In this video, we will learn how to identify adjacent angles and solve their problems.

08:31

### Video Transcript

In this lesson, we’ll learn how to identify adjacent angles and solve related problems. If we begin by recalling what the word “adjacent” means, it should help us to work out how to spot adjacent angles.

Adjacent means next to. If two things are adjacent to one another, they’re next to each other. We just have to tighten that definition up a little when considering adjacent angles. It’s not good enough just to find two angles that are next to one another.

We say that adjacent angles are two angles that have a common vertex and a common side. The vertex of an angle is the endpoint of the rays that form the sides of the angle. And when we say that adjacent angles have a common vertex and a common side, we mean that the vertex point and the side are shared by the two angles. Let’s begin by saying what this might look like, and, more importantly, we’ll then move on to the common misconceptions when it comes to adjacent angles.

Are angle one and angle two adjacent angles?

We say that angles are adjacent if they share a common vertex and a common side. The vertex is of course the endpoint of the rays that form the sides of the angle. Angle one is this one, and let’s highlight angle two in yellow. So do these two angles have a common vertex and a common side? Well, they have a vertex here that’s shared; it’s a common vertex. And in fact they have a common side here, so they share a vertex and a side, meaning yes, angle one and angle two are adjacent.

In fact, we should notice that angle one has two adjacent angles. These are two and three. Similarly angle two is adjacent to one and three, and angle three is therefore adjacent to one and two. And we can generalize and say that every angle can have two possible different angles adjacent to it, one on either side.

Are angle one and angle two adjacent angles?

We might begin by recalling that adjacent means next to and looking to see whether angles one and two are next to one another. That’s not quite enough though. We actually say that adjacent angles have a common vertex and a common side, where the vertex of an angle is the endpoint of the rays that form the sides of the angle.

Let’s begin by finding angle one in our diagram. It’s this one here. Angle two is here. We ended up actually having to overlap our pink and yellow lines, and so we see that they have a common side. It’s this one. But do they share a common vertex? Well, no, angle one’s vertex is here whereas angle two’s vertex is all the way up here. And so we say no, they are not adjacent angles.

Let’s consider another example of this form.

Are angle one and angle two adjacent angles?

We know that adjacent angles are two angles that have a common side and a common vertex, where the vertex of the angle is the endpoint of the rays that form its sides. Angle one is this angle here and angle two is over here. We had to overlap our pink and yellow lines, and that tells us that angle one and angle two have a common side. It’s this one. But do they share a common vertex? Well, no, angle one’s vertex is here and angle two’s vertex is here. And so we say no, angle one and angle two are not adjacent angles.

We can however find a pair of adjacent angles in our diagram. We notice that angle three and angle two share a common side. It’s this one. They also share a common vertex here, so angle two and angle three are adjacent angles.

In our next example, we’ll consider an extra definition.

Determine whether the angles, angle five and angle six, are adjacent, vertical, or neither adjacent nor vertical.

We know that adjacent angles are two angles that have a common vertex and a common side, where the vertex is the endpoint of the rays that form the sides of the angle. Vertical angles are angles which are opposite one another when two lines intersect at a vertex. For example, 𝑎 and 𝑏 in this diagram are vertical angles or sometimes you say vertically opposite. Let’s begin by identifying angle five. Angle five is this one. Then, angle six is here. We see they do share a common vertex. It’s this one right at the center of our diagram. They also share a common side; that’s this one. And so we see that angles five and angle six are adjacent.

If we were to find some vertical or vertically opposite angles in our diagram, we could say that five and two are vertical angles. It’s worth noting that angles cannot be both adjacent to one another and vertical or vertically opposite from one another.

In our next example, we’ll look at how we can solve problems by identifying pairs of adjacent angles.

Find the sum of the two adjacent angles from the given angles in the diagram.

Remember, adjacent angles are two angles that have a common vertex and a common side. We see we have one vertex in our diagram; it’s right at the center here. And then there are three angles to find. We have 22 degrees, 64 degrees, and 88 degrees. Each of these angles shares a common vertex. So we need to find the two that share also a common side. Well, we can see that there are two angles which share a common side, and that side is this one. These angles are 64 degrees and 88 degrees. The sum therefore of the two adjacent angles in our diagram is 64 plus 88, which is 152 or 152 degrees. The sum of our adjacent angles is 152 degrees.

In our final example, we’ll look at the relationship between the angle bisector and adjacent angles.

In the figure below, the measure of angle 𝐵𝐴𝐶 is equal to 30 degrees. If the ray from 𝐴 through 𝐶 is an angle bisector, what is the measure of angle 𝐵𝐴𝐷?

Let’s begin by identifying first angle 𝐵𝐴𝐶. It’s the angle enclosed by the line segments between 𝐵 and 𝐴 and 𝐴 and 𝐶. And so this angle here is 30 degrees. Now, we’re told that the line from 𝐴 and passing through 𝐶 is an angle bisector. Now, a bisector cuts something in half, so an angle bisector divides an angle exactly in half. And this therefore means that angle 𝐶𝐴𝐷 must be equal to angle 𝐵𝐴𝐶. It’s also 30 degrees.

Now we’re looking to find angle 𝐵𝐴𝐷. That’s this one on our diagram. We can say that it’s equal to the measure of angle 𝐵𝐴𝐶 plus the measure of angle 𝐶𝐴𝐷. Or alternatively since those angles are equal, it’s two times the measure of angle 𝐵𝐴𝐶. That’s two times 30, which is equal to 60 degrees. The measure of angle 𝐵𝐴𝐷 is 60 degrees. In general, we can say that an angle’s bisector divides it into two equally sized adjacent angles.

In this video, we learned that adjacent angles are two angles that have a common vertex and a common side. Vertical or vertically opposite angles are the angles opposite each other when two lines intersect at a vertex. Finally, we saw that an angle’s bisector divides it into two equally sized adjacent angles.