Parallel and Transversal lines

Take a look at the picture of the railway tracks above.

It looks as if the tracks are going to meet after a certain point.

But we as we keep walking along those lines, they never appear to meet.

We are going to explore a similar concept here.

Parallel lines

Now consider these tracks as two lines that never meet.

In geometry, such lines are called parallel lines.

By definition, two lines in the same plane that do not touch or intersect at any point are called parallel lines.

But how do we find out if the two lines intersect or not?

For example, in the two diagrams given above, both the lines do not intersect.

But are they both parallel?

No! The first set of lines is parallel while the second one is not.

The second set of lines appears to be non-intersecting but those will eventually meet if we extend them further.

Now, let us look at another type of line called the transversal line.

Transversal lines

When a line passes through two lines in the same plane at two distinct points, it is called a transversal line.

The two lines that it passes through may or may not be parallel.

A transversal passing through two parallel lines.

A transversal passing through two non-parallel lines.

Now, when a transversal passes through the two lines, there are 8 angles formed between the lines.

We will study the angles formed when a line passes through two parallel lines.

Observe the figure given above.

Angles 1, 2, 3 and 4 are the angles formed when the line AB is intersected by PQ.

And the angles 5, 6, 7 and 8 are the angles formed when the line CD is intersected by PQ.

The angles 1, 2, 7 and 8 are exterior angles.

And the angles 3, 4, 5 and 6 are interior angles.

Now angles 2 and 6 are said to be corresponding angles.

Similarly, angles 1 and 5 are also corresponding angles.

Corresponding angles can be formed between angles 3 and 7 and also between angles 4 and 8.

Now, look at angles 4 and 5.

Both these are interior angles but they are on the alternate sides of the transversal line.

Thus, they are alternate interior angles.

Angles 3 and 6 are also alternate interior angles.

We can similarly pair the alternate exterior angles.

For example, exterior angles 1 and 8 are on the alternate sides of the transversal.

Thus, angles 1 and 8 are called alternate exterior angles.

Angles 2 and 7 are also alternate exterior angles.

Lastly, we need to know one fun fact regarding the angles in the same diagram.

The sum of the measure of angles 4 and 6 is 180 degrees and so is the measure of the sum of the angles 3 and 5.

You can derive this property on your own. Now let us revise all that we have learnt.

Revision

Two lines in the same plane that do not touch or intersect at any point are called parallel lines.

When a line passes through two lines in the same plane at two distinct points, it is called a transversal line.

When a transversal passes through the two lines, there are 8 angles formed between the lines.

The interior angles that are formed on the alternate sides of the transversal are alternate interior angles.

The exterior angles that are formed on the alternate sides of the transversal are alternate exterior angles.

An interior and exterior angle that are formed on the same side of the transversal, but have different vertices, are corresponding angles.

Two interior angles lying on the same side of the transversal have their sum as 180 degrees.

The End