# 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 **Â **

**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.

\Â Through a point P outside m, there are two lines, l and n, both parallel to m.

This is impossible.

âˆ´ 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 Ã‡ m Â¹ Ï† andÂ n | | m.

Â

Â Â Â Â

**To Prove: ** l intersects n ( i.e. l Ã‡ n â‰ Ï† )

**Proof: **Suppose l does not intersect n (i.e. Â l Ã‡ n = Ï†)

Lines l and n are co-planar and l Ã‡ n = Ï† , therefore l should be parallel to n.

But m | | nÂ l | | n and n | | m Ãž l | | m Â Ãž l Ã‡ m = Ï†

But this is a contradiction to the hypothesis that

l Ã‡ m Â¹ Ï† . Hence our supposition is wrong.

Therefore, l Ã‡ n Â¹ Ï† , hence l intersects n.

Â

**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*Â such that *p *| | *l*, *q* | | *m*Â and l Ã‡ m = Ï† .

**To Prove:** p Ã‡ q â‰ Ï†

**Proof:** Suppose p does not intersect q

Â Â Â Â Â Â Â Â Â i.e. p Ã‡ q Â¹ Ï†

Since p Ã‡Â q â‰ Ï† and *p*, *q* lies in the same plane

âˆ´ *p* | | *q* and also *p* | | *l*

âˆ´Â *q* | | *l*

But *q* | | *m*

âˆ´Â *l* | | *m*

which is against the given condition. Thus our supposition is wrong. Hence *p* and *q *intersect each other

( i.e. *p* Ã‡ *q* â‰ Ï† ).