Question Video: Determining If Angles Are Adjacent in a Figure Mathematics

Are ∠1 and ∠2 adjacent angles?

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Video Transcript

Are angle one and angle two adjacent angles?

In this question, we are given a figure containing three angles. And we are asked to determine if two of the angles on the figure are adjacent.

We can start by recalling that an angle is the combination of two rays that start at the same point called the vertex of the angle and the rotation required to rotate the rays about the vertex so that the sides of the angle are coincident. We can use this to mark angles one and two on the diagram and the vertices of each angle, where we note that since the size of the angles are not specified, we mean the angles with smaller measure.

To determine if the two angles are adjacent, we can recall that we say that two angles are adjacent if they satisfy three criteria. First, we need both angles to share the same vertex. Second, we need both angles to share a single common side. Third, we need the distinct sides of the angle to lie on opposite sides of the side common to both angles.

We have already shown that the two angles do not share the same vertex, so they cannot be adjacent. Hence, the answer is no, angles one and two are not adjacent angles, since they have distinct vertices.

Although this is enough to answer the question, it can be useful to see how to use these criteria to verify that two angles are adjacent. To do this, let’s mark angle three on the diagram and verify that it is adjacent to angle two. We can first note that both angles share a vertex. Remember, this is the initial point of the two rays that make up the sides of the angle. Next, we can see that the two angles share a common side that we have highlighted.

Finally, we want to check that the distinct sides of the angle lie on opposite sides of the common side. This is really just a check to make sure that the angles are not overlapping. Instead, we want them to be adjacent as shown. This confirms that angle two is adjacent to angle three. However, the answer to this question is no, angle one and angle two are not adjacent angles.

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