{"id":10753,"date":"2017-06-05T15:46:50","date_gmt":"2017-06-05T15:46:50","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/prealgebra\/?post_type=chapter&#038;p=10753"},"modified":"2024-04-30T16:46:46","modified_gmt":"2024-04-30T16:46:46","slug":"summary-using-the-properties-of-angles-triangles-and-the-pythagorean-theorem","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/summary-using-the-properties-of-angles-triangles-and-the-pythagorean-theorem\/","title":{"raw":"Summary: Using the Properties of Angles, Triangles, and the Pythagorean Theorem","rendered":"Summary: Using the Properties of Angles, Triangles, and the Pythagorean Theorem"},"content":{"raw":"<h2>Key Concepts<\/h2>\r\n<ul id=\"eip-655\">\r\n \t<li><strong>Supplementary and Complementary Angles<\/strong>\r\n<ul id=\"eip-id1170324960756\">\r\n \t<li>If the sum of the measures of two angles is 180\u00b0, then the angles are supplementary.<\/li>\r\n \t<li>If angle [latex]A[\/latex] and angle [latex]B[\/latex] are supplementary, then [latex]m\\angle{A}+m\\angle{B}=180\u00b0[\/latex] .<\/li>\r\n \t<li>If the sum of the measures of two angles is [latex]90^\\circ[\/latex], then the angles are complementary.<\/li>\r\n \t<li>If angle [latex]A[\/latex] and angle [latex]B[\/latex] are complementary, then [latex]m\\angle{A}+m\\angle{B}=90\u00b0[\/latex] .<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li><strong>Solve Geometry Applications<\/strong>\r\n<ol id=\"eip-id1170326094648\" class=\"stepwise\">\r\n \t<li>Read the problem and make sure you understand all the words and ideas. Draw a figure and label it with the given information.<\/li>\r\n \t<li>Identify what you are looking for.<\/li>\r\n \t<li>Name what you are looking for and choose a variable to represent it.<\/li>\r\n \t<li>Translate into an equation by writing the appropriate formula or model for the situation. Substitute in the given information.<\/li>\r\n \t<li>Solve the equation using good algebra techniques.<\/li>\r\n \t<li>Check the answer in the problem and make sure it makes sense.<\/li>\r\n \t<li>Answer the question with a complete sentence.<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li><strong>Sum of the Measures of the Angles of a Triangle<\/strong><\/li>\r\n<\/ul>\r\n<p style=\"padding-left: 90px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223759\/CNX_BMath_Figure_09_03_012.png\" alt=\"Triangle with vertices A, B, and C\" width=\"272\" height=\"154\" \/><\/p>\r\n\r\n<ul id=\"eip-id1170322666796\">\r\n \t<li>For any [latex]\\Delta ABC[\/latex], the sum of the measures is [latex]180^\\circ[\/latex]<\/li>\r\n \t<li>[latex]m\\angle{A}+m\\angle{B}=180\u00b0[\/latex]<\/li>\r\n<\/ul>\r\n<ul id=\"eip-655\">\r\n \t<li><strong>Right Triangle<\/strong>\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223646\/CNX_BMath_Figure_09_03_015_img.png\" alt=\"Right triangle with 90 degrees label near the boxed corner.\" width=\"109\" height=\"86\" \/>\r\n<ul id=\"eip-id1170323029556\">\r\n \t<li>A right triangle is a triangle that has one [latex]90\u00b0[\/latex] angle, which is often marked with a [latex]\\angle[\/latex] symbol.<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li><strong>Properties of Similar Triangles<\/strong>\r\n<ul id=\"eip-id1170324076236\">\r\n \t<li>If two triangles are similar, then their corresponding angle measures are equal and their corresponding side lengths have the same ratio.<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<h2>Glossary<\/h2>\r\n<dl id=\"fs-id1648881\" class=\"definition\">\r\n \t<dt><strong>angle<\/strong><\/dt>\r\n \t<dd id=\"fs-id1327370\">An angle is formed by two rays that share a common endpoint. Each ray is called a side of the angle.<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1327374\" class=\"definition\">\r\n \t<dt><strong>complementary angles<\/strong><\/dt>\r\n \t<dd id=\"fs-id1544988\">If the sum of the measures of two angles is [latex]90^\\circ [\/latex] , then they are called complementary angles.<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170322667662\" class=\"definition\">\r\n \t<dt><strong>hypotenuse<\/strong><\/dt>\r\n \t<dd id=\"fs-id1170325427844\">The side of the triangle opposite the [latex]90^\\circ[\/latex] angle is called the hypotenuse.<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1444791\" class=\"definition\">\r\n \t<dt><strong>legs of a right triangle<\/strong><\/dt>\r\n \t<dd id=\"fs-id1540654\">The sides of a right triangle adjacent to the right angle are called the legs.<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1172591862696\" class=\"definition\">\r\n \t<dt><strong>right triangle<\/strong><\/dt>\r\n \t<dd id=\"fs-id1172596661577\">A right triangle is a triangle that has one [latex]90^\\circ [\/latex] angle.<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1172591572065\" class=\"definition\">\r\n \t<dt><strong>similar figures<\/strong><\/dt>\r\n \t<dd id=\"fs-id1236955\">In geometry, if two figures have exactly the same shape but different sizes, we say they are similar figures.<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1300144\" class=\"definition\">\r\n \t<dt><strong>supplementary angles<\/strong><\/dt>\r\n \t<dd id=\"fs-id1434532\">If the sum of the measures of two angles is [latex]180^\\circ [\/latex] , then they are called supplementary angles.<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1334464\" class=\"definition\">\r\n \t<dt><strong>triangle<\/strong><\/dt>\r\n \t<dd id=\"fs-id1172599767842\">A triangle is a geometric figure with three sides and three angles.<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1172599767847\" class=\"definition\">\r\n \t<dt><strong>vertex of an angle<\/strong><\/dt>\r\n \t<dd id=\"fs-id1402834\">When two rays meet to form an angle, the common endpoint is called the vertex of the angle.<\/dd>\r\n<\/dl>","rendered":"<h2>Key Concepts<\/h2>\n<ul id=\"eip-655\">\n<li><strong>Supplementary and Complementary Angles<\/strong>\n<ul id=\"eip-id1170324960756\">\n<li>If the sum of the measures of two angles is 180\u00b0, then the angles are supplementary.<\/li>\n<li>If angle [latex]A[\/latex] and angle [latex]B[\/latex] are supplementary, then [latex]m\\angle{A}+m\\angle{B}=180\u00b0[\/latex] .<\/li>\n<li>If the sum of the measures of two angles is [latex]90^\\circ[\/latex], then the angles are complementary.<\/li>\n<li>If angle [latex]A[\/latex] and angle [latex]B[\/latex] are complementary, then [latex]m\\angle{A}+m\\angle{B}=90\u00b0[\/latex] .<\/li>\n<\/ul>\n<\/li>\n<li><strong>Solve Geometry Applications<\/strong>\n<ol id=\"eip-id1170326094648\" class=\"stepwise\">\n<li>Read the problem and make sure you understand all the words and ideas. Draw a figure and label it with the given information.<\/li>\n<li>Identify what you are looking for.<\/li>\n<li>Name what you are looking for and choose a variable to represent it.<\/li>\n<li>Translate into an equation by writing the appropriate formula or model for the situation. Substitute in the given information.<\/li>\n<li>Solve the equation using good algebra techniques.<\/li>\n<li>Check the answer in the problem and make sure it makes sense.<\/li>\n<li>Answer the question with a complete sentence.<\/li>\n<\/ol>\n<\/li>\n<li><strong>Sum of the Measures of the Angles of a Triangle<\/strong><\/li>\n<\/ul>\n<p style=\"padding-left: 90px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223759\/CNX_BMath_Figure_09_03_012.png\" alt=\"Triangle with vertices A, B, and C\" width=\"272\" height=\"154\" \/><\/p>\n<ul id=\"eip-id1170322666796\">\n<li>For any [latex]\\Delta ABC[\/latex], the sum of the measures is [latex]180^\\circ[\/latex]<\/li>\n<li>[latex]m\\angle{A}+m\\angle{B}=180\u00b0[\/latex]<\/li>\n<\/ul>\n<ul id=\"eip-655\">\n<li><strong>Right Triangle<\/strong><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223646\/CNX_BMath_Figure_09_03_015_img.png\" alt=\"Right triangle with 90 degrees label near the boxed corner.\" width=\"109\" height=\"86\" \/><\/p>\n<ul id=\"eip-id1170323029556\">\n<li>A right triangle is a triangle that has one [latex]90\u00b0[\/latex] angle, which is often marked with a [latex]\\angle[\/latex] symbol.<\/li>\n<\/ul>\n<\/li>\n<li><strong>Properties of Similar Triangles<\/strong>\n<ul id=\"eip-id1170324076236\">\n<li>If two triangles are similar, then their corresponding angle measures are equal and their corresponding side lengths have the same ratio.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1648881\" class=\"definition\">\n<dt><strong>angle<\/strong><\/dt>\n<dd id=\"fs-id1327370\">An angle is formed by two rays that share a common endpoint. Each ray is called a side of the angle.<\/dd>\n<\/dl>\n<dl id=\"fs-id1327374\" class=\"definition\">\n<dt><strong>complementary angles<\/strong><\/dt>\n<dd id=\"fs-id1544988\">If the sum of the measures of two angles is [latex]90^\\circ[\/latex] , then they are called complementary angles.<\/dd>\n<\/dl>\n<dl id=\"fs-id1170322667662\" class=\"definition\">\n<dt><strong>hypotenuse<\/strong><\/dt>\n<dd id=\"fs-id1170325427844\">The side of the triangle opposite the [latex]90^\\circ[\/latex] angle is called the hypotenuse.<\/dd>\n<\/dl>\n<dl id=\"fs-id1444791\" class=\"definition\">\n<dt><strong>legs of a right triangle<\/strong><\/dt>\n<dd id=\"fs-id1540654\">The sides of a right triangle adjacent to the right angle are called the legs.<\/dd>\n<\/dl>\n<dl id=\"fs-id1172591862696\" class=\"definition\">\n<dt><strong>right triangle<\/strong><\/dt>\n<dd id=\"fs-id1172596661577\">A right triangle is a triangle that has one [latex]90^\\circ[\/latex] angle.<\/dd>\n<\/dl>\n<dl id=\"fs-id1172591572065\" class=\"definition\">\n<dt><strong>similar figures<\/strong><\/dt>\n<dd id=\"fs-id1236955\">In geometry, if two figures have exactly the same shape but different sizes, we say they are similar figures.<\/dd>\n<\/dl>\n<dl id=\"fs-id1300144\" class=\"definition\">\n<dt><strong>supplementary angles<\/strong><\/dt>\n<dd id=\"fs-id1434532\">If the sum of the measures of two angles is [latex]180^\\circ[\/latex] , then they are called supplementary angles.<\/dd>\n<\/dl>\n<dl id=\"fs-id1334464\" class=\"definition\">\n<dt><strong>triangle<\/strong><\/dt>\n<dd id=\"fs-id1172599767842\">A triangle is a geometric figure with three sides and three angles.<\/dd>\n<\/dl>\n<dl id=\"fs-id1172599767847\" class=\"definition\">\n<dt><strong>vertex of an angle<\/strong><\/dt>\n<dd id=\"fs-id1402834\">When two rays meet to form an angle, the common endpoint is called the vertex of the angle.<\/dd>\n<\/dl>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-10753\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>Prealgebra. <strong>Provided by<\/strong>: OpenStax. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":17533,"menu_order":6,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Prealgebra\",\"author\":\"\",\"organization\":\"OpenStax\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757\"}]","CANDELA_OUTCOMES_GUID":"0abe40ae7c7b4d9bb490a70e0ab41de2","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-10753","chapter","type-chapter","status-publish","hentry"],"part":7831,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/10753","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/wp\/v2\/users\/17533"}],"version-history":[{"count":14,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/10753\/revisions"}],"predecessor-version":[{"id":20408,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/10753\/revisions\/20408"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/parts\/7831"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/10753\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/wp\/v2\/media?parent=10753"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapter-type?post=10753"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/wp\/v2\/contributor?post=10753"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/wp\/v2\/license?post=10753"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}