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

In these lessons, we will learn

The following figure shows some examples of alternate interior and alternate exteriors angle pairs. Scroll down the page if you need more examples and explanations about alternate interior angles and alternate exterior angles.

### Alternate Interior Angles

#### Alternate Interior Angles

(Angles found in a **Z**-shaped figure)

*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°

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.

### Proof of the Alternate Interior Angle

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.

### Proof of the Converse of the Alternate Interior Angles

### Alternate Exterior Angles

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 of the Alternate Exterior Angle

Proof and definition of alternate and exterior angles with a transversal and parallel lines.
### Proof of the Alternate Exterior Angle Converse

You can use the free Mathway widget below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

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

The following figure shows some examples of alternate interior and alternate exteriors angle pairs. Scroll down the page if you need more examples and explanations about alternate interior angles and alternate exterior angles.

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:Therefore,

Step 2:

Therefore,

Step 3:

Therefore,

Step 4:

Therefore,

Step 5:

Therefore,

Step 6:

Therefore,

Step 7:

Therefore,

Answer:

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.

How to Find an Angle Using Alternate Interior Angles.

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

This video shows a proof of the alternate interior angle converse.If two lines are cut by a transversal and the alternate interior angles are congruent, then the lines are parallel.

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

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.

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

This video shows a proof of the alternate exterior angle converse.If two lines are cut by a transversal and the alternate angles are congruent, then the lines are parallel.

Rotate to landscape screen format on a mobile phone or small tablet to use the **Mathway** widget, a free math problem solver that **answers your questions with step-by-step explanations**.

You can use the free Mathway widget below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

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