Difference between revisions of "Mock Geometry AIME 2011 Problems/Problem 2"
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WLOG, let one of the lines be horizontal. <math> 11 </math> angles are formed above it by <math> 10 </math> lines, and all of these angles must sum to <math> 180^\circ </math>. Thus, the smallest angle must be less than <math> \frac{180^\circ}{11} </math>, since otherwise they would sum to more than <math> 180^\circ </math>. There are <math> 16 </math> positive integers less than <math> \frac{180}{11} </math>, so there are <math> \boxed{016} </math> possible measures for the smallest angle. | WLOG, let one of the lines be horizontal. <math> 11 </math> angles are formed above it by <math> 10 </math> lines, and all of these angles must sum to <math> 180^\circ </math>. Thus, the smallest angle must be less than <math> \frac{180^\circ}{11} </math>, since otherwise they would sum to more than <math> 180^\circ </math>. There are <math> 16 </math> positive integers less than <math> \frac{180}{11} </math>, so there are <math> \boxed{016} </math> possible measures for the smallest angle. | ||
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| + | ==Video Solution by Punxsutawney Phil== | ||
| + | https://youtube.com/watch?v=OEkt8HbUVbQ&t=419s | ||
Latest revision as of 21:06, 23 December 2022
Problem
Eleven nonparallel lines lie on a plane, and their pairwise intersections meet at angles of integer degree. How many possible values are there for the smallest of these angles?
Solution
Consider two intersecting lines. One line can be translated across the plane, without rotation, and the angle between the two lines will remain the same.
Now consider the full eleven lines. These lines can be translated so that they are all concurrent at a point, and still have the same equal pairwise angles.
WLOG, let one of the lines be horizontal. 
angles are formed above it by 
lines, and all of these angles must sum to 
. Thus, the smallest angle must be less than 
, since otherwise they would sum to more than . There are 
positive integers less than 
, so there are 
possible measures for the smallest angle.
