 ## Transversal Definition

What is a transversal in geometry? A transversal is any line crossing another line or lines. When it crosses two parallel lines, the resulting eight angles have interesting properties. ## Transversal Lines

You have probably ridden in a car on a street that crossed railroad tracks. As you crossed the tracks, you completed a transversal. A transversal is a line that crosses other lines. Usually we work with transversals when they cross parallel lines, like the two tracks of a railroad. ### Parallel Lines Cut By A Transversal

Let's construct a transversal to see how they interact with parallel lines. Use a straightedge and pencil to draw parallel lines $BE$ and $AR$, so that $BE$ is horizontal and at the top, with $AR$ horizontal and at the bottom.

Use a straightedge and pencil to draw a line cutting from above $BE$ to below $AR$. Label it $OW$. You see? It is never a good idea to cross a bear. 🐻 ## Transversal Angles

Our transversal $OW$ created eight angles where it crossed $BE$ and $AR$. These are called supplementary angles. ### Supplementary Angles

Supplementary angles are pairs of angles that add up to $180°$. Because all straight lines are $180°$, we know $\angle Q$ and $\angle S$ are supplementary (adding to $180°$). Together, the two supplementary angles make half of a circle. Supplementary angles are not limited to just transversals.

In this example, the supplementary angles are $QS$, $QT$, $TU$, $SU$, and $VX$, $VY$, $YZ$, $VZ$. Here are all the other pairs of supplementary angles:

### Exterior Angles

Think back to those railroad tracks. If you were between the train tracks, you would be inside the lines. If you stepped across the tracks, you would be outside the lines. The same is true with parallel lines $BE$ and $AR$ and their transversal $OW$. The angles above and below the parallel lines are outside and are called exterior angles. Your drawing has four exterior angles:

### Interior Angles

Your drawing also has four interior angles, or angles inside (between) the parallel lines: ### Vertical Angles

Angles in your transversal drawing that share the same vertex are called vertical angles. Do not confuse this use of "vertical" with the idea of straight up and down. You have four pairs of vertical angles:

### Corresponding Angles

The two parallel lines are creating corresponding angles. To be corresponding angles:

1. The two angles must be on the same side of the transversal
2. One angle must be interior and the other exterior Notice that $\angle Q$ is congruent to $\angle V$. $\angle Q$ is an exterior angle on the left side of transversal $OW$, and $\angle V$ is an interior angle on the same side of the transversal line.

All the pairs of corresponding angles are:

### Alternating Angles

Alternating angles are pairs of angles in which both angles are either interior or exterior. They appear on opposite sides of the transversal and are congruent. The four pairs of alternating angles in our drawing are:

## Lesson Summary

Transversals are lines that intersect two parallel lines at an angle. You can also construct a transversal of parallel lines and identify all eight angles the transversal forms. You can classify angles as supplementary angles (that add up to 180 degrees, vertical angles, corresponding angles, alternating angles, interior angles, or exterior angles.

## What you learned:

After working through this material, you will learn to:

• Define and identify a transversal in parallel lines
• Classify angles as supplementary, vertical, corresponding, alternate, interior or exterior
• Identify the eight angles created by a transversal in parallel lines Instructor: Malcolm M.
Malcolm has a Master's Degree in education and holds four teaching certificates. He has been a public school teacher for 27 years, including 15 years as a mathematics teacher.
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