Consecutive Angles – Definition, Types, Theorem, Examples, FAQs

What Are Consecutive Angles?

Consecutive angles are formed on the same side of the transversal when two parallel lines are crossed by a transversal. Depending on their relative positions, either between or outside the parallel lines, they are categorized as consecutive interior angles and consecutive exterior angles, respectively. 

In geometry, parallel lines can be defined as two lines in the same plane that are at equal distances from each other and never meet. They may be horizontal lines or vertical lines. Transversal is a line that intersects two parallel lines in two distinct points.                      

Consecutive Angles: Definition

When two parallel lines are intersected by a transversal, then the pairs of angles formed on the same side of the transversal (either in the interior or the exterior region of two parallel lines) are called “consecutive angles.”

In the given figure, the pairs of consecutive angles are

• ∠c and ∠e
• ∠d and ∠f
• ∠b and ∠h
• ∠a and ∠g

Types of Consecutive Angles

Based on the intersecting position of parallel lines and the transversal, there are two types of consecutive angles.

• Consecutive interior angles
• Consecutive exterior angles

Consecutive Interior Angles

The angles which lie on the interior part of the two parallel lines (between two parallel lines), on the same side of the transversal are known as consecutive interior angles. The consecutive interior angles are also referred to as “co-interior angles” or “same-side interior angles.”

In the given figure, the pair of consecutive interior angles are highlighted in the same colors.

• ∠3 and ∠5
• ∠4 and ∠6

Consecutive Exterior Angles

The angles which lie on the exterior part of the two parallel lines (outside the two parallel lines) and on the same side of the transversal are known as consecutive exterior angles. 

The consecutive exterior angles are also referred to as co-exterior angles or same-side exterior angles. Consecutive exterior angles are supplementary.

In the given figure, the pairs of consecutive exterior angles are highlighted in the same colors.

• ∠1 and ∠7
• ∠2 and ∠8

Consecutive Angles Theorem

Consecutive angles theorem states that if two parallel lines are cut by a transversal, then the pairs of consecutive interior angles formed are supplementary. 

From the above figure, we can write

∠3 + ∠5 = 180^o 

∠4 + ∠6 = 180^o

Converse of Consecutive Angles Theorem

Consecutive Angles in a Parallelogram

In a parallelogram, each pair of adjacent angles represent consecutive angles.

Here, the pair of adjacent angles are as follows

• ∠A and ∠B
• ∠C and ∠D
• ∠A and ∠D
• ∠B and ∠C

In a parallelogram, opposite sides are parallel. 

Two parallel lines AB and CD are cut by two transversals AD and BC. 

Also, two parallel lines AD and BC are cut by transversals AB and CD.

If two parallel lines are cut by a transversal, then the pairs of consecutive interior angles formed are supplementary. Thus, the sum of adjacent angles of a parallelogram is equal to 180.

•  ∠A + ∠B = 180^o
• ∠C + ∠D = 180^o
• ∠A + ∠D = 180^o
• ∠B + ∠C = 180^o

Facts about Consecutive Angles

Conclusion

In this article, we learned about the consecutive angles, their types, theorems, and facts. We also learned about consecutive angles in a parallelogram. Let’s solve a few examples and MCQs for better comprehension.

Solved Examples on Consecutive Angles

1. Identify the pairs of consecutive interior and exterior angles in the given figure.

Solution:

Pairs of consecutive interior angles: ∠3 and ∠5; ∠4 and ∠6.

Pairs of consecutive exterior angles: ∠1 and ∠7; ∠2 and ∠8.

2. A pair of consecutive interior angles formed by lines p and q are 106 and  73. Check whether two lines ‘p’ and ‘q’ are parallel or not.

Solution:

For the lines p and q to be parallel, the given consecutive interior angles must satisfy the converse of the consecutive angles theorem.

In simple words, if the sum of the measures of given angles is 180^o, the lines are parallel.

Here, 106^o + 73^o = 179^o ≠ 180^o.

Hence, the lines p and q are not parallel.

3. In the given figure, the lines l and m are two parallel lines. Find the value of ‘x’.

Solution:

In the figure, angles x and 57^o form pairs of consecutive exterior angles. Thus, their sum must be equal to 180.

x + 57^o = 180^o

x = 180^o − 57^o

Therefore, x = 123^o

4. In the given figure, the lines l and m are two parallel lines, find the value of ‘y’.

Solution:

In the figure, angles y and 123^o form a pair of consecutive interior angles. Thus, they add up to 180^o.

y + 123^o = 180^o

y = 180^o − 123^o

Therefore, x = 57^o.

4. If the two consecutive interior angles of a parallelogram are given by (x + 60)^o and  (2x + 30)^o, then which type of parallelogram is it?

Solution:

We know that the sum of interior angles of a parallelogram = 180

So, (x + 60)^o + (2x + 30)^o = 180^o

3x + 90 = 180

3x = 180 − 90

3x = 90

Therefore, $x = \frac{90}{3} = 30$

Thus, (x + 60)^o = (30 + 60)^o = 90^o

(2x + 30)^o = [2(30) + 30]^o = 90^o

Opposite angles of a parallelogram are equal. Thus, all interior angles are right angles.

Thus, it is a rectangle.

Practice Problems on Consecutive Angles

Frequently Asked Questions about Consecutive Angles