{"id":261,"date":"2017-08-23T04:34:05","date_gmt":"2017-08-23T04:34:05","guid":{"rendered":"http:\/\/americanboard.org\/Subjects\/elementary-education\/?page_id=261"},"modified":"2017-09-25T21:02:24","modified_gmt":"2017-09-25T21:02:24","slug":"proofs","status":"publish","type":"page","link":"https:\/\/americanboard.org\/Subjects\/elementary-education\/proofs\/","title":{"rendered":"Proofs"},"content":{"rendered":"<div class=\"twelve columns\" style=\"margin-top: 10%;\">\n<div class=\"advance\"><a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/three-dimensional-figures-or-solids\">\u2b05 Previous Lesson<\/a>\u00a0<a class=\"button\" href=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/geometry-measurement\">Workshop Index<\/a>\u00a0<a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/representational-systems\">Next Lesson \u27a1<\/a><\/div>\n<p><!-- CONTENT BEGINS HERE --><\/p>\n<h1 id=\"title\">Proofs<\/h1>\n<h4>Objective<\/h4>\n<p>In the coming lesson, we&#8217;ll explore geometric proofs related to triangles and parallel lines.<\/p>\n<h4>Previously Covered<\/h4>\n<ul>\n<li>In the section above, we reviewed basic three-dimensional figures and some of their properties.<\/li>\n<\/ul>\n<section>A mathematical <abbr title=\"A demonstration that a conclusion is true based on certain given elements.\">proof<\/abbr> demonstrates that, based on one or more given facts, a statement must be true. The proof itself is a sequence of statements, each justified by a postulate or a theorem, such as the Isosceles Triangle Theorem which you will see in this lesson.On the pages that follow are sample proofs that are meant to simultaneously familiarize you with proofs and reinforce some of the concepts. Remember, the notation for similar is \u223c and the symbol for congruence is \u2245.<\/p>\n<h3>Theorems and Postulates You\u2019ll Need<\/h3>\n<p><em>Parallel Axiom:<\/em> If two lines, l and m, intersect a transversal so that the sum of the interior angles on the same side of the transversal is equal to 180\u00b0, then l and m are parallel.<\/p>\n<p><center><img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/parallelaxiom.gif\" alt=\"Parallel axiom\" \/><\/center>If two lines, l and m, intersect a transversal so that the sum of the interior angles on the same side of the transversal is less than 180\u00b0, then l and m intersect on that side of the transversal.<\/p>\n<p><center><img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/lessthan180.gif\" alt=\"Less than 180\u00b0\" \/><\/center><em>Side-Angle-Side (SAS) Triangle Congruence Theorem<\/em>: If two sides and the included angle of one triangle are congruent to the corresponding sides and angle of another triangle, then the triangles are congruent.<\/p>\n<p><center><img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/sideanglesidez.gif\" alt=\"Side-Angle-Side\" \/><\/center><img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/6.1.a.gif\" alt=\"Side-Angle-Side equation\" \/><\/p>\n<p><em>Angle-Side-Angle (ASA) Triangle Congruence Theorem<\/em>: If two angles and the included side of one triangle are congruent to the corresponding angles and side of another triangle, then the triangles are congruent.<\/p>\n<p><center><img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/anglesideangle.gif\" alt=\"Angle-Side-Angle\" \/><\/center><img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/6.1.a.gif\" alt=\"Angle-Side-Angle equation\" \/><\/p>\n<p><em>Angle-Angle (AA) Similarity Theorem<\/em>: If two angles in one triangle are congruent to two angles in another triangle, then the triangles are similar. Two figures are <em>similar<\/em> if they have the same shape, but not necessarily the same size.<\/p>\n<p><center><img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/AAz.gif\" alt=\"Angle-Angle\" \/><\/center><img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/6.1.b.gif\" alt=\"Angle-Angle equation\" \/><\/p>\n<p>The AA theorem states that only two angles must be congruent for similarity. Do you see why this is true? Since the sums of the angles in each triangle are equal, if two angles are congruent, then the third angles must also be congruent.<\/p>\n<p><em>Isosceles Triangle Theorem<\/em>: If a triangle is isosceles, then the angles opposite the congruent sides are congruent.<\/p>\n<p><center><img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/ITTz.gif\" alt=\"Isosceles Triangle Theorem\" \/><\/center><\/p>\n<p class=\"figcaption\">Isosceles Triangle Theorem<\/p>\n<p>Let\u2019s prove this one together. We\u2019ll use the isosceles triangle above (\u25b3ABC) and start by bisecting \u2220ABC with segment <span class=\"line\">BD<\/span>.<\/p>\n<table>\n<thead>\n<tr>\n<th>Statement<\/th>\n<th>Justification<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>\u25b3ABC is isosceles<br \/>\n<span class=\"line\">BD<\/span> is an angle bisector<\/td>\n<td>Given<\/td>\n<\/tr>\n<tr>\n<td>m\u2220ABD \u2245 m\u2220CBD<\/td>\n<td>Definition of Angle Bisector<\/td>\n<\/tr>\n<tr>\n<td><span class=\"line\">AB<\/span> \u2245 <span class=\"line\">BC<\/span><\/td>\n<td>Definition of Isosceles<\/td>\n<\/tr>\n<tr>\n<td><span class=\"line\">BD<\/span> \u2245 <span class=\"line\">BD<\/span><\/td>\n<td>Reflexive Property of Congruence<\/td>\n<\/tr>\n<tr>\n<td>\u25b3ABC \u2245 \u25b3CBD<\/td>\n<td>SAS<\/td>\n<\/tr>\n<tr>\n<td>\u2220BAD \u2245 \u2220BCD<\/td>\n<td>Corresponding Parts of Congruent Triangles are Congruent (CPCTC)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Converse of the Isosceles Triangle Theorem: If two angles of a triangle are congruent, then the triangle is isosceles and the sides opposite those angles are congruent.<\/p>\n<p><center><img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/converse of ITTz.gif\" alt=\"Converse of the Isosceles Triangle Theorem\" \/><\/center><\/p>\n<p class=\"figcaption\">Converse of the Isosceles Triangle Theorem<\/p>\n<p>We\u2019ll prove this one together, as well. We\u2019ll use \u25b3ABC above and assume that \u2220BAC \u2245 \u2220BCA.<\/p>\n<table>\n<thead>\n<tr>\n<th>Statement<\/th>\n<th>Justification<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>\u2220BAC \u2245 \u2220BCA<\/td>\n<td>Given<\/td>\n<\/tr>\n<tr>\n<td><span class=\"line\">AC<\/span> \u2245 <span class=\"line\">AC<\/span><\/td>\n<td>Reflexive Property of Congruence<\/td>\n<\/tr>\n<tr>\n<td>\u2220ACB \u2245 \u2220CAB<\/td>\n<td>Given<\/td>\n<\/tr>\n<tr>\n<td>\u25b3ABC \u2245 \u25b3CBA<\/td>\n<td>ASA<\/td>\n<\/tr>\n<tr>\n<td><span class=\"line\">AB<\/span> \u2245 <span class=\"line\">BC<\/span><\/td>\n<td>Corresponding Parts of Congruent Triangles are Congruent (CPCTC)<\/td>\n<\/tr>\n<tr>\n<td>\u25b3ABC is isosceles<\/td>\n<td>Definition of Isosceles<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Basically, we proved that the triangle is congruent to itself. However, this example shows that the order that the vertices of the congruent triangles are named is important. That order tells us about the congruence of corresponding parts of the congruent triangles.<\/p>\n<section class=\"question\">\n<h4>Question<\/h4>\n<p>Which pair of statements can be used to prove that \u0394BAC ~ \u0394BDE?<\/p>\n<p><center><img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/equal-trianglesz.gif\" alt=\"Equal triangles question\" \/><\/center><\/p>\n<ol>\n<li>\u2220A \u2245 \u2220D since both angles are right angles. \u2220C \u2245 \u2220E since vertical angles are congruent.<\/li>\n<li>\u2220A \u2245 \u2220D since both angles are right angles. \u2220ABC \u2245 \u2220DBE since vertical angles are congruent.<\/li>\n<li>\u2220A \u2245 \u2220D since both angles are right angles. <span class=\"line\">AC<\/span> \u2245 <span class=\"line\">ED<\/span> since both are bases of right triangles.<\/li>\n<li>\u2220ABC \u2245 \u2220BDE since vertical angles are congruent. <span class=\"line\">BC<\/span> \u2245 <span class=\"line\">BE<\/span> since each segment is the hypotenuse of a right triangle.<\/li>\n<\/ol>\n<p><a class=\"q-answer button button-primary\">Reveal Answer<\/a><\/p>\n<p class=\"q-reveal\">Choice B is correct. The two triangles can be shown to be similar by AA. This means that if two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. Each statement in choice B is true and shows that the pairs of angles are congruent. Choice A is incorrect, because \u2220C and \u2220E are not vertical angles. Choice C and Choice D contain incorrect statements.<\/p>\n<\/section>\n<h3>Review<\/h3>\n<ul>\n<li>A proof demonstrates that a statement must be true. The proof itself is a sequence of statements, each justified by a postulate or a theorem.<\/li>\n<li>The order that vertices are named tells us about the congruence of corresponding parts of the congruent triangles.<\/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\/elementary-education\/three-dimensional-figures-or-solids\">\u2b05 Previous Lesson<\/a>\u00a0<a class=\"button\" href=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/geometry-measurement\">Workshop Index<\/a>\u00a0<a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/representational-systems\">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 Proofs Objective In the coming lesson, we&#8217;ll explore geometric proofs related to triangles and parallel lines. Previously Covered In the section above, we reviewed basic three-dimensional figures and some of their properties. A mathematical proof demonstrates that, based on one or more given facts, a statement must be true. [&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-261","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/americanboard.org\/Subjects\/elementary-education\/wp-json\/wp\/v2\/pages\/261","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/americanboard.org\/Subjects\/elementary-education\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/americanboard.org\/Subjects\/elementary-education\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/elementary-education\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/elementary-education\/wp-json\/wp\/v2\/comments?post=261"}],"version-history":[{"count":8,"href":"https:\/\/americanboard.org\/Subjects\/elementary-education\/wp-json\/wp\/v2\/pages\/261\/revisions"}],"predecessor-version":[{"id":1431,"href":"https:\/\/americanboard.org\/Subjects\/elementary-education\/wp-json\/wp\/v2\/pages\/261\/revisions\/1431"}],"wp:attachment":[{"href":"https:\/\/americanboard.org\/Subjects\/elementary-education\/wp-json\/wp\/v2\/media?parent=261"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}