{"id":"cb_question_bank:2eb24e21-904d-4690-bc5a-55d9427756bd","source_id":"cb_question_bank","source_item_id":"2eb24e21-904d-4690-bc5a-55d9427756bd","external_id":"761b928d-225b-4b73-968f-a9b367e5083e","ibn":null,"question_id":"f7dbde16","program":"SAT","module":"math","domain_code":"S","domain":"Geometry and Trigonometry","skill_code":"S.B.","skill":"Lines, angles, and triangles","difficulty":"H","score_band":7,"answer_type":"mcq","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"In triangles upper L upper M upper N and upper R upper S upper T , angles upper L and upper R each have measure 60 degree , upper L upper N equals 10 , and upper R upper T equals 30 . Which additional piece of information is sufficient to prove that triangle upper L upper M upper N is similar to triangle upper R upper S upper T ?","stimulus":"","stimulus_html":null,"stem_html":"<p style=\"text-align: left;\">In triangles <math alttext=\"upper L upper M upper N\"><mrow>\n\t<mi>L</mi>\n\t<mi>M</mi>\n\t<mi>N</mi>\n</mrow>\n</math> and <math alttext=\"upper R upper S upper T\"><mrow>\n\t<mi>R</mi>\n\t<mi>S</mi>\n\t<mi>T</mi>\n</mrow>\n</math>, angles <math alttext=\"upper L\"><mi>L</mi>\n</math> and <math alttext=\"upper R\"><mi>R</mi>\n</math> each have measure <math alttext=\"60 degree\"><mn>60</mn>\n<mo>°</mo></math>, <math alttext=\"upper L upper N equals 10\"><mrow>\n\t<mrow>\n\t\t<mi>L</mi>\n\t\t<mi>N</mi>\n\t</mrow>\n\t<mo>=</mo>\n\t<mn>10</mn>\n</mrow>\n</math>, and <math alttext=\"upper R upper T equals 30\"><mrow>\n\t<mrow>\n\t\t<mi>R</mi>\n\t\t<mi>T</mi>\n\t</mrow>\n\t<mo>=</mo>\n\t<mn>30</mn>\n</mrow>\n</math>. Which additional piece of information is sufficient to prove that triangle <math alttext=\"upper L upper M upper N\"><mrow>\n\t<mi>L</mi>\n\t<mi>M</mi>\n\t<mi>N</mi>\n</mrow>\n</math> is similar to triangle <math alttext=\"upper R upper S upper T\"><mrow>\n\t<mi>R</mi>\n\t<mi>S</mi>\n\t<mi>T</mi>\n</mrow>\n</math>?</p>","rationale_html":"<p>Choice D is correct. Two triangles are similar if they have three pairs of congruent corresponding angles. It’s given that angles <math alttext=\"upper L\"><mi>L</mi>\n</math> and <math alttext=\"upper R\"><mi>R</mi>\n</math> each measure <math alttext=\"60 degree\"><mn>60</mn><mo>°</mo></math>, and so these corresponding angles are congruent. If angle <math alttext=\"upper M\"><mi>M</mi>\n</math> is <math alttext=\"70 degree\"><mn>70</mn><mo>°</mo></math>, then angle <math alttext=\"upper N\"><mi>N</mi>\n</math> must be&nbsp;<math alttext=\"50 degree\"><mn>50</mn><mo>°</mo></math> so that the sum of the angles in triangle <math alttext=\"upper L upper M upper N\"><mrow>\n\t<mi>L</mi>\n\t<mi>M</mi>\n\t<mi>N</mi>\n</mrow>\n</math> is <math alttext=\"180 degree\"><mn>180</mn><mo>°</mo></math>. If angle <math alttext=\"upper T\"><mi>T</mi>\n</math> is <math alttext=\"50 degree\"><mn>50</mn><mo>°</mo></math>, then angle <math alttext=\"upper S\"><mi>S</mi>\n</math> must be&nbsp;<math alttext=\"70 degree\"><mn>70</mn><mo>°</mo></math> so that the sum of the angles in triangle <math alttext=\"upper R upper S upper T\"><mrow>\n\t<mi>R</mi>\n\t<mi>S</mi>\n\t<mi>T</mi>\n</mrow>\n</math> is <math alttext=\"180 degree\"><mn>180</mn><mo>°</mo></math>. Therefore, if the measures of angles <math alttext=\"upper M\"><mi>M</mi>\n</math> and <math alttext=\"upper T\"><mi>T</mi>\n</math> are&nbsp;<math alttext=\"70 degree\"><mn>70</mn><mo>°</mo></math> and <math alttext=\"50 degree\"><mn>50</mn><mo>°</mo></math>, respectively, then corresponding angles <math alttext=\"upper M\"><mi>M</mi>\n</math> and <math alttext=\"upper S\"><mi>S</mi>\n</math> are both <math alttext=\"70 degree\"><mn>70</mn><mo>°</mo></math>, and corresponding angles <math alttext=\"upper N\"><mi>N</mi>\n</math> and <math alttext=\"upper T\"><mi>T</mi>\n</math> are both <math alttext=\"50 degree\"><mn>50</mn><mo>°</mo></math>. It follows that triangles <math alttext=\"upper L upper M upper N\"><mrow>\n\t<mi>L</mi>\n\t<mi>M</mi>\n\t<mi>N</mi>\n</mrow>\n</math> and <math alttext=\"upper R upper S upper T\"><mrow>\n\t<mi>R</mi>\n\t<mi>S</mi>\n\t<mi>T</mi>\n</mrow>\n</math> have three pairs of congruent corresponding angles, and so the triangles are similar. Therefore, the additional piece of information that is sufficient to prove that triangle <math alttext=\"upper L upper M upper N\"><mrow>\n\t<mi>L</mi>\n\t<mi>M</mi>\n\t<mi>N</mi>\n</mrow>\n</math> is similar to triangle <math alttext=\"upper R upper S upper T\"><mrow>\n\t<mi>R</mi>\n\t<mi>S</mi>\n\t<mi>T</mi>\n</mrow>\n</math> is that the measures of angles <math alttext=\"upper M\"><mi>M</mi>\n</math> and <math alttext=\"upper T\"><mi>T</mi>\n</math> are <math alttext=\"70 degree\"><mn>70</mn><mo>°</mo></math> and <math alttext=\"50 degree\"><mn>50</mn><mo>°</mo></math>, respectively.</p>\n<p>Choice A is incorrect. If the measures of two sides in one triangle are proportional to the corresponding sides in another triangle and the included angles are congruent, then the triangles are similar. However, the two sides given are not proportional and the angle given is not included by the given sides.</p>\n<p>Choice B is incorrect. If the measures of two sides in one triangle are proportional to the corresponding sides in another triangle and the included angles are congruent, then the triangles are similar. However, the angle given is not included between the proportional sides.</p>\n<p>Choice C is incorrect and may result from conceptual or calculation errors.</p>","correct_answer":["D"],"has_media":false,"has_mathml":true,"has_table":false,"source_created_at":"2023-08-02T20:25:59.838000+00:00","source_updated_at":"2023-08-02T20:25:59.838000+00:00","options":[{"label":"A","content_html":"<p style=\"text-align: left;\"><math alttext=\"upper M upper N equals 7\"><mrow>\n\t<mrow>\n\t\t<mi>M</mi>\n\t\t<mi>N</mi>\n\t</mrow>\n\t<mo>=</mo>\n\t<mn>7</mn>\n</mrow>\n</math> and <math alttext=\"upper S upper T equals 7\"><mrow>\n\t<mrow>\n\t\t<mi>S</mi>\n\t\t<mi>T</mi>\n\t</mrow>\n\t<mo>=</mo>\n\t<mn>7</mn>\n</mrow>\n</math></p>","ord":0},{"label":"B","content_html":"<p style=\"text-align: left;\"><math alttext=\"upper M upper N equals 7\"><mrow>\n\t<mrow>\n\t\t<mi>M</mi>\n\t\t<mi>N</mi>\n\t</mrow>\n\t<mo>=</mo>\n\t<mn>7</mn>\n</mrow>\n</math> and <math alttext=\"upper S upper T equals 21\"><mrow>\n\t<mrow>\n\t\t<mi>S</mi>\n\t\t<mi>T</mi>\n\t</mrow>\n\t<mo>=</mo>\n\t<mn>21</mn>\n</mrow>\n</math></p>","ord":1},{"label":"C","content_html":"<p style=\"text-align: left;\">The measures of angles <math alttext=\"upper M\"><mi>M</mi>\n</math> and <math alttext=\"upper S\"><mi>S</mi>\n</math> are <math alttext=\"70 degree\"><mn>70</mn>\n<mo>°</mo></math> and <math alttext=\"60 degree\"><mn>60</mn>\n<mo>°</mo></math>, respectively.</p>","ord":2},{"label":"D","content_html":"<p style=\"text-align: left;\">The measures of angles <math alttext=\"upper M\"><mi>M</mi>\n</math> and <math alttext=\"upper T\"><mi>T</mi>\n</math> are <math alttext=\"70 degree\"><mn>70</mn>\n<mo>°</mo></math>&nbsp;and <math alttext=\"50 degree\"><mn>50</mn>\n<mo>°</mo></math>, respectively.</p>","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}}