Lesson Video: Angle Relationships Mathematics

Video Transcript

In this video, we will learn how to identify different types of angles and use relationships between their measures to solve problems. There are many different ways of describing angles. For example, we can describe the measure of an angle using words such as acute, obtuse, or reflex. And we can also describe the measure of an angle using a number such as 60 degrees. These are not the only ways of describing angles however. We can also describe different relationships angles have with each other. Let’s begin by considering the following angles.

We can see from the diagram that the measure of angle 𝐴𝐵𝐷 is equal to the measure of angle 𝐴𝐵𝐶 plus the measure of angle 𝐶𝐵𝐷. The two angles 𝐴𝐵𝐶 and 𝐶𝐵𝐷 are known as adjacent angles, and we can formally define this as follows. Two angles are adjacent if they share the same vertex, have a common side, and their distinct sides lie on opposite sides of the common side. In the diagram drawn, our two angles share the same vertex 𝐵. They have a common side 𝐵𝐶. And their distinct sides 𝐵𝐷 and 𝐵𝐴 do lie on opposite sides of the common side. We can therefore conclude that angle 𝐴𝐵𝐶 and angle 𝐶𝐵𝐷 are adjacent angles. We will now look at an example where we need to identify adjacent angles in a diagram and then add their measures to find a measure of the combined angle.

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

We begin by recalling the definition of adjacent angles. Two angles are adjacent if they have the same vertex, have a common side, and their distinct sides lie on opposite sides of the common side. We can see from the diagram that we have three angles that measure 22, 64, and 88 degrees. All three angles share a common vertex. However, it is only the angles of measures 64 degrees and 88 degrees that share a common side. The distinct sides of these two angles do lie on opposite sides of the common side. We can therefore conclude that 64 degrees and 88 degrees are the adjacent angles. We are asked to find the sum of these. Since 64 plus 88 is 152, the sum of the two adjacent angles in the diagram is 152 degrees.

It is worth noting that adding the measures of the two adjacent angles is equivalent to finding the measure of the combined angle, that is, the angle between the distinct sides of the adjacent angles.

Before moving on to our next example, we will consider two further angle relationships. These are known as complementary and supplementary angles. Firstly, two angles are complementary if their measures sum to 90 degrees. In other words, they sum to form a right angle as shown. In a similar way, two angles are supplementary if their measures sum to 180 degrees. In this case, the two angles form a straight line. We will now look at an example where we need to use the properties of complementary and supplementary angles.

Determine the values of 𝑥 and 𝑦.

We begin by noticing that the two angles with measures 62 degrees and 𝑥 degrees form a right angle. This means that they are complementary angles as complementary angles sum to 90 degrees. We can write this as an equation. 𝑥 degrees plus 62 degrees is equal to 90 degrees. And since all the angles have the same unit, 𝑥 plus 62 is equal to 90. We can then subtract 62 from both sides of the equation. And this means that 𝑥 is equal to 28.

In order to determine the value of 𝑦, we note that all four angles lie on a straight line. The angle of measure 𝑦 degrees combines with the angle that is the sum of 𝑥 degrees, 62 degrees, and 37 degrees to make a straight angle. In other words, these angles are supplementary since supplementary angles sum to 180 degrees. We have the equation 𝑦 degrees plus 𝑥 degrees plus 62 degrees plus 37 degrees is equal to 180 degrees. And since 𝑥 is equal to 28, this simplifies as shown. Adding 28, 62, and 37 gives us 127. So our equation becomes 𝑦 plus 127 is equal to 180. We can then subtract 127 from both sides, giving us 𝑦 is equal to 53. The values of 𝑥 and 𝑦 are 28 and 53, respectively.

Let’s now consider two further angle relationships. Firstly, let’s consider angles around a point. The diagram drawn shows two adjacent straight angles. And since straight angles measure 180 degrees, the angles around a point must sum to 360 degrees. It is important to note that this is true of any set of angles at a point even if they don’t include straight angles. Angles on the opposite side of the intersection of two straight lines are called vertically opposite angles. Vertically opposite angles are equal in measure. In the diagram drawn, angle 𝑎 is equal to angle 𝑐. And angle 𝑏 is equal to angle 𝑑.

It is also worth noting that angles 𝑎 and 𝑏 are supplementary angles. They sum to 180 degrees. The same is true of angles 𝑎 and 𝑑, angles 𝑐 and 𝑑, and angles 𝑐 and 𝑏. We also note that all four angles 𝑎, 𝑏, 𝑐, and 𝑑 sum to 360 degrees. We will now consider an example where we need to use these properties.

What is the measure of angle 𝑅𝑀𝑆 in the following figure?

In this question, we need to calculate the measure of angle 𝑅𝑀𝑆 as shown in the diagram. To do this, we will recall a number of angle properties and relationships. Firstly, we note that 𝑄𝑇 is a straight line, and we recall that angles on a straight line sum to 180 degrees. This means that the measure of angle 𝑄𝑀𝑃 plus the measure of angle 𝑇𝑀𝑃 must equal 180 degrees. From the diagram, we see that angle 𝑇𝑀𝑃 measures 146 degrees. And subtracting this from both sides of our equation, we have the measure of angle 𝑄𝑀𝑃 is 34 degrees.

Next, we recall that vertically opposite angles are equal. This means that the measure of angle 𝑆𝑀𝑇 must be equal to the measure of angle 𝑄𝑀𝑃. And we already know this is equal to 34 degrees. We now have the measures of four of the five angles on the diagram. We will therefore use one final property. Angles at a point sum to 360 degrees. The four known angles sum to 304 degrees. And this means that the measure of angle 𝑅𝑀𝑆 plus 304 degrees is equal to 360 degrees. Subtracting 304 degrees from both sides, we have the measure of angle 𝑅𝑀𝑆 is equal to 56 degrees. And this is the final answer to this question.

We will now consider one final relationship between angles. The ray that splits an angle into two angles of equal measure is called the angle bisector. If we consider the angle 𝐴𝐵𝐶 as shown, then the line 𝐷𝐵 is known as the angle bisector if the measure of angle 𝐴𝐵𝐷 is equal to the measure of angle 𝐶𝐵𝐷. We will now consider one final example where we need to use this property.

In the following figure, find the measure of angle 𝐷𝑂𝐸.

We begin by recalling that the sum of the measures of the angles at a point is 360 degrees. And it is also important to note that we cannot assume that the line from point 𝐶 to point 𝐸 is a straight line. So we can therefore not use the properties of supplementary angles. This means that the five angles in our diagram sum to 360 degrees. We can write this as an equation as shown. The measure of angle 𝐷𝑂𝐸 plus the measure of angle 𝐴𝑂𝐸 plus 119 degrees plus 76 degrees plus 41 degrees is equal to 360 degrees. Summing the three known angles gives us 236 degrees, and we can therefore simplify our equation. We can then subtract 236 degrees from both sides such that the measure of angle 𝐷𝑂𝐸 plus the measure of angle 𝐴𝑂𝐸 is equal to 124 degrees.

At this stage, it may not be clear what to do next. However, let’s consider how the two unknown angles are labeled on the diagram. This notation tells us that the two angles are equal. The line 𝑂𝐸 is an angle bisector such that the measure of angle 𝐷𝑂𝐸 is equal to the measure of angle 𝐴𝑂𝐸. Since the two angles are equal, we can rewrite our equation as two multiplied by the measure of angle 𝐷𝑂𝐸 is equal to 124 degrees. And dividing through by two, the measure of angle 𝐷𝑂𝐸 is equal to 62 degrees. As this also means that angle 𝐴𝑂𝐸 measures 62 degrees, we could add these to our diagram and then check that all five angles sum to 360 degrees.

We will now finish this video by recapping the key points. We saw at the start of this video that two angles are adjacent if they share the same vertex, have a common side, and their distinct sides lie on opposite sides of the common side. We add the measures of adjacent angles to find the measure of the combined angle. We saw that complementary angles sum to 90 degrees. Supplementary angles sum to 180 degrees. And angles at a point or in a full turn sum to 360 degrees. We saw that angles on the opposite side of the intersection of two straight lines are called vertically opposite angles and that vertically opposite angles are equal in measure. Finally, we saw that the ray that splits an angle into two angles of equal measure is called the angle bisector.