Difference between revisions of "2009 AMC 8 Problems/Problem 19"
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| − | == | + | ==Problem== |
Two angles of an isosceles triangle measure <math> 70^\circ</math> and <math> x^\circ</math>. What is the sum of the three possible values of <math> x</math>? | Two angles of an isosceles triangle measure <math> 70^\circ</math> and <math> x^\circ</math>. What is the sum of the three possible values of <math> x</math>? | ||
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\textbf{(D)}\ 165 \qquad | \textbf{(D)}\ 165 \qquad | ||
\textbf{(E)}\ 180</math> | \textbf{(E)}\ 180</math> | ||
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| + | ==Solution== | ||
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| + | There are 3 cases: where <math> x^\circ</math> is a base angle with the <math> 70^\circ</math> as the other angle, where <math> x^\circ</math> is a base angle with <math> 70^\circ</math> as the vertex angle, and where <math> x^\circ</math> is the vertex angle with <math> 70^\circ</math> as a base angle. | ||
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| + | Case 1: <math> x^\circ</math> is a base angle with the <math> 70^\circ</math> as the other angle: | ||
| + | Here, <math> x=70</math>, since base angles are congruent. | ||
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| + | Case 2: <math> x^\circ</math> is a base angle with <math> 70^\circ</math> as the vertex angle: | ||
| + | Here, the 2 base angles are both <math> x^\circ</math>, so we can use the equation <math> 2x+70=180</math>, which simplifies to <math> x=55</math>. | ||
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| + | Case 3: <math> x^\circ</math> is the vertex angle with <math> 70^\circ</math> as a base angle: | ||
| + | Here, both base angles are <math> 70^\circ</math>, since base angles are congruent. Thus, we can use the equation <math> x+140=180</math>, which simplifies to <math> x=40</math>. | ||
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| + | Adding up all the cases, we get <math>70+55+40=165</math>, so the answer is <math> \boxed{\textbf{(D)}\ 165}</math>. | ||
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| + | == Video Solution by OmegaLearn == | ||
| + | https://youtu.be/FDgcLW4frg8?t=902 | ||
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| + | ~ pi_is_3.14 | ||
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| + | https://www.youtube.com/watch?v=EO3DphZAlDY ~David | ||
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| + | ==Video Solution 2== | ||
| + | https://youtu.be/iB0dcpVREE8 Soo, DRMS, NM | ||
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==See Also== | ==See Also== | ||
{{AMC8 box|year=2009|num-b=18|num-a=20}} | {{AMC8 box|year=2009|num-b=18|num-a=20}} | ||
| + | {{MAA Notice}} | ||
Latest revision as of 18:38, 15 April 2023
Problem
Two angles of an isosceles triangle measure 
and 
. What is the sum of the three possible values of 
?
Solution
There are 3 cases: where is a base angle with the as the other angle, where is a base angle with as the vertex angle, and where is the vertex angle with as a base angle.
Case 1: is a base angle with the as the other angle:
Here, 
, since base angles are congruent.
Case 2: is a base angle with as the vertex angle:
Here, the 2 base angles are both , so we can use the equation 
, which simplifies to 
.
Case 3: is the vertex angle with as a base angle:
Here, both base angles are , since base angles are congruent. Thus, we can use the equation 
, which simplifies to 
.
Adding up all the cases, we get 
, so the answer is 
.
Video Solution by OmegaLearn
https://youtu.be/FDgcLW4frg8?t=902
~ pi_is_3.14
https://www.youtube.com/watch?v=EO3DphZAlDY ~David
Video Solution 2
https://youtu.be/iB0dcpVREE8 Soo, DRMS, NM
See Also
| 2009 AMC 8 (Problems • Answer Key • Resources) | ||
| Preceded by Problem 18 |
Followed by Problem 20 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
| All AJHSME/AMC 8 Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
