## Consecutive Interior Angles

Table of Contents

## Definitions

When two lines are crossed by another line (Transversal), **Consecutive Interior Angles** are the pairs of angles on one side of the transversal, but inside the two lines.

In this example, these are Consecutive Interior Angles:

(*c* and *f*) and (*d* and *e*)

The 2 angles in purple (d and e) make one pair of consecutive interior angles, and the other 2 angles in red (c and f) make another pair of consecutive interior angles. Both pairs are between the 2 lines and are both on the same side of the transversal.

Transversal is the line crossing the other two lines.

**Identifying Consecutive Interior Angles**

How to identify angles that are consecutive interior angles? The problem will ask you which other angle is the consecutive interior angle to a particular angle. For example:

**Question 1**. Which other angle is the consecutive interior angle to a angle **d** ?

Look at the picture and choose the other angle that matches up with the angle in question.

In this example, the angle that pairs with angle **d** is angle **e**, because they are on the same side of the transversal and also between the 2 lines.

**Question 2**. Which other angle is the consecutive interior angle to a angle **b** ?

Look at the picture and find that angle **b** is outside the 2 lines. There is no consecutive interior angle to angle **b** because angle **b** itself is not part of a pair.

When the two lines are parallel, any pair of Consecutive Interior Angles add to 180

^{o}(supplementary).

**Proof**

Given: *1* ∥ *2* , and *3* is a transversal

Prove: *m*∠*c + m*∠*f = m*∠*d + m*∠*e =*180^{o}^{}

Step | Statement | Reason |

1 | 1 ∥ 2 , and 3 is a transversal | Given |

2 | ∠c and ∠b form a linear pair ∠ d and ∠a form a linear pair | Definition of linear pair |

3 | m∠c + m∠b =180^{o}m∠d + m∠a =180^{o} | Supplement Postulate |

4 | ∠b = ∠ f ∠ a = ∠e | Corresponding Angles Theorem |

5 | m∠c + m∠f =180^{o}m∠d + m∠e =180^{o} | Substitution Property |