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Parallel Lines and a Transversal

 

A line which intersects two or more given lines at distinct points is called a 'transversal' of the given lines.

 

In the figure, AB and CD are two lines and EF is the transversal which intersects AB at M and CD at N. The three lines (AB, CD, EF) determine eight angles, four at M and four at N. By virtue of the positions of the angles, some of them can be paired together.

The following pairs of angles are called corresponding angles.
(i)
1 and 5
(ii)
4 and 8
(iii)
2 and 6
(iv)
3 and 7

The following pairs of angles are called alternate interior angles
(i)
3 and 5 (ii) 4 and 6

The following pairs of angles are called alternate exterior angles
(i)
1 and 7 (ii) 2 and 8

The following pairs of angles are called pairs of consecutive interior angles or pairs of interior angles on the same side of the transversal
(i)
4 and 5 (ii) 3 and 6

When a transversal intersects two parallel lines,
(i) Each pair of corresponding angles are equal;
(ii) Each pair of alternate interior angles are equal;
(iii) Each pair of consecutive interior angles are supplementary.

The converse of each of the above three is easy to understand.

 

Example : If the above three statements hold good for two given lines and their transversal, then the lines will be parallel.

 

In other words, if a transversal intersects two parallel lines, each pair of corresponding angles are equal. Conversely if a transversal intersects two lines, making a pair of corresponding angles equal, then the lines are parallel.





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