# Transversal Lines, Angles, & Definition

Written by
Malcolm McKinsey
Fact-checked by
Paul Mazzola

## Transversal lines definition

AÂ transversalÂ is any line crossing another line or lines. When it crosses two parallel lines, the resulting eight angles have interesting properties.

You have probably ridden in a car on a street that crossed railroad tracks. As you crossed the tracks, you completed a transversal.Â A transversal is a line that crosses other lines.Â 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 linesÂ BEÂ andÂ AR, so thatÂ BEÂ is horizontal and at the top, withÂ ARÂ horizontal and at the bottom.

Use a straightedge and pencil to draw a line cutting from aboveÂ BEÂ to belowÂ AR. Label itÂ OW. You see? It is never a good idea to cross a bear.

## Transversal angles

OurÂ transversalÂ OWÂ created eight anglesÂ where it crossedÂ BEÂ andÂ AR. These are called supplementary angles.

### Supplementary angles

Supplementary anglesÂ are pairs of angles that add up toÂ 180Â°. Because all straight lines areÂ 180Â°, we knowÂ âˆ QÂ andÂ âˆ SÂ areÂ supplementaryÂ (adding toÂ 180Â°). Together, the two supplementary angles make half of a circle. Supplementary angles are not limited to just transversals.

In this example, the supplementary angles areÂ QS,Â QT,Â TU,Â SU, andÂ VX,Â VY,Â YZ,Â VZ.

Here are all the other pairs of supplementary angles:

• âˆ QÂ supplementaryÂ toÂ âˆ S

• âˆ QÂ supplementaryÂ toÂ âˆ T

• âˆ TÂ supplementaryÂ toÂ âˆ U

• âˆ SÂ supplementaryÂ toÂ âˆ U

• âˆ VÂ supplementaryÂ toÂ âˆ X

• âˆ VÂ supplementaryÂ toÂ âˆ Y

• âˆ YÂ supplementaryÂ toÂ âˆ Z

• âˆ VÂ supplementaryÂ toÂ âˆ Z

### 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 linesÂ BEÂ andÂ ARÂ and their transversalÂ OW. The angles above and below the parallel lines are outside and are calledÂ exterior angles.

Your drawing has four exterior angles: âˆ Q,Â âˆ S,Â âˆ YÂ andÂ âˆ Z.

### Interior angles

Your drawing also has fourÂ interior angles, or angles inside (between) the parallel lines: âˆ T,Â âˆ U,Â âˆ VÂ andÂ âˆ X.

### Vertical angles

Angles in your transversal drawing that share the same vertex are calledÂ vertical angles. Do not confuse this use of "vertical" with the idea of straight up and down.

When two parallel lines are crossed by a transversal, you get four pairs of vertical angles:

• âˆ QÂ andÂ âˆ U

• âˆ SÂ andÂ âˆ T

• âˆ VÂ andÂ âˆ Z

• âˆ YÂ andÂ âˆ X

### Corresponding angles

The two parallel lines are creatingÂ corresponding angles. To be corresponding angles they must be on the same side of the transversal and one angle must be interior and the other exterior.

Notice thatÂ âˆ QÂ is congruent toÂ âˆ V.Â âˆ QÂ is anÂ exterior angleÂ on the left side of transversalÂ OW, andÂ âˆ VÂ is anÂ interior angleÂ on the same side of the transversal line.

All the pairs of corresponding angles are:

• âˆ QÂ andÂ âˆ V

• âˆ TÂ andÂ âˆ Y

• âˆ SÂ andÂ âˆ X

• âˆ UÂ andÂ âˆ Z

### Alternating angles

Alternating anglesÂ 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:

• âˆ VÂ andÂ âˆ U (interior)

• âˆ XÂ andÂ âˆ T (interior)

• âˆ QÂ andÂ âˆ Z (exterior)

• âˆ YÂ andÂ âˆ S (exterior)

There are to pairs of alternate interior angles, and two pairs of alternate exterior angles.

Alternate interior angles have their own their theorem to help identify information about the angles themselves. Alternate exterior angles also have their own theorem.

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