Pairs of Angles

Related Angles (Complementary and Supplementary Angles)

Angles are not always considered in isolation. Sometimes, the relationship between two angles based on the sum of their measures is particularly significant. The concepts of complementary angles and supplementary angles describe such relationships, where the measures of two angles add up to specific values ($90^\circ$ or $180^\circ$). These relationships are frequently encountered in geometry problems and proofs.

Complementary Angles

Two angles are defined as complementary angles if the sum of their individual measures is exactly equal to $90^\circ$. When two angles are complementary, each angle is called the complement of the other angle.

If we have two angles, $\angle \text{A}$ and $\angle \text{B}$, they are complementary if:

For example, an angle of $30^\circ$ and an angle of $60^\circ$ are complementary because $30^\circ + 60^\circ = 90^\circ$. Here, $30^\circ$ is the complement of $60^\circ$, and $60^\circ$ is the complement of $30^\circ$. Similarly, $45^\circ$ and $45^\circ$ are complementary angles ($45^\circ + 45^\circ = 90^\circ$).

The angles do not need to be adjacent (sharing a common vertex and arm) to be complementary. They can be located anywhere in the plane as long as their measures add up to $90^\circ$.

Here is an illustration of two angles that are complementary, shown both adjacent and separate:

Supplementary Angles

Two angles are defined as supplementary angles if the sum of their individual measures is exactly equal to $180^\circ$. When two angles are supplementary, each angle is called the supplement of the other angle.

If we have two angles, $\angle \text{C}$ and $\angle \text{D}$, they are supplementary if:

For example, an angle of $70^\circ$ and an angle of $110^\circ$ are supplementary because $70^\circ + 110^\circ = 180^\circ$. Here, $70^\circ$ is the supplement of $110^\circ$, and $110^\circ$ is the supplement of $70^\circ$. A right angle ($90^\circ$) is supplementary to another right angle ($90^\circ$) because $90^\circ + 90^\circ = 180^\circ$.

Similar to complementary angles, supplementary angles do not need to be adjacent to be supplementary. They can be located anywhere in the plane as long as their measures add up to $180^\circ$.

Here is an illustration of two angles that are supplementary, shown both adjacent (forming a linear pair, which we will discuss next) and separate:

Key Distinction: Complementary vs. Supplementary

The key difference lies in the sum of the angle measures:

• Complementary: Sum of measures $= 90^\circ$. (Think "C" for Corner, $90^\circ$)
• Supplementary: Sum of measures $= 180^\circ$. (Think "S" for Straight, $180^\circ$)

Answer:

Answer:

Adjacent Angles

Beyond classifying angles by their measure or the sum of their measures (like complementary and supplementary), angles can also be related based on their spatial arrangement. Adjacent angles are a pair of angles that are "next to" each other, sharing specific elements. Understanding adjacent angles is crucial for working with angle properties on intersecting lines, within polygons, and other geometric figures.

Definition

Two angles are formally defined as adjacent angles if they satisfy all three of the following conditions simultaneously:

1. They have a common vertex. This means the point where the arms of the angles meet is the same for both angles.
2. They have a common arm. One of the rays forming the first angle is also one of the rays forming the second angle. This shared ray lies between the non-common arms.
3. Their non-common arms (the arms that are not shared) lie on opposite sides of the common arm. This ensures that the angles are "side-by-side" and do not overlap except along the common arm.

If any one of these three conditions is not met, the two angles are not adjacent.

Let's consider two angles, $\angle \text{ABC}$ and $\angle \text{CBD}$.

In this diagram:

• Both $\angle \text{ABC}$ and $\angle \text{CBD}$ share the vertex B. (Condition 1 met)
• Both $\angle \text{ABC}$ (formed by $\overrightarrow{\text{BA}}$ and $\overrightarrow{\text{BC}}$) and $\angle \text{CBD}$ (formed by $\overrightarrow{\text{BC}}$ and $\overrightarrow{\text{BD}}$) share the arm $\overrightarrow{\text{BC}}$. (Condition 2 met)
• The non-common arm of $\angle \text{ABC}$ is $\overrightarrow{\text{BA}}$, and the non-common arm of $\angle \text{CBD}$ is $\overrightarrow{\text{BD}}$. These two rays, $\overrightarrow{\text{BA}}$ and $\overrightarrow{\text{BD}}$, are clearly on opposite sides of the common arm $\overrightarrow{\text{BC}}$. (Condition 3 met)

Since all three conditions are satisfied, $\angle \text{ABC}$ and $\angle \text{CBD}$ are adjacent angles.

Properties and Special Cases of Adjacent Angles

Adjacent angles themselves do not necessarily have a fixed sum. Their sum depends on the relative positions of their non-common arms.

• Variable Sum: The sum of the measures of adjacent angles can be any value, such as less than $90^\circ$, exactly $90^\circ$, less than $180^\circ$, exactly $180^\circ$, or even more, depending on the figure.
• Adjacent Complementary Angles: If two angles are adjacent and their measures add up to $90^\circ$, then they are adjacent complementary angles. In this case, their non-common arms will form a right angle.
  

  Here, $\angle 1$ and $\angle 2$ are adjacent, and if $m\angle 1 + m\angle 2 = 90^\circ$, they are adjacent complementary.

• Adjacent Supplementary Angles (Linear Pair): A particularly important case is when two adjacent angles are supplementary, meaning their measures add up to $180^\circ$. In this specific case, their non-common arms lie on the same straight line and extend in opposite directions, forming a straight angle. This special pair of adjacent supplementary angles is called a Linear Pair (discussed in the next section).
  

  Here, $\angle \text{ABD}$ and $\angle \text{DBC}$ are adjacent. Since their non-common arms $\overrightarrow{\text{BA}}$ and $\overrightarrow{\text{BC}}$ form a straight line, $m\angle \text{ABD} + m\angle \text{DBC} = 180^\circ$. They are adjacent supplementary angles and form a linear pair.

• Angles Around a Point: If multiple angles are adjacent to each other and together they complete a full circle around a common vertex, the sum of their measures is $360^\circ$.

Example 1. In the figure below, identify if $\angle \text{PQR}$ and $\angle \text{RQS}$ are adjacent angles. State the common vertex and common arm.

Answer:

Example 2. In the figure below, are $\angle \text{ABC}$ and $\angle \text{ABD}$ adjacent angles? Justify your answer.

Answer:

Linear Pair of Angles

We have learned about adjacent angles, which share a vertex and a common arm. A particularly important and frequently occurring case of adjacent angles is the linear pair. This special configuration arises when the non-common arms of two adjacent angles form a straight line. The key property of a linear pair is that their measures always sum up to $180^\circ$, forming a straight angle.

Definition

A linear pair of angles is defined as a pair of adjacent angles whose two non-common arms are opposite rays and thus form a straight line. Since the non-common arms form a straight line, they together constitute a straight angle, which measures $180^\circ$.

Let $\angle \text{AOC}$ and $\angle \text{COB}$ be two angles sharing a common vertex O and a common arm $\overrightarrow{\text{OC}}$. They form a linear pair if the non-common arms $\overrightarrow{\text{OA}}$ and $\overrightarrow{\text{OB}}$ are opposite rays, forming the straight line AB.

Consider the following diagram:

In this figure, angles $\angle \text{AOC}$ and $\angle \angle \text{COB}$ are adjacent because they share:

• Common vertex: O
• Common arm: $\overrightarrow{\text{OC}}$
• Non-common arms ($\overrightarrow{\text{OA}}$ and $\overrightarrow{\text{OB}}$) on opposite sides of the common arm $\overrightarrow{\text{OC}}$.

Furthermore, their non-common arms, $\overrightarrow{\text{OA}}$ and $\overrightarrow{\text{OB}}$, extend in opposite directions along the same straight line, forming the straight line AB. Therefore, $\angle \text{AOC}$ and $\angle \text{COB}$ constitute a linear pair.

Linear Pair Axiom

The fundamental property of a linear pair of angles is expressed as an axiom (a statement accepted without proof) or a theorem, depending on the axiomatic system being used:

Axiom/Theorem: If a ray stands on a straight line, then the sum of the two adjacent angles so formed is $180^\circ$. Conversely, if the sum of two adjacent angles is $180^\circ$, then the non-common arms of the two angles form a straight line.

This means for any linear pair of angles, such as $\angle \text{AOC}$ and $\angle \text{COB}$ in the diagram above:

This property is often referred to as the Linear Pair Postulate or Linear Pair Property. It is a direct consequence of the fact that the angles form a straight angle, which by definition measures $180^\circ$.

Example 1. In the figure, line AB is a straight line intersecting line CD at O. Find the value of $x$.

Answer:

Vertically Opposite Angles

When two straight lines intersect each other at a single point, they create four angles around that point of intersection. These angles are related to each other in specific ways. One of the fundamental relationships between the angles formed by intersecting lines is the concept of vertically opposite angles. These are pairs of angles that are directly across from each other at the vertex where the lines cross.

Definition

When two straight lines intersect at a point, they form four angles. The pairs of angles that are located opposite to each other at the point of intersection are called vertically opposite angles. These angles share the vertex (the point of intersection) but do not share any arm.

Consider two distinct straight lines, say line AB and line CD, intersecting at point O.

In this diagram, the vertex of all four angles is the point O. The lines AB and CD create four angles: $\angle \text{AOC}$, $\angle \text{AOD}$, $\angle \text{BOC}$, and $\angle \text{BOD}$.

The pairs of vertically opposite angles are:

• $\angle \text{AOC}$ and $\angle \text{BOD}$
• $\angle \text{AOD}$ and $\angle \text{BOC}$

Notice that $\angle \text{AOC}$ and $\angle \text{BOD}$ are opposite to each other across the vertex O. Similarly, $\angle \text{AOD}$ and $\angle \text{BOC}$ are opposite to each other across the vertex O.

Theorem on Vertically Opposite Angles

A crucial property of vertically opposite angles is that they are always equal in measure. This is a significant theorem in geometry.

Theorem: If two lines intersect each other, then the vertically opposite angles are equal.

Proof:

Given: Two straight lines AB and CD intersect at point O.

To Prove:

• $\text{m}\angle \text{AOC} = \text{m}\angle \text{BOD}$
• $\text{m}\angle \text{AOD} = \text{m}\angle \text{BOC}$

Proof:

Consider the straight line AB and the ray $\overrightarrow{\text{OC}}$ standing on it.

Angles $\angle \text{AOC}$ and $\angle \text{BOC}$ form a linear pair.

By the Linear Pair Axiom, the sum of the measures of angles in a linear pair is $180^\circ$.

Now, consider the straight line CD and the ray $\overrightarrow{\text{OA}}$ standing on it.

Angles $\angle \text{AOC}$ and $\angle \text{AOD}$ form a linear pair.

By the Linear Pair Axiom, their sum is $180^\circ$.

From equation (i) and equation (ii), both sums are equal to $180^\circ$. Therefore, we can equate them:

Subtract $\text{m}\angle \text{AOC}$ from both sides of the equation:

This proves that the first pair of vertically opposite angles, $\angle \text{AOD}$ and $\angle \text{BOC}$, are equal in measure.

Similarly, let's prove the other pair is equal. Consider the straight line CD and the ray $\overrightarrow{\text{OB}}$ standing on it.

Angles $\angle \text{COB}$ and $\angle \text{BOD}$ form a linear pair.

By the Linear Pair Axiom, their sum is $180^\circ$.

From equation (i) and equation (iii), both sums are equal to $180^\circ$. Therefore, we can equate them:

Subtract $\text{m}\angle \text{BOC}$ (which is the same as $\text{m}\angle \text{COB}$) from both sides of the equation:

This proves that the second pair of vertically opposite angles, $\angle \text{AOC}$ and $\angle \text{BOD}$, are equal in measure.

Thus, the theorem is proved: vertically opposite angles formed by the intersection of two straight lines are equal.

Example 1. In the figure, lines PQ and RS intersect at O. If $\angle \text{POR} = 80^\circ$, find the measures of $\angle \text{SOQ}$, $\angle \text{POS}$, and $\angle \text{ROQ}$.

Answer:

Angles Formed by Intersecting Lines

We have already explored the concepts of adjacent angles, linear pairs, and vertically opposite angles. These concepts become particularly relevant and visible when two straight lines cross each other. The act of intersection creates a specific configuration of four angles around the point of intersection, and these angles have predictable relationships based on the definitions and theorems we've discussed.

Angles at the Intersection

Consider two distinct straight lines, let's call them $l_1$ and $l_2$, that intersect at a single point, say O. This point O is the common vertex for all the angles formed.

In the diagram above, lines $l_1$ and $l_2$ intersect at O, creating four angles around O. These angles are labelled as $\angle 1$, $\angle 2$, $\angle 3$, and $\angle 4$.

Relationships between these Angles:

Based on the definitions of adjacent angles, linear pairs, and vertically opposite angles, we can identify specific relationships among $\angle 1, \angle 2, \angle 3$, and $\angle 4$.

1. Adjacent Angles:

Adjacent angles share a common vertex and a common arm, with non-common arms on opposite sides of the common arm. Looking at the figure:

• $\angle 1$ and $\angle 2$ share vertex O and a common arm (the ray between $\angle 1$ and $\angle 2$). Their non-common arms lie on the line $l_2$ and are opposite rays. Thus, $\angle 1$ and $\angle 2$ are adjacent.
• $\angle 2$ and $\angle 3$ share vertex O and a common arm (the ray between $\angle 2$ and $\angle 3$). Their non-common arms lie on the line $l_1$ and are opposite rays. Thus, $\angle 2$ and $\angle 3$ are adjacent.
• $\angle 3$ and $\angle 4$ share vertex O and a common arm (the ray between $\angle 3$ and $\angle 4$). Their non-common arms lie on the line $l_2$ and are opposite rays. Thus, $\angle 3$ and $\angle 4$ are adjacent.
• $\angle 4$ and $\angle 1$ share vertex O and a common arm (the ray between $\angle 4$ and $\angle 1$). Their non-common arms lie on the line $l_1$ and are opposite rays. Thus, $\angle 4$ and $\angle 1$ are adjacent.

In total, there are four pairs of adjacent angles formed by two intersecting lines.

2. Linear Pairs:

A linear pair is a special case of adjacent angles where the non-common arms form a straight line. We know that the sum of angles in a linear pair is $180^\circ$. From the adjacent pairs identified above, we can see they also form linear pairs:

• $\angle 1$ and $\angle 2$: Non-common arms form line $l_2$. So, $\text{m}\angle 1 + \text{m}\angle 2 = 180^\circ$.
• $\angle 2$ and $\angle 3$: Non-common arms form line $l_1$. So, $\text{m}\angle 2 + \text{m}\angle 3 = 180^\circ$.
• $\angle 3$ and $\angle 4$: Non-common arms form line $l_2$. So, $\text{m}\angle 3 + \text{m}\angle 4 = 180^\circ$.
• $\angle 4$ and $\angle 1$: Non-common arms form line $l_1$. So, $\text{m}\angle 4 + \text{m}\angle 1 = 180^\circ$.

Thus, there are also four linear pairs formed by two intersecting lines.

3. Vertically Opposite Angles:

Vertically opposite angles are pairs of angles opposite to each other at the vertex. We know from the theorem that vertically opposite angles are equal in measure.

• $\angle 1$ and $\angle 3$ are vertically opposite. Therefore, $\text{m}\angle 1 = \text{m}\angle 3$.
• $\angle 2$ and $\angle 4$ are vertically opposite. Therefore, $\text{m}\angle 2 = \text{m}\angle 4$.

There are two pairs of vertically opposite angles formed by two intersecting lines.

Summary of Properties of Angles Formed by Intersecting Lines

When two straight lines intersect at a point:

• Four angles are formed at the point of intersection.
• There are four pairs of adjacent angles.
• There are four linear pairs of angles. The sum of the measures of angles in each linear pair is $180^\circ$.
• There are two pairs of vertically opposite angles. The angles in each pair are equal in measure.
• The sum of the measures of all four angles around the point of intersection is $360^\circ$. This is because they complete a full angle around the vertex. For example, $\text{m}\angle 1 + \text{m}\angle 2 + \text{m}\angle 3 + \text{m}\angle 4 = 360^\circ$.

These relationships allow us to find the measures of all four angles if the measure of just one of them is known.

Answer:

We are given that lines AB and CD intersect at O, and the measure of angle $\angle \text{AOD}$ is $115^\circ$.

Find $\text{m}\angle \text{BOC}$:

Observe the angles $\angle \text{AOD}$ and $\angle \text{BOC}$. These angles are opposite to each other at the vertex O formed by the intersection of lines AB and CD. Thus, they are vertically opposite angles.

By the Vertically Opposite Angles Theorem, vertically opposite angles are equal in measure.

$\text{m}\angle \text{BOC} = \text{m}\angle \text{AOD}$

(Vertically Opposite Angles)

Since $\text{m}\angle \text{AOD} = 115^\circ$,

$\mathbf{m}\angle \text{BOC} = 115^\circ$

Find $\text{m}\angle \text{AOC}$:

Consider the straight line AB and the ray $\overrightarrow{\text{OC}}$ standing on it at O. The angles $\angle \text{AOC}$ and $\angle \text{BOC}$ form a linear pair.

By the Linear Pair Axiom, the sum of the measures of angles in a linear pair is $180^\circ$.

$\text{m}\angle \text{AOC} + \text{m}\angle \text{BOC} = 180^\circ$

(Linear Pair)

We know $\text{m}\angle \text{BOC} = 115^\circ$. Substitute this value:

$\text{m}\angle \text{AOC} + 115^\circ = 180^\circ$

Subtract $115^\circ$ from both sides:

$\text{m}\angle \text{AOC} = 180^\circ - 115^\circ$

$\mathbf{m}\angle \text{AOC} = 65^\circ$

Find $\text{m}\angle \text{BOD}$:

Observe the angles $\angle \text{AOC}$ and $\angle \text{BOD}$. These angles are opposite to each other at the vertex O. Thus, they are vertically opposite angles.

By the Vertically Opposite Angles Theorem, vertically opposite angles are equal.

$\text{m}\angle \text{BOD} = \text{m}\angle \text{AOC}$

(Vertically Opposite Angles)

Since we found $\text{m}\angle \text{AOC} = 65^\circ$,

$\mathbf{m}\angle \text{BOD} = 65^\circ$

Verification (Optional):

We can verify our answers using other linear pairs or the sum of angles around a point.

• Check linear pair on CD: $\text{m}\angle \text{AOD} + \text{m}\angle \text{AOC} = 115^\circ + 65^\circ = 180^\circ$. Correct.
• Check linear pair on AB: $\text{m}\angle \text{AOD} + \text{m}\angle \text{BOD} = 115^\circ + 65^\circ = 180^\circ$. Correct.
• Check linear pair on CD: $\text{m}\angle \text{COB} + \text{m}\angle \text{BOD} = 115^\circ + 65^\circ = 180^\circ$. Correct.
• Sum of all angles around O: $m\angle \text{AOD} + m\angle \text{AOC} + m\angle \text{BOC} + m\angle \text{BOD} = 115^\circ + 65^\circ + 115^\circ + 65^\circ = 360^\circ$. Correct.

The measures of the angles are:

• $\text{m}\angle \text{BOC} = 115^\circ$
• $\text{m}\angle \text{AOC} = 65^\circ$
• $\text{m}\angle \text{BOD} = 65^\circ$