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Consecutive Interior Angles

Last Updated : 19 Sep, 2023
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Consecutive Interior Angles are situated on the same sides of the transversal and in the case of parallel lines, consecutive interior angles add up to 180°, which implies the supplementary nature of Consecutive Interior Angles.

This article explores, almost all the possibilities related to Consecutive Interior Angles which are also called co-interior angles. This article covers a detailed expiation about Consecutive Interior Angles including, its definition, other angles related to transversal, and theorems related to Consecutive Interior Angles as well.

What are Consecutive Interior Angles?

A consecutive internal angle is a pair of non-adjacent interior angles that are located on the same side of the transversal. Things that appear next to each other are said to as ‘consecutive’. On the internal side of the transversal, consecutive interior angles are situated adjacent to each other. To identify them, look at the image below and the attributes of successive inner angles.

  • The vertices of consecutive inner angles vary.
  • They are situated between two lines.
  • They are on the same transverse side.
  • They have something in common.

Consecutive Interior Angles Definition

When a transversal intersects two parallel or non-parallel lines, the pairs of angles on the same side of the transversal and inside the pair of lines are called consecutive interior angles or co-interior angles.

Consecutive Interior Angles Example

Co-Interior-Angles

In the figure given above, each pair of angles such as 3 and 6, 4 and 5 (both are highlighted with the same colour in the illustration) are examples of Consecutive Interior Angles, as these are indicated on the same side of the transversal line l and lie between the lines m and n. 

Are Consecutive Interior Angles Congruent?

For any two angles to be congruent they need to be equal in measure, but as we already know there is no such property related to Consecutive Interior Angles which states their equality. Thus, Consecutive Interior Angles are not Congruent.

Read more about Congruence of Triangles.

Consecutive Interior Angles for Parallel Lines

Pairs of angles that are on the same side of a transversal line and meet two parallel lines are known as consecutive internal angles. They have a common vertex and are situated in the middle of the parallel lines. Interior angles that follow one another are supplementary if their measurements sum to 180 degrees. This geometric idea is crucial for a number of tasks, such as calculating unknown angles and comprehending the connections between the angles created by parallel lines.

Read more about Parallel Lines.

Properties of Consecutive Interior Angles

Certainly, the following are the bulleted properties of consecutive interior angles for parallel lines crossed by a transversal:

  • Consecutive Interior Angles adds up to 180°.
  • Consecutive Interior Angles are situated between the parallel lines and on the same side of the transversal.
  • Other angles are between them along the transversal; they are not next to one another.
  • Consecutive interior angles have similar sizes if the lines are parallel.
  • They create a linear pair with the transversal, which adds to their complementary character.
  • Lines that are parallel correspond to alternate internal angles on the other side of the transversal.

Consecutive Interior Angle Theorem

The successive interior angle theorem determines the relationship between the consecutive interior angles. The ‘consecutive interior angle theorem’ asserts that if a transversal meets two parallel lines, each pair of consecutive internal angles is supplementary, which means that the sum of the consecutive interior angles equals 180°.

Consecutive Interior Angle Theorem Proof

To understand the Consecutive Interior Angle Theorem, look at the illustration below.

Co-Interior-Angles-for-Parallel-Lines-

It is assumed that n and m are parallel, and o is the transversal.

∠2 = ∠6 (corresponding angles) . . . (i)

∠2 + ∠4 = 180° (Supplementary linear pair of angles) . . . (ii)

Substituting ∠2 for ∠6 in Equation (ii) yields

∠6 + ∠4 = 180°

Similarly, we may demonstrate that ∠3 + ∠5 = 180°.

∠1 = ∠5 (corresponding angles) . . . (iii)

∠1 + ∠3 = 180° (Supplementary linear pair of angles) . . . (iv)

When we substitute ∠1 for ∠5 in Equation (iv), we obtain

∠5 + ∠3 = 180°

As may be seen, ∠4 + ∠6 = 180°, and ∠3 + ∠5 = 180°

As a result, it is demonstrated that consecutive interior angles are supplementary.

Converse of Consecutive Interior Angle Theorem

According to the converse of the consecutive interior angle theorem, if a transversal intersects two lines in such a way that a pair of successive internal angles are supplementary, then the two lines are parallel.

Proof of Converse of Consecutive Interior Angle Theorem

The proof and converse of this theorem are provided below.

Using the same illustration,

∠6 + ∠4 = 180° (Consecutive Interior Angles) . . . (i)

Because ∠2 and ∠4 make a straight line,

∠2 + ∠4 = 180° (Supplementary linear pair of angles) . . . (ii)

Because the right sides of Equations (i) and (ii) are identical, we may equate the left sides of equations (i) and (ii) and express it as:

∠2 + ∠4 = ∠6 + ∠4

We obtain ∠2 = ∠6 when we solve this, which produces a similar pair in the parallel lines.

Thus, in the above figure, one set of related angles is equal, which can only happen if the two lines are parallel. This leads to the proof of the opposite of the consecutive interior angle theorem: if a transversal crosses two lines in a such that two subsequent internal angles are supplementary,

Consecutive Interior Angles of a Parallelogram

Because opposite sides of a parallelogram are always parallel, successive interior angles of a parallelogram are always supplementary. Examine the parallelogram below, where ∠A and ∠B, ∠B and ∠C, ∠C and ∠D, and ∠D and ∠A are successive internal angles. This can be explained as follows:

Co-Interior-Angles-for-Parallelogram

If we consider AB || CD and BC as transversal, then

∠B + ∠C = 180°

If we consider AB || CD and AD as transversal, then

∠A + ∠D = 180°

If we consider AD || BC and CD as transversal, then

∠C + ∠D = 180°

If we consider AD || BC and AB as transversal, then

∠A + ∠B = 180°

Read More,

Solved Examples of Consecutive Interior Angles

Example 1: If transversal cuts two parallel lines and a pair of successive interior angles measure (4x + 8)° and (16x + 12)°, calculate the value of x and the value of both consecutive interior angles.

Solution:

Because the supplied lines are parallel, the inner angles (4x + 8)° and (16x + 12)° are consecutive. These angles are additional according to the consecutive interior angle theorem.

As a result, (4x + 8)° + (16x + 12)° = 180°

⇒ 20x + 20 = 180°

⇒ 20x = 180° – 20°

⇒ 20x = 160°

 ⇒ x = 8°

Let us now substitute x for the values of the subsequent interior angles.

Thus, 4x + 8 = 4(8) + 8 = 40° and

16x + 12 = 16(8) + 12 = 140°

Thus, value of both consecutive interior angles 40° and 140°.

Example 2: The value of 3 is 85° and ∠6 is 110°. Now, check the ‘n’ and ‘m’ lines are parallel.

Co-Interior-Angles-for-Parallel-Lines-

Solution:

If the angles 110° and 85° in the above figure are supplementary, then the lines ‘n’ and ‘m’ are parallel.

However, 110° + 85° = 195°, indicating that 110° and 85° are NOT supplementary.

As a result, the given lines are NOT parallel, according to the Consecutive Interior Angles Theorem.

Example 3: Find the missing angles ∠3, ∠5, and ∠6. In the diagram, ∠4 = 65°.

Co-Interior-Angles-for-Parallel-Lines-

Solution:

Given: ∠4 = 65°, ∠4 and ∠6 are corresponding angles, therefore;

∠6 = 65°

By supplementary angles theorem, we know;

∠5 + ∠6 = 180°

∠5 = 180° – ∠6 = 180° – 65° = 115°

Since,

∠3 = ∠6

Therefore, ∠3 = 115°.

Practice Problems on Co-Interior Angles

Problem 1: In a pair of parallel lines cut by a transversal, if one co-interior angle measures (2x – 7)° and other is (x + 1)°, then what is the measure of both co-interior angles?

Problem 2: If angle P is a co-interior angle with angle Q on a pair of parallel lines, and angle Q measures 60°, what is the measure of angle P?

Problem 3: In a pair of parallel lines intersected by a transversal, if sum of both cosecutive interior angles is (3z-8)° and one of the co-interior angle is z. Then find the value of both cosecutive interior angles.

Consecutive Interior Angles – FAQs

1. Define Consecutive Interior Angles.

Consecutive interior angles are a pair of angles formed by two parallel lines and a transversal, located on the same side of the transversal and on the inside of the parallel lines.

2. What is the Theorem of Consecutive Interior Angles?

The Consecutive Interior Angles Theorem states that when two parallel lines are intersected by a transversal line, the consecutive interior angles formed on the same side of the transversal are supplementary, meaning their measures add up to 180°.

3. Is it Always Necessary to have Consecutive Interior Angles?

No, not all successive interior angles are supplementary. They are only useful when the transversal runs along parallel lines. It should be noted that successive internal angles can also be generated when a transversal crosses over two non-parallel lines, although they are not supplementary in this situation.

4. Give an Example of a Real-World Consecutive Interior Angle.

In actual life, you may witness sequential interior angles in a variety of places, such as a window grill with vertical and horizontal rods. They are made by intersecting two horizontal rods (two parallel lines) with a vertical rod (transversal).

5. What are the Three Co-Interior Angle Rules?

Three co-interior angle rules are:

  1. A collection of angle pairs created when transversal encounters parallel lines is known as co-interior angles.
  2. Inside the parallel lines are co-interior angles.
  3. The sum of co-interior angles is 180 degrees.

6. What is the Relationship between Consecutive Interior Angles and Parallel Lines?

Consecutive interior angles are the angles created on the internal side of a transversal when it crosses two parallel lines. The successive interior angles created when the transversal travels across two parallel lines are supplementary.

7. Do Consecutive Interior Angles add up to 180°?

Yes, in case of parallel lines consecutive interior angles add up to 180°. But for non parallel lines there is no exact value which these angles add up to.

8. What are Some Differences between Consecutive and Alternate Interior Angles?

Pairs of angles on the same side of a transversal line in respect to two parallel lines are known as consecutive internal angles. Pairs of angles that are on the outside of the transversal and within the parallel lines are known as alternate interior angles.  

While alternate angles are congruent if the lines are parallel, consecutive angles add up to 180 degrees. Both types have unique geometric characteristics and are important in geometry.

9. Is Co-Interior and Consecutive Interior angles is same?

Yes, co-interior and consecutive interior angles are names of the same angle pairs.

10. What is the Property of Co-Interior angles?

The property of co-interior angles is that they add up to 180 degrees when two parallel lines are intersected by a transversal.

11. What are Consecutive Interior vs. Exterior Angles?

The key differences between both consecutive interior and exterior angles are listed as follows:

Property Consecutive Interior Angles Consecutive Exterior Angles
Location On the same side of the transversal, between the parallel lines On opposite sides of the transversal, one outside and one inside the parallel lines
Relationship Supplementary (sum equals 180 degrees) Supplementary (sum equals 180 degrees)


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