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May 3rd, 2014, 02:34 PM  #1 
Newbie Joined: Mar 2014 From: Norway Posts: 2 Thanks: 0  Geometry (triangle)
"In a triangle, all of the angles $\displaystyle A$, $\displaystyle B$ and $\displaystyle C$ are less/lower than $\displaystyle 90°$. If all of the angles are integers, and the smallest angle is $\displaystyle \frac{1}{5}$ of the biggest angle, then what is the sum of the two biggest angles?" I can't really figure this one out, as there seems to not be enough information. I tried $\displaystyle a+b+c=180$. Lets say $\displaystyle a$ is the smallest angle and $\displaystyle c$ is the biggest. Then $\displaystyle a=\frac{c}{5}$ I then get $\displaystyle \frac{c}{5}+b+c=180$. If I solve this, I end up with $\displaystyle 5b+6c=900$. I don't know how to go from here, but wolframalpha gives $\displaystyle b+c=\frac{b+900}{6}$. Link to wolfram: (5b+6c=900 what is b+c  WolframAlpha) The answer is not supposed to be in this "algebra" way, it's supposed to be a whole number. Does anyone know? The answer is supposed to be $\displaystyle b+c=163°$ If I know that $\displaystyle b+c=163°$, I can just add that into the equations, and I get $\displaystyle c=85°$, $\displaystyle b=78°$ and $\displaystyle a=17°$. But how am I supposed to find the solution without knowing the answer already ($\displaystyle b+c=163$)? Thanks in advance =) 
May 3rd, 2014, 03:11 PM  #2 
Senior Member Joined: Sep 2012 From: British Columbia, Canada Posts: 764 Thanks: 53 
We want to find $\displaystyle b+c$, where $\displaystyle a<b<c<90$ are the angles in the triangle. We know that $\displaystyle a+b+c=180$ so $\displaystyle \frac{c}{5}+b+c=\frac{6c}{5}+b=180$ Since $\displaystyle b<c$, we know that $\displaystyle \frac{6c}{5}+c=\frac{11c}{5}>180$ which gives $\displaystyle c>81.818181...$ so now we have $\displaystyle 81<c<90$ $\displaystyle c$ must be a multiple of $\displaystyle 5$, otherwise $\displaystyle a$ will not be an integer, hence $\displaystyle c=85$ and the other angles will be determined after. 
May 4th, 2014, 02:41 AM  #3 
Global Moderator Joined: Dec 2006 Posts: 18,852 Thanks: 1570 
Or (equivalently), as c = 5a, c < 90° implies 5a < 90°, and so a < 18°. Also, as b = 180°  a  5a, b < c implies 180°  6a < 5a, and so a > 180°/11 = (16 4/11)°. Hence a = 17°, which implies b =78° and c = 85°, and so b + c = 163°. 

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