Consecutive Interior Angles
In this minilesson, 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.
These 8 angles are classified into four types:
 Alternate Interior Angles
 Corresponding Angles
 Alternate Exterior Angles
 Consecutive Interior Angles
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 nonadjacent 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 "cointerior angles" or "sameside interior angles."
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.
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.
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.
 The consecutive interior angles are nonadjacent and lie on the same side of the transversal.
 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.
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?
Solution
In the given figure, 125^{o} and 60^{o} are the consecutive interior angles if they are supplementary.
But \[125^\circ+60^\circ = 185^\circ\]
Thus, 125^{o} and 60^{o} 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 1^{st} Street and 2^{nd} Street, which are parallel.
However, Maple Avenue makes a \(40^\circ\) angle with 2^{nd} Street.
What is the measure of angle \(x\)?
Solution
In the figure, 40^{o} and \(x\) are consecutive interior angles because the 1^{st} Street an 2^{nd} 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 L_{1} and L_{2} are parallel lines.
What is the value of \(\angle C\)?
Solution
Through C, draw a line parallel to L_{1} and L_{2}, as shown below.
We have:
\(\angle x\) = \(\angle \beta \) = 60^{o} (alternate interior angles)
\(\angle y\) = 180^{o} – 120^{o} (consecutive interior angles)
\(\angle y\) = 60^{o}
Thus, we get
\(\angle C\) = \(\angle x\) + \(\angle y\) = 120^{o}
Therefore,
\(\angle C =120^\circ\) 

Consider the following figure, in which L_{1} \(\parallel \) L_{2}:
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 minilesson 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 180^{o}.
2. What are consecutive interior angles?
Consecutive interior angles are the pair of nonadjacent interior angles that lie on the same side of the transversal.