When parallel lines are cut by a transversal line the alternate exterior angles are complementary angles?

Lines are parallel if they are always the same distance apart (called "equidistant"), and will never meet. Just remember:

Always the same distance apart and never touching.

The red line is parallel to the blue line in each of these examples:

Example 1		Example 2

Parallel lines also point in the same direction.

Parallel lines have so much in common. It's a shame they will never meet!

When parallel lines get crossed by another line (which is called a Transversal), you can see that many angles are the same, as in this example:

These angles can be made into pairs of angles which have special names.

Click on each name to see it highlighted:

Now play with it here. Try dragging the points, and choosing different angle types. You can also turn "Parallel" off or on:

Testing for Parallel Lines

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In Geometry, when any two parallel lines are cut by a transversal, many pairs of angles are formed. There is a relationship that exists between these pairs of angles. While some of them are congruent, the others are supplementary. Let us learn more about the angles formed when parallel lines are cut by a transversal.

What are Parallel Lines Cut by Transversal?

Parallel lines are straight equidistant lines that lie on the same plane and never meet each other. When any two parallel lines are intersected by a line (known as the transversal), the angles that are subsequently formed, have a relationship. The various pairs of angles that are formed on this intersection are Corresponding angles, Alternate Interior Angles, Alternate Exterior Angles and Consecutive Interior Angles. Observe the figure given below which shows two parallel lines 'a' and 'b' cut by a transversal 'l'.

Angles Formed by Parallel Lines Cut by Transversal

When parallel lines are cut by a transversal, four types of angles are formed. Observe the following figure to identify the different pairs of angles and their relationship. The figure shows two parallel lines 'a' and 'b' which are cut by a transversal 'l'.

Corresponding angles

When two parallel lines are intersected by a transversal, the corresponding angles have the same relative position. In the figure given above, the corresponding angles formed by the intersection of the transversal are:

• ∠1 and ∠5
• ∠2 and ∠6
• ∠3 and ∠7
• ∠4 and ∠8

It should be noted that the pair of corresponding angles are equal in measure, that is, ∠1 = ∠5, ∠2 = ∠6, ∠3 = ∠7, and ∠4 = ∠8

Alternate Interior Angles

Alternate interior angles are formed on the inside of two parallel lines which are intersected by a transversal. In the figure given above, there are two pairs of alternate interior angles.

It should be noted that the pair of alternate interior angles are equal in measure, that is, ∠3 = ∠6, and ∠4 = ∠5

Alternate Exterior Angles

When two parallel lines are cut by a transversal, the pairs of angles formed on either side of the transversal are named as alternate exterior angles. In the figure given above, there are two pairs of alternate exterior angles.

It should be noted that the pair of alternate exterior angles are equal in measure, that is, ∠1 = ∠8, and ∠2 = ∠7

Consecutive Interior Angles

When two parallel lines are cut by a transversal, the pairs of angles formed on the inside of one side of the transversal are called consecutive interior angles or co-interior angles. In the given figure, there are two pairs of consecutive interior angles.

It should be noted that unlike the other pairs given above, the pair of consecutive interior angles are supplementary, that is, ∠4 + ∠6 = 180°, and ∠3 + ∠5 = 180°.

Properties of Parallel Lines Cut by Transversal

When any two parallel lines are cut by a transversal they acquire some properties. In other words, any two lines can be termed as parallel lines if the following conditions related to the angles are fulfilled.

• Any two lines that are intersected by a transversal are said to be parallel if the corresponding angles are equal.
• Any two lines that are intersected by a transversal are said to be parallel if the alternate interior angles are equal.
• Any two lines that are intersected by a transversal are said to be parallel if the alternate exterior angles are equal.
• Any two lines that are intersected by a transversal are said to be parallel if the consecutive interior angles are supplementary.

Related Links

Check out the following links related to Parallel Lines Cut by Transversal.

• Parallel Lines
• Transversal

1. Example 1: Identify the corresponding angles in the figure which shows two parallel lines 'm' and 'n' cut by a transversal 't'.

   

   Solution: In the given figure, two parallel lines are cut by a transversal, and the corresponding angles in the figure are ∠1 and ∠3; and ∠2 and ∠5.

2. Example 2: Find the value of x in the given parallel lines 'a' and 'b', cut by a transversal 't'.

   

   Solution: The given parallel lines are cut by a transversal, therefore, the marked angles in the figure are the alternate interior angles which are equal in measure. This means, 8x - 4 = 60°, and 8x = 64, x = 8.

   Therefore, the value of x = 8.

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FAQs on Parallel Lines Cut by Transversal

Parallel lines are straight equidistant lines that lie on the same plane and never meet each other. A transversal is any line that intersects two straight lines at distinct points. When any two parallel lines are intersected by a transversal, various angles are formed. There is a relationship that exists between these pairs of angles.

What happens When Parallel Lines are Cut by a Transversal?

When any two parallel lines are cut by a transversal, there are various pairs of angles that are formed. These angles are corresponding angles, alternate interior angles, alternate exterior angles, and consecutive interior angles.

What are the Special Pairs of Angles Formed when Parallel Lines Cut by Transversal?

When parallel lines are cut by a transversal, there are 4 special types of angles that are formed - corresponding angles, alternate interior angles, alternate exterior angles, and consecutive interior angles. While the pairs of corresponding angles, alternate interior angles, alternate exterior angles are congruent, the pairs of consecutive interior angles are supplementary.

How to Calculate Angle Measures in Parallel Lines Cut by a Transversal?

The unknown angles can be easily calculated when two parallel lines are cut by a transversal. The following facts help in finding the unknown angles. When parallel lines are cut by a transversal,

When Two Parallel Lines are Cut by a Transversal, are the Corresponding Angles Congruent?

Yes, when two parallel lines are intersected by a transversal, the corresponding angles that are formed are congruent.

When Two Parallel Lines are Cut by a Transversal, are the Alternate Interior Angles Congruent?

Yes, when two parallel lines are intersected by a transversal, the alternate interior angles are congruent.

The pair of non-adjacent angles on the outer side or “outside” of the two lines but on opposite sides of the transversal are known as alternate exterior angles.

When two parallel or non-parallel lines are intersected by a transversal, they usually form eight (8) angles. Exterior angles are the angles that exist outside of the two lines. But when it comes to pairs of alternate exterior angles, keep in mind that they are not only on opposite sides of the transversal but also outside of the two lines.

In our illustration above, \angle 1 and \angle 8 are a pair of alternate exterior angles. Why? Because both angles are located outside lines s and h, are not adjacent to each other, and are on opposite sides of the transversal. The other pair of alternate exterior angles are \angle 2 and \angle 7.

Alternate Exterior Angles Theorem

If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent.

This illustration shows lines, d and m, being intersected by the transversal, t. Since lines d and m are parallel lines, the alternate exterior angles formed are congruent which means they have the same measure.

Conversely, alternate exterior angles are not congruent and do not have a relationship if two non-parallel lines are cut by a transversal.

Example Problems Involving Alternate Interior Angles

Example 1: Identify all pairs of alternate exterior angles.

First, we need to pick out our exterior angles. We see that \angle 5, \angle 6, \angle 3, and \angle 9 are all located outside lines r and e.

However, since we are asked to list the alternate exterior angles, we need to look at the pairs of angles that are not adjacent to each other and are located on opposite sides of the transversal, t.

Therefore, our alternate exterior angles are:

• \angle 5 and \angle 9
• \angle 6 and \angle 3

Example 2: Determine if the following statements are True or False.

• When two non-parallel lines are cut by a transversal, the alternate exterior angles formed are congruent.

Answer: FALSE. Alternate exterior angles are not congruent and have no relationship in any way when a transversal cuts two non-parallel lines.

• Alternate exterior angles are congruent when a transversal intersects two parallel lines.

Answer: TRUE. Alternate exterior angles are only of the same measure if the lines cut through by a transversal are parallel.

Example 3: What is the measure of \angle 5?

Right off the bat, we see that \angle 5 and \angle 2 are a pair of alternate exterior angles. And since line w and line g are parallel lines cut by the transversal, t, then the two angles are congruent or their measures are the same.

Given that the measure of \angle 2 is 60^\circ, we can conclude that the measure of \angle 5 is also 60^\circ.

You might also be interested in:

Alternate Interior Angles

Complementary Angles

Corresponding Angles

Supplementary Angles

Vertical Angles