{"id":229673,"date":"2025-06-08T11:05:12","date_gmt":"2025-06-08T11:05:12","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=229673"},"modified":"2025-06-08T11:05:15","modified_gmt":"2025-06-08T11:05:15","slug":"triangle-prs-is-isosceles-with-overlinerp-cong-overliners","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/06\/08\/triangle-prs-is-isosceles-with-overlinerp-cong-overliners\/","title":{"rendered":"\\triangle PRS is isosceles with \\overline{RP} \\cong \\overline{RS}"},"content":{"rendered":"\n

\\triangle PRS is isosceles with \\overline{RP} \\cong \\overline{RS}. \\overline{RQ} is drawn such that it bisects \\angle PRS. What additional fact can be used to prove \\triangle PRQ \\cong \\triangle SRQ by SAS in order to state that \\angle P \\cong \\angle S because they are congruent parts of congruent triangles?<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n

Correct Answer:<\/strong>
RQ \u2245 RQ<\/strong><\/p>\n\n\n\n

\n\n\n\n

Explanation <\/strong><\/p>\n\n\n\n

To determine the correct answer, let’s analyze the problem step by step.<\/p>\n\n\n\n

We are given that triangle PRS<\/strong> is isosceles with RP \u2245 RS<\/strong>. This means two sides of the triangle are congruent, indicating that angles opposite those sides will also be congruent if we can prove corresponding triangles are congruent.<\/p>\n\n\n\n

Next, a segment RQ<\/strong> is drawn such that it bisects \u2220PRS<\/strong>, which means \u2220PRQ \u2245 \u2220SRO<\/strong>. This provides one pair of congruent angles between triangles PRQ<\/strong> and SRQ<\/strong>.<\/p>\n\n\n\n

Now, we are tasked with identifying an additional fact<\/strong> needed to prove \u0394PRQ \u2245 \u0394SRQ<\/strong> using the Side-Angle-Side (SAS)<\/strong> congruence postulate. The SAS postulate requires two sides and the included angle of one triangle to be congruent to two sides and the included angle of another triangle.<\/p>\n\n\n\n

So far, we have:<\/p>\n\n\n\n

    \n
  1. RP \u2245 RS<\/strong> (Given)<\/li>\n\n\n\n
  2. \u2220PRQ \u2245 \u2220SRQ<\/strong> (Given, since RQ bisects \u2220PRS)<\/li>\n<\/ol>\n\n\n\n

    For SAS, we need to establish that the side RQ \u2245 RQ<\/strong>, which is the included side<\/strong> between those angles. This is a reflexive property<\/strong>, where a segment is always congruent to itself. This fact completes the required three components for SAS:<\/p>\n\n\n\n

      \n
    • RP \u2245 RS<\/strong> (Given)<\/li>\n\n\n\n
    • \u2220PRQ \u2245 \u2220SRQ<\/strong> (Angle bisector)<\/li>\n\n\n\n
    • RQ \u2245 RQ<\/strong> (Reflexive property)<\/li>\n<\/ul>\n\n\n\n

      With these, we can now use SAS to prove \u0394PRQ \u2245 \u0394SRQ<\/strong>.<\/p>\n\n\n\n

      Since the triangles are congruent, all their corresponding parts are congruent as well. Thus, \u2220P \u2245 \u2220S<\/strong>, because they are corresponding angles<\/strong> of congruent triangles.<\/p>\n\n\n\n

      Therefore, the correct and necessary additional fact is:
      RQ \u2245 RQ<\/strong>.<\/p>\n\n\n\n

      \"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"

      \\triangle PRS is isosceles with \\overline{RP} \\cong \\overline{RS}. \\overline{RQ} is drawn such that it bisects \\angle PRS. What additional fact can be used to prove \\triangle PRQ \\cong \\triangle SRQ by SAS in order to state that \\angle P \\cong \\angle S because they are congruent parts of congruent triangles? The Correct Answer and Explanation […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-229673","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/229673","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=229673"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/229673\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=229673"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=229673"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=229673"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}