Identifying and Understanding Consecutive Interior Angles

Consecutive interior angles are angles formed between a transversal line and two parallel lines. Knowing how to identify pairs of consecutive interior angles will help you in solving for unknown angles found within parallel lines.

What Are Consecutive Interior Angles?

Consecutive interior angles are angles formed when a transversal line cuts through two parallel lines. These are interior angles formed within the inner region of the two parallel lines. At the same, they lie non-adjacent to each other and are found on the same side. Use this quick guide when identifying consecutive interior angles:

• The pair of interior angles that lie within the parallel lines.
• They must not share the same vertex.
• They, however, lie on the same side of the parallel lines’ inner region.

Take a look at the two parallel lines, m and n, cut through by a transversal line shown above. You can always find four interior angles inside the parallel lines’ region such as the following: ∠3, ∠4, ∠5, and ∠6. Consecutive interior angles lie on the same side, so we have the following pairs of consecutive interior angles: ∠3 & ∠5 and ∠4 & ∠6.

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After learning how to identify consecutive interior angles, it’s time to test your knowledge by applying what you’ve learned so far!

Example 1

A transversal line passes through the pair of parallel lines, s and t. Eight angles are formed as shown above. Which of the following pair of angles are consecutive interior angles?

A. ∠a and ∠e

B. ∠a and ∠g

C. ∠d and ∠f

D. ∠c and ∠f

First, identify the four interior angles that you can find from the parallel lines. These are the angles found within their inner region. Hence, we have the following angles: ∠c, ∠d, ∠e, and ∠f. This means that ∠a & ∠e and ∠a & ∠g couldn’t possibly be pairs of consecutive interior angles eliminating A and B.

While the pair of angles, ∠c and ∠f, are interior angles, they are not found on the same side. Now, taking a look at C: ∠d and ∠f are interior angles lying on the same side confirming that the two are consecutive interior angles. This shows that C is the correct answer.

Consecutive Interior Angle Theorem

The consecutive interior angle theorem states that any pair of consecutive interior angles are supplementary (or add up to 180°).

This means that given the consecutive interior angles, ∠3 & ∠5 and ∠4 & ∠6, each pair will add up to 180°.

∠3 + ∠5 = 180°

∠4 + ∠6 = 180°

Keep this property in mind when solving problems involving parallel lines and their angles. Remember that any pair of consecutive interior angles will always be supplementary.

Example 2

If ∠c has an angle measure of 70°, what is the measure of ∠e?

The angles, ∠c and ∠e, are consecutive interior angles, so they are supplementary. This means that the sum of their angle measures is equal to 180°.

Hence, ∠e has an angle measure of 110°.