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In geometry, pairs of angles can relate to each other in several ways.

In these lessons, we will learn

- alternate interior angles
- proof of the alternate interior angle theorem
- proof of the converse of the alternate interior angle theorem
- alternate exterior angles
- proof of the alternate exterior angle theorem
- proof of the converse of the alternate exterior angle theorem

Related Topics: Other Types of Angles in Geometry

When a line (called a transversal) intersects a pair of parallel lines alternate interior angles are formed. Alternate interior angles are equal to each other.

The **Alternate Interior Angles Theorem **states that

When two parallel lines are cut by a transversal, the resulting alternate interior angles are congruent.

One way to find the alternate interior angles is to draw a **zigzag
line** on the diagram. In the above diagrams, ** d**
and

Example 1:
Given the diagram below, determine the values of the angles *b,
c, d, e, f, g *and *h.*

Solution:

Step 1: *b * is a supplement
of 60°.

Therefore,
*b* + 60° =180° ⇒ *b*
= 180° – 60° = 120°

Step 2: *b* and *c* are vertical angles.

Therefore, *c* = *b* = 120°

Step 3: *d *and 60° are
vertical angles.

Therefore, *d* = 60°

Step 4: *d *and *e*
are alternate interior angles.

Therefore, *e* = *d* = 60°

Step 5: *f* and *e* are supplementary
angles.

Therefore, *f* + 60° =180° ⇒
*f* = 180° – 60° = 120°

Step 6: *g* and *f* are
vertical angles.

Therefore, *g *= *f* = 120°

Step 7: *h* and *e* are
vertical angles.

Therefore, *h *= *e *= 60°

Answer: *b* = 120°,
*c* = 120°, d = 60°, *e* = 60°, *f* = 120°, *g* = 120° and *h*
= 60°

From the above example, you may notice that either an angle is 60° or it is 120°. Actually, all the small angles are 60° and all the big angles are 120°.

In general, the diagram will be as shown below. The small and big pair of angles are supplementary (i.e. small + big = 180°). Therefore, given any one angle you would be able to work out the values of all the other angles.

This video shows how to identify alternate interior angles and their properties.

How to Find an Angle Using Alternate Interior Angles.

The following video gives an example of how to use alternate interior angles to find the measures of angles.

This video will prove that Alternate Interior Angles Are Congruent by using the Corresponding Angle Postulate.

This video shows a proof of the Alternate Interior Angle theorem showing that when lines are parallel, alternate interior angles are congruent.

The **Converse of the Alternate Interior Angle** states that

If two lines are cut by a transversal and the alternate interior angles are congruent, then the lines are parallel.

This video shows a proof of the alternate interior angle converse.

One way to remember
alternate exterior angles is that they are the vertical
angles of the alternate interior angles.
Alternate exterior angles** **are equal to one another.

The **Alternate Exterior Angles Theorem **states that

When two parallel lines are cut by a transversal, the resulting alternate exterior angles are congruent.

** a **and

** b** and

Alternate Exterior Angles definition and properties.

This video shows how to identify alternate exterior angles and their properties.

When two lines are crossed by a transversal, the opposite angle pairs on the outside of the lines are alternate exterior angles. The two lines do not have to be parallel. Find out how to locate alternate exterior angles and the characteristics of alternate exterior angles.

This video shows how to find an angle using alternate exterior angle.

Proof and definition of alternate and exterior angles with a transversal and parallel lines.

The **Converse of the Alternate Exterior Angle Theorem **states that

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

This video shows a proof of the alternate exterior angle converse.