# The Parallel Postulate: Definition & Examples

## What Is the Parallel Postulate?

Thousands of years ago, a Greek mathematician named Euclid laid out the foundations of plane geometry in five **postulates**, which are statements assumed to be true without proof. In today's lesson, we will cover his fifth postulate, called the **parallel postulate**, which states that if a straight line intersects two straight lines forming two interior angles on the same side that add up to less than 180 degrees, then the two lines, if extended indefinitely, will meet on that side on which the angles add up to less than 180 degrees.

Confused? You probably are, and that's okay. Let's simplify what that technical mess of a definition actually means. Suppose we have two lines, as shown here on screen.

We draw a third line that intersects the other two and label the interior angles *alpha* and *beta*, as shown here.

Imagine extending the two lines indefinitely. In our figure, it can be clearly seen that they would eventually intersect.

What the parallel postulate claims is that these lines would intersect on the side of the angles *alpha* and *beta* only if the sum of these angles is less than 180 degrees. Note that the potential intersection occurs on the side of the angles *alpha* and *beta*. That is, we are talking about this side:

And not this side:

Also note that the postulate states that the intersection would only occur if the sum of *alpha* and *beta* is less than 180 degrees. If you're still confused, then why don't we take a closer look at the parallel postulate with some examples? Hands-on stuff never hurts, right?

## The Parallel Postulate - Examples

We can have three general cases with respect to the parallel postulate, starting out with drawing two parallel lines labeled *m* and *l* and a transversal labeled *n*. We place the angles *alpha* and *beta* to the right of the transversal. In the resulting figure, it's easy to see that lines *m* and *l* would never cross each other.

Oh, and here's a quick and important note: recall from geometry that when *m* and *l* are parallel, the angles *alpha* and *beta* are supplementary. That is, they sum up to 180 degrees. Now suppose that we rotate line *m* clockwise. Our rotation reduces the angle measure of *alpha*, meaning that the sum of *alpha* and *beta* also become less than 180 degrees. In this case, it's easy to see that the lines *m* and *l* would eventually cross to the right of transversal *n*.

The same scenario would have resulted if we kept line *m* fixed and instead rotated line *l* counter clockwise.

Now let's go back to lines *m* and *l* being parallel. We now rotate line *m* counter clockwise, as shown on the screen.

This rotation increases the angle measure of *alpha*, making the sum of *alpha* and *beta* more than 180 degrees. In this case, we can see that lines *m* and *l* would never cross to the right of the transversal, in accordance with the parallel postulate.

Speaking of which, the same scenario would have resulted if we kept line *m* fixed and rotated line *l* clockwise. As a final note, if we had placed the angles *alpha* and *beta* on the other side of the transversal, the parallel postulate would still hold. However, this time we'd be looking at whether lines *m* and *l* cross on the left side of the transversal, because *alpha* and *beta* are now on the left side of the transversal.

## Lesson Summary

Hopefully, we're all squared away here. (Catch that geometry joke?) Okay, a **postulate** is a statement assumed to be true without proof. The **parallel postulate** states that if a straight line intersects two straight lines forming two interior angles on the same side that add up to less than 180 degrees, then the two lines, if extended indefinitely, will meet on that side on which the angles add up to less than 180 degrees. We have looked at what this definition means with visual examples and, in our analysis, the parallel postulate held in every scenario. You should now feel pretty confident in understanding this important postulate.

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