What are Consecutive Angles?

Consecutive angles are a pair of angles that are formed in specific geometric configurations.

To understand what consecutive angles are, you need (1) two parallel lines, and (2) a third line that cuts across these two lines. This third line is called a transversal.

In the below picture, lines l1 and l2 are the parallel lines and t is the transversal.

When a transversal intersects two parallel lines, consecutive angles are formed. There are two types of consecutive angles: consecutive interior angles and consecutive exterior angles. Let us look at them in more detail.

Consecutive Interior Angles

Consecutive interior angles are the pair of angles on the same side of the transversal and inside the parallel lines (hence the name interior angles).

In the above figure, angles 1 and 2 are consecutive interior angles. Similarly, angles 3 and 4 are consecutive interior angles.

An interesting property of consecutive interior angles is that they add upto 180 degrees. In other words, they are supplementary.

Why are they supplementary? One way to convince yourself that this is true is to use the concept of alternate angles. You can see that angle 1 is equal to angle 4, and angle 2 is equal to angle 3, because they are alternate angle pairs. Further, angles 1 and 3 add up to 180, and angles 2 and 4 add up to 180. Thus, it must be true that angles 1 and 2 are supplementary (since angle 2 is the same as angle 3). Similarly, angles 3 and 4 are supplementary.

Consecutive Exterior Angles

On the other hand, consecutive exterior angles are the pair of angles on the same side of the transversal and outside the parallel lines, and they are also supplementary.

Let us continue marking consecutive exterior angles in the above diagram. We get:

Here, angles 5 and 6 are consecutive exterior angles. Similarly, angles 7 and 8 are consecutive exterior angles. Again, they add up to 180. You can convince yourself that this must be true with a similar idea. Angle 5 is equal to angle 3. Similarly, angle 6 is equal to angle 4. But we already know that angles 3 and 4 are supplementary! (because they are consecutive interior angles!). Thus the consecutive exterior angles 5 and 6 are supplementary. Similarly, consecutive exterior angles 7 and 8 are also supplementary.

Adjacent Angles in a Parallelogram = Consecutive Angles

In a parallelogram, all the adjacent angles are consecutive angles, and their sum is always 180 degrees.

Why are all adjacent angles consecutive? Because in a parallelogram, every side can be viewed as a transversal intersecting two parallel lines, and all the angles of a parallelogram are consecutive angles. Thus, we obtain a picture like this:

Thus, angles 1 and 2 are supplementary, and so are the pairs (2,3), (3,4), (4,1).

Because adjacent angles in a parallelogram are supplementary, we can verify that all of them add upto 360 degrees.

In general, the sum of the angles in a quadrilateral is always 360 degrees. This is a fundamental property of quadrilaterals, and it holds true for all types of quadrilaterals, regardless of whether they are a parallelogram or not. But in the case of a parallelogram, you also have the additional property that adjacent angles add upto 180 degrees.

For a square or a rectangle, you have an even additional property that all angles are 90 degrees.

For more general quadrilaterals, it is not true that adjacent angles must add upto 180 degrees but all angles must still add upto 360 degrees. For example, if the consecutive angles of a quadrilateral are given in the ratio form of 5:7:11:13, the sum of the angles can be found by using the general formula of 5x + 7x + 11x + 13x = 360, where x is the common factor. By solving for x, the measures of the consecutive angles can be derived. In this specific case, the sum of the consecutive angles is 360 degrees, and the individual angle measures are 50, 70, 110, and 130 degrees, respectively.Note that not all adjacent angle pairs add upto 180 degrees (only two pairs, do, namely 70+110=180, and 130+50=180, but other pairs do not).

To summarize, there are two types of consecutive angles: consecutive interior angles and consecutive exterior angles. Consecutive interior angles are the pair of non-adjacent interior angles that lie on the same side of the transversal, while consecutive exterior angles are the pair of angles on the same side of the transversal and outside the parallel lines. The consecutive interior angles are supplementary, meaning their sum is 180 degrees. The same is true for consecutive exterior angles.

Therefore, consecutive angles are not necessarily equal to each other, but their sum is 180 degrees. This concept is used to solve various problems in geometry and is applicable in real-life scenarios involving parallel lines and transversals

If you liked learning about consecutive angles, learn about quadrants in the 2D plane!

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