How Many Acute Angles from Three Intersecting Lines?

Date: 10/14/2002 at 16:02:38
From: Brian
Subject: Making acute angles from intersecting lines

What is the greatest number of acute angles you can make with three 
intersecting lines?

Date: 10/14/2002 at 17:04:31
From: Doctor Ian
Subject: Re: Making acute angles from intersecting lines

Hi Brian,

With two lines, two acute angles is the best you can do:

  \  a /
   \  /
    \/
  o /\ o          a = acute, o = obtuse
   /  \
  /  a \

(If the lines intersect at right angles, you get _no_ acute angles.) 

So the question is, how many can you add with that third line? 

If you make the third line parallel to one of the first two, 

        \  a /
         \  /
          \/
        o /\ o
         /  \
        /  a \
  \  a / 
   \  /   
    \/
 o  /\ o          a = acute, o = obtuse
   /  \
  /  a \

you get just two more. If you use the third line to make an acute 
triangle, 

         \  a /
          \  /
           \/ 
        o  /\ o          a = acute, o = obtuse
          /  \
         /  a \
        /      \
       /        \
    o / a      a \  o
  ---------------------
  a / o         o  \ a
   /                \

you get four more. Note that you can get also get four extras by 
intersecting all the lines at a single point:

   \  a /
    \  /
   a \/ a
 ----------- 
   a /\ a          a = acute, o = obtuse
    /  \
   /  a \

So that seems to cover the cases where the number of points of 
intersection is one, two, or three. Can you get four intersections 
from three lines? If not, then the answer to your question is 6.  

Does this help? 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/