Consecutive Interior Angles

Consecutive Interior Angles

In this mini-lesson, we will explore the world of consecutive interior angles by learning the meaning of consecutive interior angles.

We will also solve some example problems on consecutive interior angles by using the consecutive interior angles theorem.

Let's begin!

Look at the figure below. Can you observe two parallel lines?

When two parallel lines are intersected by a transversal, 8 angles are formed.

parallel lines cut by a transversal

These 8 angles are classified into four types:

We will study more about "Consecutive Interior Angles" here.

Lesson PLan

What Is the Meaning of Consecutive Interior Angles?

Consecutive Interior Angles Definition

Consecutive interior angles are the pair of non-adjacent interior angles that lie on the same side of the transversal.

Consecutive interior angles:

  • have different vertices
  • lie between two lines
  • and are on the same side of the transversal

"Consecutive interior angles" are also known as "co-interior angles" or "same-side interior angles."

Interior angles of parallel lines intersected by a transversal

In the above figure, \(L_1\) and \(L_2\) are parallel and \(L\) is the transversal.

By the consecutive interior angles definition,

Here, the pairs of consecutive interior angles in the above figure are:

  • 1 and 4
  • 2 and 3

What Are the Types of Intersection Angles?

When a transversal crosses a pair of parallel lines, you end up with 4 types of intersection angles.

Two parallel lines and a transversal

If you want to learn about each type of angles in detail, click on the "+" symbol that is against each heading.

1. Corresponding Angles

Corresponding angles are angles that:

  • have different vertices
  • lie on the same side of the transversal and lie above or below the lines
  • are always equal

When a transversal intersects two parallel lines, the corresponding angles formed are always equal.

In the above figure, \({\angle 1}\) & \({\angle 5}\), \({\angle 2}\) & \({\angle 6}\), \({\angle 4}\) & \({\angle 8}\), \({\angle 3}\) & \({\angle 7}\) are all pairs of corresponding angles.

2. Alternate Interior Angles

Alternate interior angles are angles that:

  • have different vertices
  • lie on the alternate sides of the transversal
  • lie between the interior of the two lines

When a transversal intersects two parallel lines, the alternate interior angles formed are always equal.

Here,\({\angle 4}\) & \({\angle 6}\) and \({\angle 3}\) & \({\angle 5}\) are pairs of alternate interior angles.

3. Alternate Exterior Angles

Alternate exterior angles are those angles that:

  • have different vertices
  • lie on the alternate sides of the transversal
  • are exterior to the lines

When a transversal intersects two parallel lines, alternate exterior angles formed are always equal.

Here,\({\angle 1}\) & \({\angle 7}\) and \({\angle 2}\) & \({\angle 8}\) are pairs of alternate exterior angles.

4. Consecutive interior Angles

Consecutive interior angles are those angles that:

  • have different vertices
  • lie between two lines
  • and are on the same side of the transversal

When a transversal intersects two parallel lines, the consecutive interior angles are always supplementary.

Here,\({\angle 4}\) & \({\angle 5}\) and \({\angle 3}\) & \({\angle 6}\) are pairs of consecutive interior angles.

Consecutive Interior Angle Theorem

The relation between the consecutive interior angles is determined by the consecutive interior angle theorem.

The "consecutive interior angle theorem" states that if a transversal intersects two parallel lines, each pair of consecutive interior angles are supplementary (their sum is 180\(^\circ\)).

Conversely, if a transversal intersects two lines such that a pair of consecutive interior angles are supplementary, then the two lines are parallel.

Here is the proof of this theorem and its converse.

Consecutive Interior Angle Theorem - Proof

Look at the following figure.

L1 and L2 are parallel; L is a transversal

We know that the two lines are parallel, thus we have:

\[ \begin{align} \angle 1& = \angle 5 \;\;\;\text{ (corresponding angles)} \\[0.3cm]\angle 5 + \angle4& = 180^\circ \;\text{ (linear pair)}\end{align} \]

From the above two equations, \[\angle 1 + \angle4 = 180^\circ\]

Similarly, we can show that \[\angle 2 + \angle 3 = 180^\circ \]

Converse of Consecutive Interior Angle Theorem - Proof

Let us assume that

\[ \begin{align}\angle 1 + \angle4 &= 180^\circ & \rightarrow (1) \end{align}\]

Since \(\angle 5\) and \(\angle 4\) forms linear pair,

\[ \begin{align}\angle 5 + \angle4 &= 180^\circ & \rightarrow (2) \end{align}\]

From (1) and (2),

\[ \angle 1 = \angle 5\]

Thus, a pair of corresponding angles is equal, which can only happen if the two lines are parallel.

Hence, the converse of consecutive interior angle theorem is proved.

Would you like to observe visually how the consecutive interior angles are supplementary?

Here is an illustration for you to try.

Click on "Go" to see how the "Consecutive Interior Angles Theorem" is true.

 
important notes to remember
Important Notes
  1. The consecutive interior angles are non-adjacent and lie on the same side of the transversal. 
  2. Two lines are parallel if and only if the consecutive interior angles are supplementary.

How to Find Consecutive Interior Angles?

We use the consecutive interior angles theorem to find the consecutive interior angles. Here is an example.

Example

In the following figure, \(l\| m\).

Find the consecutive angles labeled in the following figure.

Consecutive interior theorem examples: Two angles (2x+4) and (12x+8) are consecutive interior angles

Solution

Since \(l \| m\), \((2x+4)^\circ\) and \((12x+8)^\circ\) are consecutive interior angles.

By the consecutive interior angle theorem, these angles are supplementary.

Thus, \[ \begin{align} (2x+4)+(12x+8) &=180\\[0.2cm] 14x +12&=180\\[0.2cm] 14x&=168\\[0.2cm] x&=12 \end{align}\]

Thus, the values of the consecutive angles are:

\[\begin{align} 2x+4&= 2(12)+4 = 28^\circ\\[0.2cm]
12x+8&= 12(12)+8=152^\circ
\end{align}\]


Solved Examples

Example 1

 

 

Are the following lines \(l\) and \(m\) parallel?

lines l and m are parallel

Solution

In the given figure, 125o and 60o are the consecutive interior angles if they are supplementary.

But \[125^\circ+60^\circ = 185^\circ\]

Thus, 125o and 60o are NOT supplementary.

Thus, by the "Consecutive Interior Angle Theorem," the given lines are NOT parallel.

\(\therefore\) \(l\) and \(m\) are NOT parallel.
Example 2

 

 

Sixth Avenue runs perpendicular to the 1st Street and 2nd Street, which are parallel.

However, Maple Avenue makes a \(40^\circ\) angle with 2nd Street.

What is the measure of angle \(x\)?

Sixth Avenue runs perpendicular to the 1st Street and 2nd Street, which are parallel. However, Maple Avenue makes a 40 degree angle with 2nd Street.

Solution

In the figure, 40o and \(x\) are consecutive interior angles because the 1st Street an 2nd Street are parallel.

By the consecutive interior angle theorem, \(x\) and \(40^\circ\) are supplementary.

\[\begin{align} x+40&=180\\[0.2cm]
x&=140 \end{align} \]

Thus,

\(x=140^\circ\)
Example 3

 

 

Consider the following figure, in which L1 and L2 are parallel lines.

Alternate interior angles - parallel lines intersected by transversal

What is the value of \(\angle C\)?

Solution

Through C, draw a line parallel to L1 and L2, as shown below.

Co-interior angles - parallel lines intersected by transversal

We have:

\(\angle x\) = \(\angle \beta \) = 60o (alternate interior angles)

\(\angle y\) = 180o – 120o (consecutive interior angles)

\(\angle y\) = 60o

Thus, we get

\(\angle C\) = \(\angle x\) + \(\angle y\) = 120o

Therefore,

\(\angle C =120^\circ\)
 
Challenge your math skills
Challenging Questions
  1. Consider the following figure, in which L1 \(\parallel \) L2:

    parallel lines intersected by transversal - solved examples on transversals

    The angle bisectors of \(\angle BAX\) and \(\angle ABY\) intersect at C, as shown. Find the value of \(\angle ACB\).

Interactive Questions

Here are a few activities for you to practice.

Select/Type your answer and click the "Check Answer" button to see the result.

 
 
 
 
 

Let's Summarize

The mini-lesson targeted the fascinating concept of Consecutive Interior Angles. The math journey around Consecutive Interior Angles starts with what a student already knows, and goes on to creatively crafting a fresh concept in the young minds. Done in a way that not only it is relatable and easy to grasp, but also will stay with them forever. Here lies the magic with Cuemath.

We hope you enjoyed learning about Consecutive Interior Angles with the simulations and practice questions. Now you will be able to easily solve problems on consecutive interior angles, consecutive interior angles theorem, and consecutive interior angles definition.

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Frequently Asked Questions (FAQs)

1. Are consecutive interior angles congruent?

Consecutive interior angles are NOT congruent.

They are supplementary. It means that they add up to 180o.

2. What are consecutive interior angles?

Consecutive interior angles are the pair of non-adjacent interior angles that lie on the same side of the transversal.

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