Consecutive angles are the angles that formed when a transversal intersects two parallel lines. Each angle from a pair of consecutive angles lies on each of the parallel lines on any one side of the transversal (either interior or exterior). Let us learn more about it in detail in this article.
|1.||Consecutive Angles Definition|
|2.||Types of Consecutive Angles|
|3.||Consecutive Angles Theorem|
|4.||Consecutive Angles in a Parallelogram|
|5.||FAQs on Consecutive Angles|
Consecutive Angles Definition
When two parallel lines are crossed over by a transversal, there are two sets of angles formed on each of the parallel lines. Those sets of angles can be further divided into two sections 1 and 2, as shown in the figure below.
Here 'l' and 'm' represent the parallel lines and 'q' represents the transversal. The pair of supplementary angles on the same relative position (either on the interior side of the transversal or on the exterior side) on any one of the sections of the transversal is known as consecutive angles.
The pair of consecutive angles for section 2 in the above figure can be named as (∠A,∠B) and (∠C,∠D). Similarly, for section 1, (∠E,∠F) and (∠G,∠H) are the pair of consecutive angles.
Types of Consecutive Angles
There are two types of consecutive angles according to their position with relation to parallel lines and the transversal. i.e.,
- Consecutive interior angles
- Consecutive exterior angles
Consecutive Interior Angles
These consecutive angles lie on the interior region of the two parallel lines and on the same side of the transversal. They are also known as same side interior angles or co-interior angles. The pair of consecutive interior angles for the two sections in the above figure can be named (∠A,∠B) and (∠E,∠F).
Note: Interior consecutive angles are supplementary angles, i.e., they add up to 180°. This can be proved by the consecutive interior angles theorem which states that "If a transversal intersects two parallel lines, each pair of interior consecutive angles are supplementary (their sum is 180°)."
Consecutive Exterior Angles
These consecutive angles lie on the outside or exterior region of the two parallel lines and on the same side of the transversal. The pair of consecutive exterior angles for the two sections in the above figure can be named (∠C,∠D) and (∠G,∠H).
Consecutive Angles Theorem
Consecutive angles theorem or consecutive interior angles theorem states that "Two consecutive interior angles are always supplementary angles". In other words, two interior consecutive angles always add to 180°.
Let us look at the proof of consecutive angles theorem.
Refer to the above figure, we have,
∠E = ∠H (corresponding angles)
∠F + ∠H = 180° (linear pair)
From the above two equations, ∠E + ∠F = 180°.
Similarly, we can show that ∠A + ∠B = 180°.
► Can you figure out whether the converse of the consecutive angles theorem is true or not?
Consecutive Angles in a Parallelogram
All the pairs of adjacent angles in a parallelogram are consecutive angles. The opposite sides of a parallelogram are equal and parallel to each other. Refer to the figure given below, PQ || RS, so we can say that QR and PS act as transversals on two parallel line segments PQ and RS. It makes ∠QPS and ∠PSR a pair of consecutive angles. Similarly, we can state that all the adjacent angles in a parallelogram are consecutive angles. Look at the image given below to know all the consecutive angles in a parallelogram.
Related Articles on Consecutive Angles
Check these interesting articles related to consecutive angles in geometry.
Consecutive Angles Examples
Example 1: Two consecutive angles of a parallelogram are in the ratio of 1:8. Can you find out the value of the smaller angle?
Solution: Let the smaller angle be 'x', the bigger angle be '8x'. Since ∠A and ∠B are consecutive angles, ∠A+∠B=180°.
This implies, x + 8x = 180°.
9x = 180°
x = 20°
Therefore, the value of the smaller angle is 20°.
Example 2: Sean wants to find if the two lines 'p' and 'q' shown in the figure are parallel or not?
Solution: In the given figure, both the marked angles are interior angles. If p and q are parallel lines, then they must follow the consecutive angles theorem, which means the angles must be supplementary. But 150° + 20° ≠ 180°. This shows that they are not supplementary angles. Therefore, p and q are not parallel.
Example 3: Mark was given a parallelogram with one of the angles as ∠W=(a−10)°, where a = 70. Can you help Mark find out the value of ∠Y?
Solution: Given, ∠W=(a−10)° and a=70. Since ∠W and ∠X are consecutive angles, we can write,
∠W + ∠X = 180°
On substituting the given values, we get,
(a−10) + ∠X = 180
70 - 10 + ∠X = 180
∠X = 180−60
∠X = 120°
Similarly, ∠X and ∠Y are consecutive angles, hence we can write, ∠X + ∠Y=180°. On substituting the values, we get,
120° + ∠Y=180°
∠Y = 180° - 120°
∠Y = 60°
[An interesting result is seen here. i.e., ∠W=∠Y.]
Therefore, the value of ∠Y is 60°.
FAQs on Consecutive Angles
What are Consecutive Angles?
A pair of angles formed on each side of the transversal when it interacts with two parallel lines are known as consecutive angles. The angles in the same region (either interior or exterior) on each of the parallel lines form consecutive angles pairs.
Do Consecutive Angles Equal Each Other?
No, consecutive angles are not equal to each other. Their sum is 180 degrees. The only case when they can be equal is in a rectangle or when a transversal interacts with two parallel lines at 90 degrees angle. In that case, each of the angles in a consecutive angles pair is 90 degrees.
What are Consecutive Angles in a Parallelogram?
In a parallelogram, all the adjacent angles are consecutive angles. Their sum is always 180 degrees. So, in any parallelogram, we can have four pairs of consecutive angles.
Do Consecutive Angles Add up to 180?
Yes, as per the consecutive interior angles theorem, the sum of two interior consecutive angles is always 180 degrees. So, we can say that consecutive interior angles add up to 180 degrees.
How to Find the Measure of Consecutive Angles?
As we discussed above, the sum of two consecutive interior angles is 180 degrees. So, if any one of the angles is given, we can easily find the other consecutive angle by subtracting the given angle from 180.
Are Consecutive Angles Equal to 90?
No, it is not true that consecutive angles are equal to 90 degrees. Their sum is 180 degrees. It is possible with square and rectangle where the measurement of each of the consecutive angles is 90 degrees.
What are Interior Consecutive Angles?
When two parallel lines are crossed by a transversal, then the pairs of angles on one side of the transversal but inside the two lines are called consecutive interior angles.
What are Exterior Consecutive Angles?
When two parallel lines are crossed by a transversal, then the pairs of angles on one side of the transversal but outside the two lines are called consecutive exterior angles.