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2011 AMC 10B Problems/Problem 7 - Revision history
2024-03-28T19:03:06Z
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Nathan wailes at 17:11, 4 July 2013
2013-07-04T17:11:36Z
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Nathan wailes
https://artofproblemsolving.com/wiki/index.php?title=2011_AMC_10B_Problems/Problem_7&diff=42997&oldid=prev
Gina: /* Solution */
2011-11-13T19:10:28Z
<p><span dir="auto"><span class="autocomment">Solution</span></span></p>
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 19:10, 13 November 2011</td>
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<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>The sum of two angles in a triangle is <math>\frac{6}{5}</math> of a right angle <math>\longrightarrow \frac{6}{5} \times <del class="diffchange diffchange-inline">9 </del>= 108</math></div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>The sum of two angles in a triangle is <math>\frac{6}{5}</math> of a right angle <math>\longrightarrow \frac{6}{5} \times <ins class="diffchange diffchange-inline">90 </ins>= 108</math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>If <math>x</math> is the measure of the first angle, then the measure of the second angle is <math>x+30</math>.</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>If <math>x</math> is the measure of the first angle, then the measure of the second angle is <math>x+30</math>.</div></td></tr>
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Gina
https://artofproblemsolving.com/wiki/index.php?title=2011_AMC_10B_Problems/Problem_7&diff=39173&oldid=prev
Gina at 21:12, 4 June 2011
2011-06-04T21:12:14Z
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 21:12, 4 June 2011</td>
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<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The sum of two angles of a triangle is <math>\frac{6}{5}</math> of a right angle, and one of these two angles is <math>30^{\circ}</math> larger than the other. What is the degree measure of the largest angle in the triangle?</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The sum of two angles of a triangle is <math>\frac{6}{5}</math> of a right angle, and one of these two angles is <math>30^{\circ}</math> larger than the other. What is the degree measure of the largest angle in the triangle?</div></td></tr>
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Gina
https://artofproblemsolving.com/wiki/index.php?title=2011_AMC_10B_Problems/Problem_7&diff=39155&oldid=prev
Gina: see also
2011-06-04T20:44:20Z
<p>see also</p>
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 20:44, 4 June 2011</td>
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<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><cmath>x + x + 30 = 108 \longrightarrow 2x = 78 \longrightarrow x = 39</cmath></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><cmath>x + x + 30 = 108 \longrightarrow 2x = 78 \longrightarrow x = 39</cmath></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Now we know the measure of two angles are <math>39^{\circ}</math> and <math>69^{\circ}</math>. By the Triangle Sum Theorem, the sum of all angles in a triangle is <math>180^{\circ},</math> so the final angle is <math>72^{\circ}</math>. Therefore, the largest angle in the triangle is <math>\boxed{\<del class="diffchange diffchange-inline">textbf</del>{(B)} 72}</math></div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Now we know the measure of two angles are <math>39^{\circ}</math> and <math>69^{\circ}</math>. By the Triangle Sum Theorem, the sum of all angles in a triangle is <math>180^{\circ},</math> so the final angle is <math>72^{\circ}</math>. Therefore, the largest angle in the triangle is <math>\boxed{\<ins class="diffchange diffchange-inline">mathrm</ins>{(B) <ins class="diffchange diffchange-inline">\ </ins>} 72}</math></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">== See Also==</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">{{AMC10 box|year=2011|ab=B|num-b=6|num-a=8}}</ins></div></td></tr>
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Gina
https://artofproblemsolving.com/wiki/index.php?title=2011_AMC_10B_Problems/Problem_7&diff=38707&oldid=prev
Gina: Created page with '== Problem 7 == The sum of two angles of a triangle is <math>\frac{6}{5}</math> of a right angle, and one of these two angles is <math>30^{\circ}</math> larger than the other. W…'
2011-05-25T23:48:26Z
<p>Created page with '== Problem 7 == The sum of two angles of a triangle is <math>\frac{6}{5}</math> of a right angle, and one of these two angles is <math>30^{\circ}</math> larger than the other. W…'</p>
<p><b>New page</b></p><div>== Problem 7 ==<br />
<br />
The sum of two angles of a triangle is <math>\frac{6}{5}</math> of a right angle, and one of these two angles is <math>30^{\circ}</math> larger than the other. What is the degree measure of the largest angle in the triangle?<br />
<br />
<math> \textbf{(A)}\ 69 \qquad\textbf{(B)}\ 72 \qquad\textbf{(C)}\ 90 \qquad\textbf{(D)}\ 102 \qquad\textbf{(E)}\ 108 </math><br />
<br />
== Solution ==<br />
<br />
The sum of two angles in a triangle is <math>\frac{6}{5}</math> of a right angle <math>\longrightarrow \frac{6}{5} \times 9 = 108</math><br />
<br />
If <math>x</math> is the measure of the first angle, then the measure of the second angle is <math>x+30</math>.<br />
<cmath>x + x + 30 = 108 \longrightarrow 2x = 78 \longrightarrow x = 39</cmath><br />
<br />
Now we know the measure of two angles are <math>39^{\circ}</math> and <math>69^{\circ}</math>. By the Triangle Sum Theorem, the sum of all angles in a triangle is <math>180^{\circ},</math> so the final angle is <math>72^{\circ}</math>. Therefore, the largest angle in the triangle is <math>\boxed{\textbf{(B)} 72}</math></div>
Gina