Alternate Interior Angles & Alternate External Angles

In geometry, pairs of angles can relate to each other in several ways. In this lesson, we will be describing alternate interior angles and alternate exterior amgles.

Alternate Interior 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.

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

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.

Alternate Exterior Angles

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.

a and h are alternate exterior angles and they are equal to one another.

b and g are alternate exterior angles and they are equal to one another.

How to identify alternate interior angles and their properties

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

How to identify alternate exterior angles and their properties

How to find an angle using alternate exterior angle

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