Parallel Lines

After understanding about intersection of lines and the different types of angles, we are now ready to study parallel lines

Parallel lines seem a very easy concept from our everyday experiences. The railway track on which trains travel is one of the earliest experiences of this idea for children. But “geometrically accurate” & parallel lines are not easy to construct on a piece of paper.

In a previous chapter 22.8 we had asked students to take a piece of paper and fold 2 parallel lines. They were not expected to take the reference of any of the sides. They should also be able to prove geometrically that the 2 are parallel; mere appearance is not sufficient.

The idea was to make students realize that the only way to fold 2 (geometrically correct) parallel lines on a sheet of paper is to fold a line, then its perpendicular and then the perpendicular to the 2ndline.

A moment’s reflection will show that this is equivalent to the Fifth Postulate!

Recalling the results proved in Chapter 22.17, we can state that two lines intersected by a transversal would be parallel only under the following conditions.

1. A pair of the corresponding angles are equal OR

2. A pair of Alternate Interior angles are equal OR

3. A pair of Alternate Exterior angles are equal OR

4. If the sum of the interior angles (on the same side of the transversal) add up to a straight angle (180 degrees)

Euclid’s Fifth Postulate.

Euclid assumed Statement (4) above as one of his main postulates on which his plane geometry was based. This has a very interesting history which is fully told in chapter 29.5. Parallel Lines

If two lines on a plane are not parallel then they must intersect. The point of intersection may be outside the piece of paper on which they are drawn. But they will intersect somewhere on the ‘infinite’ plane.

Lines which coincide or overlap with each other are called Coincident Lines. They are a special case of parallel lines, where the distance between the lines has reduced to 0. We can say that the points on both the lines are collinear. A familiar example is the position of the hour & minute hands of a clock at 12!

We can also think of parallel lines representing a zero angle with arms at a certain finite distance.