to show that two triangles are\u00a0congruent.<\/p>\n<h4><strong>Angle -Angle-Side (AAS) postulate<\/strong><\/h4>\n<blockquote><p>If two angles\u00a0and a non-included side of one triangle are congruent to two\u00a0angles and the non-included side of another triangle, then the two triangles are congruent.<\/p><\/blockquote>\n<h3>Review of New Vocabulary and Concepts<\/h3>\n<ul>\n<li>A figure has <em><strong>symmetry\u00a0<\/strong><\/em>if there is an isometry of the plane mapping the figure to\u00a0itself.<\/li>\n<li>There are three main types of\u00a0symmetry: <strong><em>reflectional<\/em><\/strong><em>, <strong>point<\/strong><\/em>,\u00a0and <strong><em>rotational symmetry<\/em><\/strong><em>.<\/em><\/li>\n<li>A polygon is <strong><em>convex\u00a0<\/em><\/strong>if no two points of it lie on opposite sides of a\u00a0line containing a side of the polygon or that a line sontaining a\u00a0side of the polygon does not cross the interior of the polygon.<\/li>\n<li>A polyhedron is <strong><em>convex\u00a0<\/em><\/strong>if no two points of it lie on opposite sides of a\u00a0line containing a side of the polygon or if a segment connecting\u00a0any two points is on the polyhedron or the interior of the\u00a0polyhedron.<\/li>\n<li>Any polyhedron satisfies\u00a0<strong><em>Euler\u2019s Formula<\/em><\/strong>:\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/s4_p7_clip_image002.gif\" width=\"86\" height=\"11\" name=\"graphics3\" align=\"BOTTOM\" border=\"0\" \/>,\u00a0where <em>F\u00a0<\/em>= number of\u00a0faces, <em>E <\/em>=\u00a0number of edges, and <em>V\u00a0<\/em>= number of\u00a0vertices .<\/li>\n<li><strong><em>Perspective drawing\u00a0<\/em><\/strong>uses the notion of parallel lines and vanishing point to give the\u00a0viewer a sense of three dimensions.<\/li>\n<li>A <strong><em>vanishing point\u00a0<\/em><\/strong>is a point on the horizon line where parallel lines\u00a0appear to meet.<\/li>\n<li>In <strong><em>one-point\u00a0perspective<\/em><\/strong> drawing, parallel lines appear to meet\u00a0at one point in the horizon.<\/li>\n<li>In <strong><em>two-point\u00a0perspective<\/em><\/strong> drawing, parallel lines appear to meet\u00a0at two distinct points in the horizon.<\/li>\n<li>If a<strong><em> transversal\u00a0<\/em><\/strong>intersects two parallel lines, then any pair of\u00a0alternate interior angles is congruent.<\/li>\n<li>If a transversal intersects two\u00a0lines and any pair of alternate interior angles is congruent,\u00a0then the other pair or alternate interior angles is congruent.<\/li>\n<li><strong><em>The Triangle Angle\u00a0Sum Theorem<\/em> :<\/strong> <em>The\u00a0sum of the angles of a triangle is 180 \u00b0. <\/em><\/li>\n<li>A <em><strong>correspondence\u00a0between two triangles<\/strong><\/em> is a correspondence between\u00a0its vertices.<\/li>\n<li>A correspondence <em>ABC\u00a0<\/em><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/double%20arrow.gif\" width=\"21\" height=\"14\" name=\"graphics4\" align=\"ABSMIDDLE\" border=\"0\" \/><em>DEF\u00a0<\/em>between two triangles is called a <strong>congruence<\/strong> if\u00a0every pair of corresponding sides is congruent and every pair of\u00a0angles is congruent.<\/li>\n<li>The <strong><em>SSS<\/em><\/strong><em>, <strong>SAS<\/strong>,\u00a0<strong>ASA<\/strong><\/em>, and <strong><em>AAS\u00a0<\/em><\/strong>postulates give sufficient conditions for two triangles to be\u00a0congruent.<\/li>\n<\/ul>\n<\/section>\n<p><!-- CONTENT ENDS HERE --><\/p>\n<div class=\"advance\"><a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/two-dimensional-representations-of-three-dimensional-objects\">\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\/similarity-ratio\">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 Triangles Objective In this lesson, you will study some of these basic properties of transversals and triangles in this section and give\u00a0sufficient conditions for the congruence of triangles. You will also prove some of these theorems as simple\u00a0corollaries of others. Previously Covered: \u2022 A polygon is the union of [&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-134","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/134","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=134"}],"version-history":[{"count":16,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/134\/revisions"}],"predecessor-version":[{"id":753,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/134\/revisions\/753"}],"wp:attachment":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/media?parent=134"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}