Consecutive Interior Angles

Consecutive interior angles are formed on the inner sides of the transversal and are also known as co-interior angles or same-side 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. Let us learn more about consecutive interior angles on this page.

Consecutive interior angles are defined as the pair of non-adjacent 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. Observe the following figure and the properties of consecutive interior angles to identify them.

• Consecutive interior angles have different vertices.
• They lie between two lines.
• They are on the same side of the transversal.
• They share a common side.

In the figure given above, L₁ and L₂ are parallel lines and T is the transversal. By the consecutive interior angles definition, the pairs of consecutive interior angles in the figure are:

• ∠1 and ∠4
• ∠2 and ∠3

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.

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 is supplementary, that is, the sum of the consecutive interior angles is 180°.

Proof of Consecutive Interior Angle Theorem

Observe the following figure to understand the Consecutive Interior Angle Theorem.

It is given that the two lines L₁ and L₂ are parallel, and T is the transversal. Since L₁ // L₂, it can be said that:

• ∠1 = ∠5 (corresponding angles with // lines) ----- (Equation 1)
• ∠1 + ∠4 = 180° (Linear pair of angles are supplementary) ---- (Equation 2)
• Substituting ∠1 as ∠5 in Equation (2), we get, ∠5 + ∠4 = 180°
• Similarly, we can show that, ∠3 +∠6 = 180°.
• ∠2 = ∠6 (corresponding angles with // lines) ----- (Equation 3)
• ∠2 + ∠3 = 180° (Linear pair of angles are supplementary) ---- (Equation 4)
• Substituting ∠2 as ∠6 in Equation (4), we get, ∠6 + ∠3 = 180°
• So, it can be seen that ∠4 + ∠5 = 180°; and ∠3 + ∠6 = 180°

Therefore, it is proved that consecutive interior angles are supplementary.

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, ∠5 + ∠4 = 180° (Consecutive Interior angles) ---------- (Equation 1)

Since ∠1 and ∠4 form a linear pair of angles,

∠1 + ∠4 = 180° ---------- (Linear pair of angles are supplementary) -----------(Equation 2)

Since the right-hand sides of Equation 1 and Equation 2 are equal, we can equate the left-hand side of the equations (1) and (2) and write it as:

∠1 + ∠4 = ∠5 + ∠4

If we solve this, we get ∠1 = ∠5 which forms a corresponding pair in the parallel lines.

Thus, one pair of corresponding angles is equal in the given figure, which can only happen if the two lines are parallel. Hence, the converse of consecutive interior angle theorem is proved, which says that if a transversal intersects two lines such that a pair of consecutive interior angles are supplementary, then the two lines are parallel.

Consecutive Interior Angles of a Parallelogram

We know that the opposite sides of a parallelogram are parallel, therefore, the consecutive interior angles of a parallelogram are always supplementary. Observe the following parallelogram in which ∠A and ∠B, ∠B and ∠C, ∠C and ∠D, and ∠D and ∠A are consecutive interior angles. This can be understood as follows:

• If we take AB // CD and BC as the transversal, then ∠B + ∠C = 180°
• If we take AB // CD and AD as the transversal, then ∠A + ∠D = 180°
• If we take AD // BC and CD as the transversal, then ∠C + ∠D = 180°
• If we take AD // BC and AB as the transversal, then ∠A + ∠B = 180°

Tips on Consecutive Interior Angles

Here are some important points to remember about consecutive interior angles.

• The consecutive interior angles are non-adjacent 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

• Exterior Angle Theorem
• Pairs of Angles
• Supplementary Angles
• Intersecting and Non-Intersecting Lines

What are Consecutive Interior Angles?

Consecutive interior angles are formed when a transversal passes through a pair of parallel lines or non-parallel 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 to Identify Consecutive Interior Angles?

Consecutive Interior angles can be identified with the help of the following properties:

• Consecutive interior angles are formed when any two straight lines are intersected by a transversal.
• They have different vertices.
• They lie between two lines.
• They are on the same side of the transversal.
• They share a common side.
• If the consecutive angles are formed between two parallel lines cut by a transversal, then they are supplementary.

What is the Consecutive Interior Angles Theorem?

The consecutive interior angles theorem states that if a transversal passes through two parallel lines, it makes two pairs of consecutive interior angles that are supplementary. In other words, the consecutive interior angles that are formed by two parallel lines intersected by a transversal add up to 180°.

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 non-parallel 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°.

What is Another Name for Consecutive Interior Angles?

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

What are the Other Angles Formed apart from Consecutive Interior Angles when a Transversal Passes Through Two Parallel Lines?

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.

What is the Difference Between Alternate Interior Angles and Consecutive Interior Angles?

The Alternate Interior Angles and Consecutive Interior Angles are different pairs of angles formed when two parallel lines are cut by a transversal. Alternate interior angles are located between two intersecting lines, but they are on the opposite sides of the transversal. Whereas, Consecutive interior angles are located on the inside of two lines on the same side of the transversal.

How are Consecutive Interior Angles Related?

If the consecutive interior angles are formed between two parallel lines cut by a transversal, then they are supplementary. This means they would add up to 180°.

How to find Consecutive Interior Angles?

We know the rule that if the consecutive interior angles are formed between two parallel lines cut by a transversal, then they are supplementary. This means if we know one of the consecutive angles in a pair, the other angle can be easily calculated by subtracting it from 180°. In other cases, if we have the consecutive angles as (20x + 5)° and (24x - 1)°, we can find the value of x with the following method. Since the consecutive angles are supplementary we can write it as (20x + 5)° + (24x - 1)° = 180°. This can be solved as, 44x + 4 = 180, and the value of x = 4. Now, the value of x can be substituted in the given expressions and the consecutive angles will be (20 × 4) + 5 = 85°, and (24 × 4) - 1 = 95°.

Give an example of Consecutive Interior Angles in Real Life.

In real life, consecutive interior angles can be seen in various places, for example, in a window grill with vertical and horizontal rods. They are formed when two horizontal rods (two parallel lines) are intersected by a vertical rod (transversal).

What do Consecutive Interior Angles Look Like?

Consecutive interior angles form a figure, somewhat like the letter U, where the inner angles are the consecutive interior angles.

How are Consecutive Interior 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.