# Parallel Lines and a Transversal

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

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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) âˆ 1 and âˆ 5

(ii) âˆ 4 and âˆ 8

(iii) âˆ 2 and âˆ 6

(iv) âˆ 3 and âˆ 7

The following pairs of angles are called alternate interior angles

(i) âˆ 3 and âˆ 5 (ii) âˆ 4 and âˆ 6

The following pairs of angles are called alternate exterior angles

(i) âˆ 1 and âˆ 7 (ii) âˆ 2 and âˆ 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) âˆ 4 and âˆ 5 (ii) âˆ 3 and âˆ 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.

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

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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.