# Lesson Explainer: Angle Relationships Mathematics

In this explainer, 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 in degrees, such as . These are not the only ways of describing angles; we can also describe different relationships angles have with each other.

We first recall that an angle is defined as the rotation required to move one ray onto another starting at the same point. We call the rays the sides of the angle and the shared point the vertex of the angle.

Let’s now consider the following angles.

If we wanted to measure , we could note that the rotation required to move onto is exactly the same as the rotation required to move onto added to the rotation required to move onto . In other words,

This occurs when a ray cuts the angle into smaller angles. Alternatively, we can think of the two smaller angles as sharing a side since both and contain side .

However, we need to be careful; we also need these angles to share a vertex, and we also want the distinct sides of the angles to be on opposite sides of the common side. We call these types of angles adjacent angles, but we need to check all of the criteria is met.

For example, and are not adjacent because they do not have a common vertex.

Similarly, and are not adjacent since their distinct sides lie on the same side of .

We can define adjacent angles formally as follows.

We say that two angles are adjacent if they share the same vertex and have a common side and their distinct sides lie on opposite sides of the common side.

This is a useful definition since we can add the measures of adjacent angles to determine the measure of the rotation of one side of the first angle to the nonadjacent side of the second angle.

Let’s now see an example of identifying adjacent angles in a diagram and adding their measures to find the measure of the combined angle.

### Example 1: Finding the Measure of the Angle Made of Two Adjacent Angles

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

We first recall that we call two angles adjacent if they share the same vertex and have a common side and their distinct sides lie on opposite sides of the common side.

We can see that we are given three angles in the diagram and that only the angles of measures and share a common side.

We can also see from the diagram that the angles share the same vertex and that their distinct sides are on opposite sides of the common side. Hence, the angles are adjacent.

We can add their measures together to get

In the previous example, we added the measures of two adjacent angles together. We can note that this is equivalent to finding the measure of the combined angle, that is, the angle between the distinct sides of the adjacent angles.

We can use this property in the reverse direction. For example, imagine we are given the measures of the combined angle and one of the adjacent angles as shown.

Since is adjacent to , we know that the sum of their measures is the measure of the combined angle:

We are given in the diagram that and that , so we can substitute these values into the equation and evaluate to get

There are other examples of related angles that are useful to consider. For example, if the sum of the measures of two angles is , then we say that the angles are complementary. In our above example, we can see that and that , so their sum has a measure of . Thus, these are complementary angles. It is worth noting that the angles do not need to be adjacent to be complementary. We only require that the sum of their measures be . However, we can always find complementary angles by splitting a right angle into two adjacent angles.

We also give a name to two angles whose measures sum to ; we call these supplementary angles. Once again, we do not need the angles to be adjacent. We only need their measures to sum to . For example, consider the following angles.

These angles are not adjacent; however, their measures sum to , so they are supplementary. We can always find supplementary angles by splitting a straight angle into two adjacent angles. This is equivalent to saying that two angles formed by a straight line, a point on the line, and a ray from this point are supplementary.

We define these terms formally as follows.

### Definition: Complementary and Supplementary Angles

We say that two angles are complementary if their measures sum to .

We say that two angles are supplementary if their measures sum to .

Let’s see an example of determining the measure of an angle using the measure of a complementary angle.

### Example 2: Finding the Measure of an Angle given Its Complementary Angle’s Measure

Given that , what is ?

We begin by noting that and are adjacent since they have a common vertex and share side and their distinct sides are on opposite sides of . We then recall that the sum of the measures of adjacent angles is equal to the measure of the combined angle: .

We can see in the diagram that , so and are complementary since their measures sum to . We have

Let’s now see an example involving the properties of adjacent, complementary, and supplementary angles.

### Example 3: Finding the Measures of Two Angles Using the Properties of Supplementary and Complementary Angles

Determine the values of and .

We want to determine the measures of the two unknown angles in the diagram. We can start by checking for relationships between the angles. For example, we can see in the diagram that the following angle is a right angle.

Since this right angle is split into two adjacent angles, the sum of their measures must be . In other words, the angle of measure and the angle of measure are complementary. Thus,

So, .

We can also notice that the angle of measure combines with the following angle to make a straight angle.

In other words, these angles are supplementary, so their measures sum to . We have

We know that , so we get

Hence, and .

It is also worth noting that two adjacent straight angles will combine to give an angle of measure . We can combine this with our knowledge of adjacent angles to show that the sum of the measures of adjacent angles around a point is . For example, the measures of the four angles in the following diagram sum to since they combine to make a full turn.

### Definition: Angles around a Point

The sum of the measures of the accumulative angles around a point is .

Another useful property worth noting is that if two adjacent angles are also supplementary then the distinct sides of the two angles must form a straight line.

Before we move on to our next example, there are two more angle relationships we will discuss. Let’s start by considering the measures of angles at a point of intersection between any two lines.

If we measure the angles, we will find that the opposite angles have equal measure. This is always true, and we can prove this result using our knowledge of adjacent and supplementary angles.

We can note that the angles of measures and are supplementary and that the angles of measures and are also supplementary. Hence,

For both equations to be true, we must have . The same result is true for the angles of measures and . We have

So, .

We call angles on the opposite side of the intersection of two straight lines “vertically opposite angles,” and this means we have just proven that vertically opposite angles are equal in measure.

### Definition: Vertically Opposite 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.

Let’s see an example of using the equality in measure of vertically opposite angles to determine the measure of an angle.

### Example 4: Finding the Measure of an Angle Using the Properties of Vertically Opposite Angles

What is in the following figure?

We want to determine the measure of from the diagram. To do this, let’s start by looking for relationships between the given angles. First, we can note that there is a vertex shared by all of the angles in the diagram, and we can recall that the sum of the measures of the angles around a point is . We can also note that is a straight angle, so its measure is , and that is a right angle, so its measure is . We can also see that and are vertically opposite, and this means that they are equal in measure.

We see that and are adjacent. We have

We know that and that . Thus,

Subtracting from sides of the equation gives

One final relationship between angles we want to discuss is when two adjacent angles are equal in measure. We can equivalently think of this as splitting an angle into two angles of equal measure. We call these bisections of an angle and we call the ray which bisects the angle an angle bisector.

### Definition: Angle Bisector

The ray that splits an angle into two angles of equal measure is called the angle bisector.

In our final example, we will determine the measure of an angle using the properties of angle bisectors and the properties of angles around a point.

### Example 5: Finding the Measure of One Angle Using the Properties of Angle Bisectors and Angles around a Point

In the following figure, find .

We begin by recalling that the sum of the measures of angles around a point is and that the angle bisector will split an angle into two angles of equal measure. We see that , so must bisect . Therefore, .

We can find the measure of using the sum of angles around a point. We have

We can then halve this value to find the measure of :

Let’s finish by recapping some of the important points from this explainer.

### Key Points

• We say that two angles are adjacent if they share the same vertex and have a common side and their distinct sides lie on opposite sides of the common side.
• We can add the measures of adjacent angles to determine the measure of the combined angle.
• We say that two angles are complementary if their measures sum to .
• We say that two angles are supplementary if their measures sum to .
• An angle that is a full turn has a measure of . The sum of accumulative angles around a point is .
• Angles on the opposite side of the intersection of two straight lines are called vertically opposite angles.
• Vertically opposite angles are equal in measure.
• The ray that splits an angle into two angles of equal measure is called the angle bisector.