एक ही रेखा के समानांतर रेखाएँ: Difference between revisions

Revision as of 13:37, 28 June 2024

Fig. 1 - Transversal Line

If two lines are parallel to the same line, will they be parallel to each other? Let us verify.

In the fig.1 line $m||$ line $l$ and line $n||$ line $l$ .

Let us draw a line $t$ transversal for the lines $l,m,n$

We know that line $m||$ line $l$ and line $n||$ line $l$ .

Hence $\angle 1=\angle 2$ and $\angle 1=\angle 3$ (Corresponding angles axiom)

But $\angle 2=\angle 3$ as they are corresponding angles

Therefore, we can say that line $m||$ line $n$ (Converse of corresponding angles axiom)

This result can be stated in the form of the following theorem:

Theorem 1: Lines which are parallel to the same line are parallel to each other.

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+[[File:Transversal Line.jpg|alt=Fig. 1 - Transversal Line|thumb|Fig. 1 - Transversal Line]]
+If two lines are parallel to the same line, will they be parallel to each other? Let us verify.
+In the fig.1 line <math>m || </math> line <math>l </math> and line <math>n || </math> line <math>l </math> .
+Let us draw a line <math>t </math> transversal for the lines <math>l,m,n </math>
+We know that  line <math>m || </math> line <math>l </math> and line <math>n || </math> line <math>l </math> .
+Hence <math>\angle 1=\angle 2 </math> and <math>\angle 1=\angle 3 </math> (Corresponding angles axiom)
+But <math>\angle 2=\angle 3 </math> as they are corresponding angles
+Therefore, we can say that  line <math>m || </math> line <math>n </math> (Converse of corresponding angles axiom)
+This result can be stated in the form of the following theorem:
+'''Theorem 1''': Lines which are parallel to the same line are parallel to each other.
 [[Category:रेखाएँ और कोण]][[Category:कक्षा-9]][[Category:गणित]]
-Lines parallel to the same line