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 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

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.

The Alternate Interior Angles Theorem states that

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

alt interior angles  alt interior angles

One way to find the alternate interior angles is to draw a zigzag line on the diagram. In the above diagrams, d and e are alternate interior angles. Similarly, c and f are also alternate interior angles.

Alternate Interior 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.

alt interior angles

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.

alt interior angles - big & small





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

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.






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.

The Alternate Exterior Angles Theorem states that

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

alt exterior angles

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.

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

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.








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