Consecutive Interior Angles
When a transversal crosses any two parallel lines, it forms many angles like alternate interior angles, corresponding angles, alternate exterior angles, consecutive interior angles. Consecutive interior angles are formed on the inner sides of the transversal and are also known as cointerior angles or sameside interior angles.
Consecutive Interior Angles Definition
Consecutive interior angles are defined as the pair of nonadjacent interior angles that lie on the same side of the transversal. The word 'consecutive' refers to things that appear next to each other. Consecutive interior angles are located next to each other on the internal side of a transversal. Consecutive interior angles:
 Have different vertices
 Lie between two lines
 Are on the same side of the transversal
 Share a common side
In the above figure, \(L_1\) and \(L_2\) are parallel and \(L\) is the transversal.
By the consecutive interior angles definition, the pairs of consecutive interior angles in the above figure are:
 1 and 4
 2 and 3
Angles Formed by a Transversal
When a transversal crosses a pair of parallel lines, many pairs of angles are formed other than consecutive interior angles. They are corresponding angles, alternate interior angles, and alternate exterior angles. Observe the following figure and relate to the various pairs of angles and their properties given in the table.
The following table lists the properties of all the types of angles formed when a transversal crosses two parallel lines. Refer to the figure given above to relate to the angles.
Types of Angles  Properties  Name of the Angles in the Figure 

Corresponding Angles 
Corresponding angles are those angles that:
When a transversal intersects two parallel lines, the corresponding angles formed are always equal. 
In the above figure, ∠1 & ∠5, ∠2 &∠6, ∠4 & ∠8, ∠3 & ∠7 are all pairs of corresponding angles. 
Alternate Interior Angles 
Alternate interior angles are those angles that:
When a transversal intersects two parallel lines, the alternate interior angles formed are always equal. 
In the above figure,∠4 & ∠6 and ∠3 & ∠5 are pairs of alternate interior angles. 
Alternate Exterior Angles 
Alternate exterior angles are those angles that:
When a transversal intersects two parallel lines, the alternate exterior angles formed are always equal. 
In the above figure, ∠1 & ∠7 and ∠2 & ∠8 are pairs of alternate exterior angles. 
Consecutive Interior Angles 
Consecutive interior angles are those angles that:
When a transversal intersects two parallel lines, the consecutive interior angles are always supplementary. 
In the above figure, ∠4 & ∠5 and ∠3 & ∠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, that is, their sum is 180°.
Proof of Consecutive Interior Angle Theorem
Look at the following figure.
We know that the two lines are parallel, thus we have:
∠1 = ∠5 (Corresponding angles)  (1)
∠5 + ∠4 = 180° (Linear pair of angles)  (2)
Substituting ∠1 = ∠5 in equation (2), we get,
∠1+∠4=180°.
Similarly, we can show that, ∠2+∠3 = 180°.
Therefore, it is proved that consecutive interior angles are supplementary.
Converse of Consecutive Interior Angle Theorem
The converse of consecutive interior angle theorem states that if a transversal intersects two lines such that a pair of consecutive interior angles are supplementary, then the two lines are parallel. The proof of this theorem and its converse is shown below.
Referring to the same figure,
It is given, ∠1 + ∠4 = 180°  (3)
Since ∠5 and ∠4 form a linear pair of angles,
∠5 + ∠4 = 180°.  (4)
Equating the lefthand side of the equations (3) and (4), since their righthand sides are equal, we get,
∠1 + ∠4 = ∠5 + ∠4
Therefore, ∠1 = ∠5.
Thus, a pair of corresponding angles are equal, which can only happen if the two lines are parallel. Hence, the converse of consecutive interior angle theorem is proved.
Important Notes
Here are some important points to remember about consecutive interior angles.
 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.
Related Articles on Consecutive Interior Angles
Check out these interesting articles to know more about consecutive interior angles and their related topics.
 Line Segment
 Parallel Lines Formula
 Supplementary Angles
 Intersecting and NonIntersecting Lines
 Parallel lines
Consecutive Interior Angles Examples

Example 1: Are the following lines 'l' and 'm' parallel?
Solution:
In the given figure, if the angles 125° and 60° are supplementary, then it can be proved that the lines 'l' and 'm' are parallel.
But 125° + 60° = 185°, which means that 125° and 60° are NOT supplementary.
Thus, as per the "Consecutive Interior Angle Theorem," the given lines are NOT parallel. Therefore the lines 'l' and 'm' are NOT parallel.

Example 2: Using the consecutive interior angles theorem, find the value of angle 'x' if line 1 and line 2 are parallel?
Solution:
In the figure, 40° and ∠x are consecutive interior angles because the 'Line 1' and 'Line 2' are parallel.
By the consecutive interior angle theorem, ∠x and 40° are supplementary.
∠x + 40° = 180°
∠x = 180°  40°Therefore, ∠x = 140°.

Example 3: In the following figure, line 'l' is parallel to the line 'm'. Find the consecutive angles labeled in the following figure.
Solution
Since l  m, (2x + 4)° and (12x + 8)° are consecutive interior angles. By the consecutive interior angle theorem, these angles are supplementary.
Thus, (2x + 4) + (12x + 8) =180°
14x + 12 = 180°
14x = 180°  12°
14x = 168°
x = 12°Thus, the values of the consecutive interior angles are:
2x + 4 = 2(12) + 4 = 28°
12x + 8 = 12(12) + 8 = 152°.
FAQs on Consecutive Interior Angles
What are Consecutive Interior Angles?
Consecutive interior angles are formed when a transversal passes through a pair of parallel lines or nonparallel lines. They are formed on the interior sides of the two crossed lines at the point where the transversal intersects the two lines. If the lines that the transversal crosses are parallel, then, the pair of consecutive interior angles are supplementary.
How are Consecutive Angles Related to Parallel Lines?
Consecutive interior angles are the angles that are formed on the internal side of a transversal when it crosses two lines that are parallel. When the transversal passes through two parallel lines, then the consecutive interior angles that are formed are supplementary.
What is the Consecutive Interior Angles Theorem?
The consecutive interior angles theorem states that the transversal passing through two parallel lines makes two pairs of consecutive interior angles that are supplementary.
What is the Converse of Consecutive Interior Angles Theorem?
The converse of the consecutive interior angles theorem states that if a transversal intersects two lines such that a pair of consecutive interior angles are supplementary, then the two lines are parallel.
Are Consecutive Interior Angles Always Supplementary?
No, consecutive interior angles are not always supplementary. They are supplementary only when the transversal passes through parallel lines. It is to be noted that consecutive interior angles can also be formed when a transversal passes through two nonparallel lines, although in this case, they are not supplementary.
Are Consecutive Interior Angles Congruent?
Consecutive interior angles are NOT congruent. They are supplementary if a transversal passes through two parallel lines. It means that they add up to 180^{o}.
How are Consecutive Interior Angles Otherwise Called?
Consecutive interior angles are also known as "cointerior angles" or "sameside interior angles."
Are there any Other Angles Formed apart from Consecutive Interior Angles when a Transversal Passes Through Two Parallel Lines?
Yes, when a transversal passes through two parallel lines, there are other angles formed apart from consecutive interior angles, like, corresponding angles, alternate interior angles, alternate exterior angles.