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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 60∘. 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 ∠𝐴𝐡𝐷, we could note that the rotation required to move 𝐡𝐴 onto 𝐡𝐷 is exactly the same as the rotation required to move 𝐡𝐴 onto οƒͺ𝐡𝐢 added to the rotation required to move οƒͺ𝐡𝐢 onto 𝐡𝐷. In other words, π‘šβˆ π΄π΅π·=π‘šβˆ π΄π΅πΆ+π‘šβˆ πΆπ΅π·.

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 ∠𝐴𝐡𝐢 and ∠𝐢𝐡𝐷 contain side οƒͺ𝐡𝐢.

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, ∠𝐴𝐡𝐢 and ∠𝐴𝐷𝐸 are not adjacent because they do not have a common vertex.

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

We can define adjacent angles formally as follows.

Definition: Adjacent Angles

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.

Answer

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 64∘ and 88∘ 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 64+88=152∘∘∘

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 ∠𝐴𝐡𝐢 is adjacent to ∠𝐢𝐡𝐷, we know that the sum of their measures is the measure of the combined angle: π‘šβˆ π΄π΅π·=π‘šβˆ π΄π΅πΆ+π‘šβˆ πΆπ΅π·.

We are given in the diagram that π‘šβˆ π΄π΅π·=70∘ and that π‘šβˆ π΄π΅πΆ=20∘, so we can substitute these values into the equation and evaluate to get 70=20+π‘šβˆ πΆπ΅π·π‘šβˆ πΆπ΅π·=70βˆ’20=50.∘∘∘∘∘

There are other examples of related angles that are useful to consider. For example, if the sum of the measures of two angles is 90∘, then we say that the angles are complementary. In our above example, we can see that π‘šβˆ π΄π΅π·=70∘ and that π‘šβˆ π΄π΅πΆ=20∘, so their sum has a measure of 90∘. 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 90∘. 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 180∘; we call these supplementary angles. Once again, we do not need the angles to be adjacent. We only need their measures to sum to 180∘. For example, consider the following angles.

These angles are not adjacent; however, their measures sum to 180∘, 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 90∘.

We say that two angles are supplementary if their measures sum to 180∘.

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 π‘šβˆ π΄π‘‚π΅=75∘, what is π‘šβˆ π΅π‘‚πΆ?

Answer

We begin by noting that βˆ π΄π‘‚π΅ and βˆ π΅π‘‚πΆ are adjacent since they have a common vertex and share side οƒͺ𝑂𝐡 and their distinct sides are on opposite sides of οƒͺ𝑂𝐡. We then recall that the sum of the measures of adjacent angles is equal to the measure of the combined angle: βˆ π΄π‘‚πΆ.

We can see in the diagram that π‘šβˆ π΄π‘‚πΆ=90∘, so βˆ π΄π‘‚π΅ and βˆ π΅π‘‚πΆ are complementary since their measures sum to 90∘. We have π‘šβˆ π΄π‘‚πΆ=π‘šβˆ π΄π‘‚π΅+π‘šβˆ π΅π‘‚πΆ90=75+π‘šβˆ π΅π‘‚πΆπ‘šβˆ π΅π‘‚πΆ=90βˆ’75=15.∘∘∘∘∘

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 π‘₯ and 𝑦.

Answer

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 90∘. In other words, the angle of measure π‘₯∘ and the angle of measure 62∘ are complementary. Thus, π‘₯+62=90π‘₯=90βˆ’62=28.∘∘∘∘∘∘∘

So, π‘₯=28.

We can also notice that the angle of measure π‘¦βˆ˜ combines with the following angle to make a straight angle.

In other words, these angles are supplementary, so their measures sum to 180∘. We have 𝑦+π‘₯+62+37=180.∘∘∘∘∘

We know that π‘₯+62=90∘∘∘, so we get 𝑦+90+37=180𝑦=180βˆ’90βˆ’37=53.∘∘∘∘∘∘∘∘∘

Hence, π‘₯=28 and 𝑦=53.

It is also worth noting that two adjacent straight angles will combine to give an angle of measure 360∘. 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 360∘. For example, the measures of the four angles in the following diagram sum to 360∘ 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 360∘.

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 π‘Ž and 𝑐 are supplementary and that the angles of measures 𝑐 and 𝑏 are also supplementary. Hence, π‘Ž+𝑐=180,𝑏+𝑐=180.∘∘

For both equations to be true, we must have π‘Ž=𝑏. The same result is true for the angles of measures 𝑐 and 𝑑. We have π‘Ž+𝑐=180,π‘Ž+𝑑=180.∘∘

So, 𝑐=𝑑.

We call angles on the opposite side of the intersection of two straight lines β€œvertically 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 π‘šβˆ π‘…π‘€π‘† in the following figure?

Answer

We want to determine the measure of βˆ π‘…π‘€π‘† 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 360∘. We can also note that βˆ π‘„π‘€π‘‡ is a straight angle, so its measure is 180∘, and that βˆ π‘„π‘€π‘… is a right angle, so its measure is 90∘. We can also see that βˆ π‘ƒπ‘€π‘‡ and βˆ π‘„π‘€π‘† are vertically opposite, and this means that they are equal in measure.

We see that βˆ π‘„π‘€π‘… and βˆ π‘…π‘€π‘† are adjacent. We have π‘šβˆ π‘„π‘€π‘†=π‘šβˆ π‘„π‘€π‘…+π‘šβˆ π‘…π‘€π‘†.

We know that π‘šβˆ π‘„π‘€π‘†=146∘ and that π‘šβˆ π‘„π‘€π‘…=90∘. Thus, 146=90+π‘šβˆ π‘…π‘€π‘†.∘∘

Subtracting 90∘ from sides of the equation gives π‘šβˆ π‘…π‘€π‘†=146βˆ’90=56.∘∘∘

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 π‘šβˆ π·π‘‚πΈ.

Answer

We begin by recalling that the sum of the measures of angles around a point is 360∘ and that the angle bisector will split an angle into two angles of equal measure. We see that π‘šβˆ π΄π‘‚πΈ=π‘šβˆ π·π‘‚πΈ, so οƒͺ𝑂𝐸 must bisect βˆ π΄π‘‚π·. Therefore, π‘šβˆ π΄π‘‚πΈ=π‘šβˆ π·π‘‚πΈ=12Γ—π‘šβˆ π΄π‘‚π·.

We can find the measure of βˆ π΄π‘‚π· using the sum of angles around a point. We have 360=41+76+119+π‘šβˆ π΄π‘‚π·π‘šβˆ π΄π‘‚π·=360βˆ’41βˆ’76βˆ’119=124.∘∘∘∘∘∘∘∘∘

We can then halve this value to find the measure of βˆ π·π‘‚πΈ: π‘šβˆ π·π‘‚πΈ=12Γ—124=62.∘∘

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 90∘.
  • We say that two angles are supplementary if their measures sum to 180∘.
  • An angle that is a full turn has a measure of 360∘. The sum of accumulative angles around a point is 360∘.
  • 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.

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