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For some value 'd', an isosceles triangle has at least one angle of

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nick1816
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For some value 'd', an isosceles triangle has at least one angle of [#permalink] New post   19 May 2019, 01:55
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For some value 'd', an isosceles triangle has at least one angle of (2d-40) degrees and at least one angle of (d+20) degrees. For each of such triangles, let x denote the smallest angle in degrees. Then what is the sum of all possible x's?

A. 50
B. 56
C. 106
D. 126
E. 150
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chetan2u
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Re: For some value 'd', an isosceles triangle has at least one angle of [#permalink] New post   19 May 2019, 04:07
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nick1816 wrote:
For some value 'd', an isosceles triangle has at least one angle of (2d-40) degrees and at least one angle of (d+20) degrees. For each of such triangles, let x denote the smallest angle in degrees. Then what is the sum of all possible x's?

A. 50
B. 56
C. 106
D. 126
E. 150


Three cases


1) angle that is twice is 2d-40
So 2(2d-40)+d+20=180...5d=240.....d=48
So angles are 2d-40=2*48-40=56 and d+20=48+20=68..
Least angle or X is 56..
2) angle that is twice is d+20
So (2d-40)+2(d+20)=180...4d=180......d=45
So angles are 2d-40=2*45-40=50 and d+20=45+20=65.
Least angle or X is 50..
3) angle that is twice is 2d-40 and d+20
So (2d-40)=d+20......d=60.
So angles are 2d-40=2*60-40=80 and d+20=60+20=80, so third angle is 180-80-80=20
Least angle or X is 20.

Sum becomes 56+50+20=126

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Fdambro294
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Re: For some value 'd', an isosceles triangle has at least one angle of [#permalink] New post   23 Oct 2020, 21:33
I think everyone forgot the 3rd Case.....


Great Question!
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For some value 'd', an isosceles triangle has at least one angle of [#permalink] New post   24 Oct 2020, 18:09
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Small note that there is a "4th" case, in case (3) when 2d - 40 = d + 20 we get 80 = 80 but the angle 80 only needs to appear "at least once", therefore we can have an 80 - 80 - 20 triangle or 80 - 50 - 50 triangle. However, the 4th case has an x of 50 so it does not contribute a new value, hence the answer is still the same as if we don't consider the 4th case.
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For some value 'd', an isosceles triangle has at least one angle of [#permalink]
24 Oct 2020, 18:09
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