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Lines Parallel to the Same Line

Theorem 6 - Parallel Lines
Lines which are parallel to the same line, are parallel to each other.
Given: Three lines l, m and n in a plane such that l
| | m and n | | m.

To Prove: l
| | n  

Proof: Suppose l is not parallel to n, then l, n intersect in a unique point, say P. Now P is not on m, since P is on l and l | | m.
Through a point P outside m, there are two lines, l and n, both parallel to m.
This is impossible.

Our supposition is wrong. Hence l | | n.


Theorem 7
If l, m, n are lines in the same plane such that l intersects m and n | | m, then l intersects n also.
Given: Three lines l, m, n are in the same plane l, such that l
Ç m ¹ φ and n | | m.
 

    

To Prove: l intersects n ( i.e. l Ç n φ )

Proof: Suppose l does not intersect n (i.e.  l
Ç n = φ)

Lines l and n are co-planar and l
Ç n = φ , therefore l should be parallel to n.
But m
| | n l | | n and n | | m Þ l | | m  Þ l Ç m = φ
But this is a contradiction to the hypothesis that
l
Ç m ¹ φ . Hence our supposition is wrong.
Therefore, l
Ç n ¹ φ , hence l intersects n.
 

Example : If l and m are intersecting lines, p | | l and q | | m, show that p and q also intersect.


Given : Four lines l, m, p, q such that p | | l, q | | m and l Ç m = φ .
To Prove: p
Ç q φ
Proof: Suppose p does not intersect q
          i.e. p
Ç q ¹ φ

Since p Ç q φ and p, q lies in the same plane
p | | q and also p | | l
  q | | l
But q | | m

 l | | m
which is against the given condition. Thus our supposition is wrong. Hence p and q intersect each other
( i.e. p
Ç q φ ).





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