SOLUTION: Can someone please help me with the following two questions. Been working on both for hours, just can't get them. Two lines in a plane can intersect forming four angles, (some m

Algebra ->  Angles -> SOLUTION: Can someone please help me with the following two questions. Been working on both for hours, just can't get them. Two lines in a plane can intersect forming four angles, (some m      Log On


   

Question 703977: Can someone please help me with the following two questions. Been working on both for hours, just can't get them.
Two lines in a plane can intersect forming four angles, (some may have the same measure). What is the greatest number of angles we can form using three lines.
AND:
The same question but using four lines.
Thankyou ahead of time for any help I can get. If can,lease explain how you got what you did.
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
I thought hard too, and this is what I came up with.
At one crossing of 2 lines, there are four angles, and the wording hints that we should count all four.

With 3 lines, each line has at most 3-1=2 crossings
(less if one or both of the other lines are parallel).
If we multiply 3 (lines) times 2 crossings per line,
you get at most 6 crossings, but each crossing is counted twice,
one time from the point of view of one of the crossing lines,
and another time from the point of view of the other crossing line.
So there are really at most 6%2F2=3 crossings
(less if all 3 lines were to cross together at the same point).
That would mean at most 3%2A4=highlight%2812%29 angles.


With four lines there are at most 4%2A%284-1%29%2F2=6 crossings.
That would give you at most 6%2A4=highlight%2824%29 angles.


EXTRA:
n lines could have at most n%28n-1%29%2F2 crossings and
that would make a maximum of 4n%28n-1%29%2F2=2n%28n-1%29 angles
NOTE:
Maybe you should try the artofproblemsolving.com forums.
Some of the people there are future Math Olympics competitors.