{"id":"cb_question_bank:705adb84-79f7-45c0-b607-6be2e786044d","source_id":"cb_question_bank","source_item_id":"705adb84-79f7-45c0-b607-6be2e786044d","external_id":null,"ibn":"11114-DC","question_id":"bd7f6e30","program":"SAT","module":"math","domain_code":"S","domain":"Geometry and Trigonometry","skill_code":"S.B.","skill":"Lines, angles, and triangles","difficulty":"H","score_band":6,"answer_type":"mcq","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"In the figure above, R T equals, T U . What is the value of x ?","stimulus":"The figure presents triangle R T U, with R U horizontal, and U to the right of R. A point V is on R U, such that V is closer to R than it is to U. Vertex T is above R U. The angle at T is labeled 114 degrees. Side U T is extended to point S, above vertex R, and line segment S V is drawn. The angle to the right of S V and below S T is labeled 31 degrees. The angle to the left of S V and above R V is labeled x degrees.","stimulus_html":"<div class=\"stimulus_reference \">\n\t<div class=\"standalone_image \"><img src=\"/api/assets/40a20d2e1be2a85cad9c5dab88b47f40730547c05e2ef92a1e3a624cbc05b7c1\" alt=\"The figure presents triangle R T U, with R U horizontal, and U to the right of R. A point V is on R U, such that V is closer to R than it is to U. Vertex T is above R U. The angle at T is labeled 114 degrees. Side U T is extended to point S, above vertex R, and line segment S V is drawn. The angle to the right of S V and below S T is labeled 31 degrees. The angle to the left of S V and above R V is labeled x degrees.\" width=\"154\" height=\"115\"></div>\n</div>\n","stem_html":"<p class=\"stem_paragraph \">In the figure above, <span class=\"math_expression \"><span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/e7932b30c9c6bdc9a147c9d4b1752d4db74641c8c403bb626af72a33d800c7b6\" alt=\"R T equals, T U\"></span></span>. What is the&nbsp;value&nbsp;of&nbsp;<span class=\"italic\">x</span> ?</p>\n","rationale_html":"<p>Choice C is correct. Since <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/e7932b30c9c6bdc9a147c9d4b1752d4db74641c8c403bb626af72a33d800c7b6\" alt=\"the length of side R T equals the length of side T U\"></span>, it follows that <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/49be5bf54dcec06917f4d28b38d112d925e9426d8ff234b3b4072c0c60145932\" alt=\"triangle R T U\"></span> is an isosceles triangle with base <span class=\"italic\">RU</span>. Therefore, <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/1adeb262537fdbaa9ec1f5406891b46c0279c7f289cadbc1de07ac19793225d6\" alt=\"angle T R U\"></span> and <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/59c52bdddba813616e14a7a33c00c841cd2170173ecdbf135eff55f2f5e8e5e8\" alt=\"angle T U R\"></span> are the base angles of an isosceles triangle and are congruent. Let the measures of both <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/1adeb262537fdbaa9ec1f5406891b46c0279c7f289cadbc1de07ac19793225d6\" alt=\"angle T R U\"></span> and <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/59c52bdddba813616e14a7a33c00c841cd2170173ecdbf135eff55f2f5e8e5e8\" alt=\"angle T U R\"></span> be <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/37a12ddf5c127f4824310a436fe0671fa8fd562d18858841e15253d6536d427e\" alt=\"t degrees\"></span>. According to the triangle sum theorem, the sum of the measures of the three angles of a triangle is <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/ec0041af7817f5d2c927cbdf134a73bc148941d5c3e5ddb80665e5fac696f97e\" alt=\"180 degrees\"></span>. Therefore, <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/29ef8ee6d034f9762b5d608c5e3fd4075d07ee9757ce5d38f19ae3d8e73ad001\" alt=\"114 degrees plus 2 t degrees, equals 180 degrees\"></span>, so <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/d1273e2e7be98712cb96751bc590ecab92388e95787b502cc2921744892a1e04\" alt=\"t equals 33\"></span>.</p><p>Note that <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/59c52bdddba813616e14a7a33c00c841cd2170173ecdbf135eff55f2f5e8e5e8\" alt=\"angle T U R\"></span> is the same angle as <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/aab088ba5a75df06f387f3da730a06b3ce145d9d295bcae2da2430114d5520bf\" alt=\"angle S U V\"></span>. Thus, the measure of <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/aab088ba5a75df06f387f3da730a06b3ce145d9d295bcae2da2430114d5520bf\" alt=\"angle S U V\"></span> is <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/1c1f9d0d18a47a8d541ee9d853b1f1591f6f090836a8d68db00d41f43c70e6b6\" alt=\"33 degrees\"></span>. According to the triangle exterior angle theorem, an external angle of a triangle is equal to the sum of the opposite interior angles. Therefore, <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/5877f887e14ac31b9a2697d829281b52cfb818895e7bd181e0390c0984a409d0\" alt=\"x degrees\"></span> is equal to the sum of the measures of <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/1b049b0c4f9ec5bfd3e0cd16c6f9b65f6f0e95da88105b4063eb42577ab6a6e9\" alt=\"angle V S U\"></span> and <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/aab088ba5a75df06f387f3da730a06b3ce145d9d295bcae2da2430114d5520bf\" alt=\"angle S U V\"></span>; that is, <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/1f549c33be375eda70a9b9e5602504c5d5470106a083ae10c9e447cd0cf33359\" alt=\"31 degrees plus 33 degrees, equals 64 degrees\"></span>. Thus, the value of <span class=\"italic\">x</span> is 64.</p><p>Choice B is incorrect. This is the measure of <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/45391a3482ad1e6ffad48f6c491566413b79249b4acac4f9d4fb8fc654122a06\" alt=\"angle S T R\"></span>, but <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/45391a3482ad1e6ffad48f6c491566413b79249b4acac4f9d4fb8fc654122a06\" alt=\"angle S T R\"></span> is not congruent to <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/499296f36442210bcf9fad14cee2f6a567a989e47b13b7f81b14c6b2f1f57a36\" alt=\"angle S V R\"></span>. Choices A and D are incorrect and may result from a calculation error.</p><p>&nbsp;</p><p></p>\n","correct_answer":["C"],"has_media":true,"has_mathml":false,"has_table":false,"source_created_at":"2023-08-02T20:25:59.638000+00:00","source_updated_at":"2023-08-02T20:25:59.638000+00:00","options":[{"label":"A","content_html":"<p>72</p>\n","ord":0},{"label":"B","content_html":"<p>66</p>\n","ord":1},{"label":"C","content_html":"<p>64</p>\n","ord":2},{"label":"D","content_html":"<p>58</p>\n","ord":3}],"attribution":{"source_id":"cb_question_bank","source_name":"College Board digital SAT question bank (community dump)","rights_holder":"College Board","canonical_url":"https://satsuitequestionbank.collegeboard.org/","retrieved_from":"https://raw.githubusercontent.com/mdn522/sat-question-bank/main/data/cb-digital-questions.json","attribution":"SAT practice questions © College Board, from the official SAT Suite Question Bank. Retrieved via the community dump at github.com/mdn522/sat-question-bank.","disclaimer":"SAT® and PSAT/NMSQT® are trademarks registered by the College Board, which is not affiliated with, does not sponsor, and does not endorse this project.","license":"College Board copyright; source repository carries no licence. Local use only.","redistributable":false,"item_count":2017}}