Which pair of angles are adjacent angles?

The word “adjacent” means “next to,” so it makes sense that two angles are considered adjacent if they are next to each other!

Here, we can see that \(\angle COB\) is next to \(\angle BOA\), making them adjacent. Formally, two requirements must be met for two angles to be considered adjacent. First, the angles must share the same vertex, or corner point. Here, \(O\) is that shared vertex. Second, the two angles must share one common side that lies between them. In this case, that common side is the ray from \(O\) to \(B\).

Angles that overlap cannot be adjacent. For example, angles a and b shown here are not adjacent because they overlap.

Let’s take a look at some more angles. In this diagram, which pairs of angles are adjacent?

The angle generated by the ray between its initial and final positions is the measure of rotation of a ray when rotated about its terminus. Two rays that are joined end-to-end make an angle.

A pair of angles is sometimes used in geometry. Complementary angles, adjacent angles, linear pairs of angles, opposite angles, and so on are all examples of pairs of angles. We’ll go over the definitions of adjacent angles and vertical angles in-depth in this article.

When two angles have the same vertex and side, they are said to be neighbouring angles. Vertex of the angle is a point where the rays meet to form the sides of the angle. When two adjacent angles share the same vertex and side, they might be complementary or supplementary angles.

Example of Adjacent Angles

Consider a wall clock. The minute hand and second hand of the clock make one angle, denoted by ∠AOC, and the hour hand and second hand form another, denoted by ∠COB. Both of these pairs of angles, ∠AOC and ∠COB, are known as adjacent angles since they are near to each other. The uncommon arms are on either side of the common arms in ∠AOC and ∠COB. They share a common vertex and a common arm. Adjacent angles are those that are adjacent to one another.

Properties of Adjacent Angles

The following are some of the essential properties of adjacent angles:

Two angles are adjacent if they are in the same plane.

Vertical angles, also known as vertically opposite angles, are generated when two lines intersect. The sum of two vertically opposite angles is always the same. A vertical angle and its adjacent angle are also called supplementary angles since they sum up to 180 degrees. When two lines connect and form an angle, such as X=45°, the opposite angle is also 45°. And the angle next to angle X will be 180 – 45 = 135 degrees.

A pair of non-adjacent angles formed by the intersection of two straight lines is known as a vertical angle. In simple terms, vertical angles are placed in the corners of the “X” created by two straight lines, across from one another. Because they are perpendicular to each other, they are also known as vertically opposite angles.

Theorem Related to Vertical Angles

Angles that are vertically opposite angles theorem Two opposed vertical angles created when two lines intersect one another are always equal (congruent) to each other, according to the theorem.

Proof:- considering two intersecting lines and at the intersecting point four angles are formed. Let us say them ∠1, ∠2,∠3,∠4 where (∠1, ∠2),( ∠2,∠3),(∠3,∠4),(∠1, ∠4) are the adjacent angles such that sum of all these adjacent angles are 180° as they are linear pair of angles.

∠1 + ∠2 = 180° (both are the linear pair of angles) ——— (1)

∠1 +∠4 = 180° (both are the linear pair of angles) ——— (2)

From equations (1) and (2), ∠1 + ∠2 = 180° = ∠1 +∠4.

The two angles are said to be adjacent angles when they share the common vertex and side. The endpoint of the rays, forming the sides of an angle, is called the vertex of an angle. Adjacent angles can be a complementary angle or supplementary angle when they share the common vertex and side.

Adjacent angles are an important concept to understand in maths. They are a key concept in geometry and are usually introduced in 4th grade maths. Although kids study angles in their math courses throughout their time at school, it’s often a difficult concept to grasp. If your child is struggling with understanding not only angles, but any other concepts in maths, you may want to consider tutoring courses.

In order to help you or your child on your journey to understanding angles, we have put together this little guide to walk you through the key concepts, definitions and FAQs surrounding adjacent angles.

Adjacent angles are two angles that have a common side and a common vertex (corner point) but do not overlap in any way. When you break down the phrase adjacent angles, it becomes easy to visualise exactly what it is; they are two angles that are next to each other.

Being able to identify a common side and a common vertex is the simplest way to identify an adjacent angle. If two angles share one side and both derive from the same corner (vertex) point, then they are adjacent angles.

It’s important to remember that adjacent angles must have BOTH a common side and common vertex. Therefore, if you see two angles that are coming from the same corner but there is another angle in the middle, it means that they do not share any sides. This means that they are not adjacent angles as they don’t share a side AND a vertex.

Identifying adjacent angles becomes easier with practice and seeing examples will help you understand what you are looking for.

Identifying the difference between adjacent angles and vertical angles is an important skill to master in geometry. The best way to visualize the difference between these two types of angles is to imagine two straight lines intersecting each other to form a cross.

When a cross is formed, four angles are formed. We know how to identify the adjacent angles, because they have a common side and a common vertex. But how do we identify a vertical angle? Identifying a vertical angle is equally as easy as finding an adjacent angle. Similarly to adjacent angles, a set of vertical angles will share a vertex point. However, they do not need to share a common side.

When thinking about a cross, the vertical angles are the angles that are opposite each other. This is why they are sometimes called vertically opposite angles.

In order to further help you visualize what adjacent angles look like, here’s a quick list of their properties:

In order to understand what a linear pair looks like, you must imagine a cross. When two lines intersect, four angles are created.

If you take a look at the picture to the right, you can see that there are four angles labelled 1, 2, 3, and 4. In this image, the linear angles are 1 and 3, 3 and 2, 2 and 4, 4 and 1.

You can triple check that two angles are a linear pair by seeing if they add up to 180 degrees. All linear pairs of angles are supplementary and therefore always add up to 180 degrees. If the angles are adjacent and add up to 180 degrees you can be confident in making the assertion that they are a linear pair of adjacent angles.

Vertically opposite angles are technically not adjacent angles, but where you find adjacent angles, you will likely also find some vertically opposite angles.

Vertical angles have already been explored, but to clarify, vertical angles share the same vertex but do not share any of the same sides. If we take the above picture, 3 and 4 and 1 and 2 are considered vertically opposite angles.

A key property of vertically opposite angles is that they measure exactly the same. For example, if angle 1 was 30 degrees, angle 2 would also measure as 30 degrees.

Put simply, adjacent angles are angles that share a common side and a common vertex (corner point).

This is TRUE in some cases! Supplementary adjacent angles always add up to 180. This is because the two angles sit next to each other on a straight line and all angles on a straight line add up to 180.

However, if the adjacent angles are not linear pairs and another angle is in the mix, the two adjacent angles will not add up to 180.

As vertical and adjacent angles can often exist in a small area together, many people believe that vertical angles can also be adjacent angles. This is FALSE. Vertical angles do not share any of the same sides, meaning they cannot be adjacent.

YES! Adjacent angles can be linear pairs. As linear pairs share both a common side and a common vertex, they can be considered adjacent angles. However, not all adjacent angles are linear pairs.

This was a quick run through of adjacent angles to help you get to grips with this integral part of the geometry syllabus. However, there’s always more that you can do to ensure you achieve the grade you want.

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Each of us has neighbors. Some have a common wall between their apartments, some have a common fence. It can be said that apartments or fences are adjacent to each other. Adjacent can be many things, including angles.

When we look at a watch with an hour, minute, and second hand, we see a pair of adjacent angles.

When we look at an open book its covers and one of the pages form a pair of adjacent angles.

When we ride a bike, the three adjacent spokes of the wheel form a pair of adjacent angles.

Two roads at the T-junction, a tram rail and a sleeper, two cross-street signs, open scissors, all these real-life objects form adjacent angles.

Having such a variety of adjacent angles in everyday life, it is worth studying and understanding which properties these angles have and what can be done with using these properties.

The word “adjacent” means “next” or “neighboring”. We can treat adjacent angles as angles that are next to each other.

Adjacent angles are a pair of angles that share a common side and vertex. Three things that need to be done to keep the angles adjacent:

The following diagram shows two adjacent angles 1 and 2 with the common arm $\vec{OB}$ and common vertex O.

Adjacent angles never overlap

Every angle has one vertex and two arms, so it can have two possible adjacent angles, one attaching to each arm. In the diagram below, angles 2 and 3 are adjacent to angle 1: angles 1 and 2 share the common vertex O and common arm $\vec{OB}$, angles 1 and 3 share the common vertex O and common arm $\vec{OC}$.

ANGLE ADDITION POSTULATE:

The angle addition postulate states that if point B is in the interior of AOC, then

m∠AOB+m∠ BOC=m∠ AOC

This postulate can be applied to any pair of adjacent angles.

Are there any restrictions on the measures of the adjacent angles? Let us think. If two adjacent angles never overlap, then their total measure cannot be greater than the measure of a full angle. So, if angles 1 and 2 are adjacent, then

m∠1+m∠2≤360°

Thus, we can obtain the following combinations of adjacent angles:

Two reflex angles can never form a pair of adjacent angles. Each of two arbitrary reflex angles has a measure greater than 180°, so the sum of the measures of these two reflex angles is always greater than 360° and angles overlap.

We already understood what the adjacent angles are. Let us consider the following example. Will the angles AOC and AOB shown in the figure be adjacent? Why?

Formally, all the conditions of the definition are held:

Are they adjacent? No, because they are overlapping. So, do not forget that two adjacent angles never overlap.

Determining whether two angles are adjacent or not, you should clearly remember three things that these angles should have: be a pair of angles, have a common vertex, and a common arm. It is best to learn to understand whether the angles are adjacent or not on specific examples.

EXAMPLE: In each case determine whether the angles 1 and 2 are adjacent or not.

SOLUTION: a) Pair of angles ∠1 and ∠2 share the common vertex O and common arm $\vec{OB}$, so these angles are adjacent.

b) Pair of angles ∠1 and ∠2 share the common vertex O but do not share the common arm, so these angles are not adjacent.

c) Pair of angles ∠1 and ∠2 share the common vertex O and common arm $\vec{OB}$, so these angles are adjacent.

d) Pair of angles ∠1 and ∠2 share the common arm $\vec{OB}$ but do not share the common vertex, so these angles are not adjacent.

e) Pair of angles ∠1 and ∠2 share the common vertex O and common arm $\vec{OB}$, so these angles are adjacent.

f) Pair of angles ∠1 and ∠2 share the common vertex O and common arm $\vec{OB}$, so these angles are adjacent.

g) Pair of angles ∠1 and ∠2 share the common vertex O but do not share the common arm, so these angles are not adjacent.

A linear pair of angles is a pair of adjacent angles formed when two lines intersect.

The above diagram shows two intersecting lines AC⃡ and BD⃡ which form four linear pairs of angles:

When two lines intersect, there are always four linear pairs of angles.

What is the same between a pair of adjacent angles and a linear pair of angles?

Supplementary angles are two angles whose measures add up to 180°.

Supplementary angles are not necessarily adjacent angles. But every two adjacent angles whose measures add up to 180° are supplementary. Using this fact, we can state that

each linear pair of angles is a pair of supplementary angles

Take an arbitrary linear pair of angles, for example, angles 1 and 2. These two angles together form the straight angle DOB. The measure of a straight angle is always 180°. Therefore, by angle addition postulate,

m∠1+m∠2=m∠DOB

If m∠DOB=180°, then

m∠1+m∠2=180°

By the definition of supplementary angles, angles 1 and 2 are supplementary angles. In the same way, we can prove that each linear pair of angles is a pair of supplementary angles.

Now, we can outline the following basic properties of a linear pair of angles:

Note that the interior angle of the triangle and the corresponding exterior angle of the triangle together form a linear pair of angles and are always supplementary.

Complementary angles are two angles whose measures add up to 90°.

Complementary angles are not necessarily adjacent angles. But every two adjacent angles whose measures add up to 90° are complementary.

In geometry, there are two types of complementary angles:

Two complementary angles with a common vertex and a common arm are called adjacent complementary angles.

In the figure given below, AOB and BOC are adjacent angles as they have a common vertex O and a common arm $\vec{OB}$.

They also together form the right angle AOC. Thus, the measures of angles AOB and BOC add up to 90° and these two angles are adjacent complementary angles.

Two complementary angles which are not adjacent are called non-adjacent complementary angles.

In the figure given below, AOB and COD are non-adjacent angles as they have a common vertex O and do not have a common arm.

The measures of angles AOB and COD add up to 90°, so, these two angles are complementary. Therefore, angles AOB and COD are non-adjacent complementary angles.

The following properties of adjacent angles help us identify whether the angles are adjacent and investigate these angles.

PROPERTIES:

When two lines intersect, two pairs of opposite angles are formed. These angles are called vertical angles.

The above diagram shows two intersecting lines AC⃡ and BD⃡ which form two pairs of vertical angles:

This statement is known as the vertical angle theorem. Let us prove that angle 1 is congruent to angle 3.

Since angles 1 and 2 form a linear pair of angles, they are supplementary angles and

m∠1+m∠2=180°

Since angles 2 and 3 form a linear pair of angles, they are supplementary angles and

m∠2+m∠3=180°

By the transitive property, if m∠1+m∠2=180° and m∠2+m∠3=180° , then

m∠1+m∠2=m∠2+m∠3

Subtract m∠2 from both sides of the above equality:

m∠1=m∠3

Using the definition of congruent angles,

∠1≅∠3

In the same way, we can prove that the remaining two vertical angles are congruent too.

1. Which angles are called adjacent?

Adjacent angles are a pair of angles that share a common side and vertex. Three things that need to be done to keep the angles adjacent: adjacent angles go in pairs, adjacent angles share the common arm, and adjacent angles have the same vertex.

2. Could three angles be adjacent?

Three angles can be adjacent pairwise, but all together they are not adjacent, because adjacent angles are pairs of angles.

3. Can two obtuse angles be adjacent?

Any two obtuse angles that have a common vertex and a common arm are adjacent – the sum of their measures is less than 360°.

4. Can two reflex angles be adjacent?

Any two reflex angles with a common vertex and a common arm are not adjacent – the sum of their measures is greater than 360°.

5. Is the sum of the measures of two adjacent angles 180°?

No, the sum of the measures of two adjacent angles could be an arbitrary number of degrees, not necessarily 180°. To 180 degrees add up only two adjacent supplementary angles.

6. When two lines intersect, is there a relationship between two vertical angles and a linear pair of angles?

A linear pair of angles are always adjacent angles that add up to 180°. Two vertical angles are always opposite congruent angles, one of these angles belongs to a linear pair of angles. If you know the measures of one of the angles formed when two lines intersect, then you know the measures of all four angles.

7. Are the vertical angles adjacent?

Never. Vertical angles only have a common vertex but never have a common arm.

8. What is the difference between complementary and supplementary angles?

The measures of two complementary angles add up to 90° while the measures of two supplementary angles add up to 180°.

SOLUTION: If two angles are congruent, then they are both of the same type. We already know that two reflex angles cannot form a pair of adjacent angles as the sum of their measures is greater than 360°.

If these angles are not reflex, then consider the following by size angles – adjacent straight angles. When we take two congruent straight angles each of them has the measure of 180° and they add up to 360°. This is possible, so the maximum measure of each of two congruent adjacent angles could be 180°.

ANSWER: 180°

SOLUTION: a) Pair of angles ∠1 and ∠2 share the common vertex O but do not share the common arm, so these angles are not adjacent.

b) Pair of angles ∠1 and ∠2 share the common vertex O but they overlap, so these angles are not adjacent.

c) Pair of angles ∠1 and ∠2 share the common vertex O and common arm $\vec{OB}$, so these angles are adjacent.

d) Pair of angles ∠1 and ∠2 share the common vertex O and common arm $\vec{OB}$, so these angles are adjacent. Moreover, these angles are complementary.

ANSWER: a) not adjacent b) not adjacent c) adjacent d) adjacent

a) m∠1=45°, m∠2=135°;

b) m∠1=105°, m∠2=95°.

SOLUTION: Add the measures of the given angles. If the sum of the measures is 180°, then the given angles are supplementary, if the sum of the measures is not 180°, then the given angles are not supplementary.

a) m∠1+m∠2=45°+135°=180°

In this case, angles 1 and 2 are supplementary angles.

b) m∠1+m∠2=105°+95°=200°≠180°

In this case, angles 1 and 2 are not supplementary angles.

ANSWER: a) supplementary b) not supplementary

a) m∠1=64°, m∠2=36°;

b) m∠1=25°, m∠2=65°.

SOLUTION: Add the measures of the given angles. If the sum of the measures is 90°, then the given angles are complementary, if the sum of the measures is not 90°, then the given angles are not complementary.

a) m∠1+m∠2=64°+36°=100°≠90°

In this case, angles 1 and 2 are not complementary angles.

b) m∠1+m∠2=25°+65°=90°

In this case, angles 1 and 2 are not complementary angles.

ANSWER: a) not complementary b) complementary

SOLUTION: If two lines a and b intersect, then they form four linear pairs of angles: ∠1 and ∠2, ∠2 and ∠3, ∠3 and ∠4, ∠4 and ∠1.

If every two adjacent angles are congruent, then

∠ 1≅ ∠2, ∠2≅∠3, ∠3≅∠4, ∠4≅∠1

By the transitive property,

∠1≅∠2≅∠3≅∠4

All these four angles form the full angle at the common vertex. Using the angle addition postulate, the measures of all four angles add up to 360°, so

m∠1+m∠2+m∠3+m∠4=360°

Since all four angles are congruent, they all have the same measures and the measure of each such angle is $\frac{1}{4}$360°=90°.

ANSWER: 90°

The word “adjacent” means “next to,” so it makes sense that two angles are considered adjacent if they are next to each other!

Here, we can see that \(\angle COB\) is next to \(\angle BOA\), making them adjacent. Formally, two requirements must be met for two angles to be considered adjacent. First, the angles must share the same vertex, or corner point. Here, \(O\) is that shared vertex. Second, the two angles must share one common side that lies between them. In this case, that common side is the ray from \(O\) to \(B\).

Angles that overlap cannot be adjacent. For example, angles a and b shown here are not adjacent because they overlap.

Let’s take a look at some more angles. In this diagram, which pairs of angles are adjacent?

Starting from the left, we can see that angles 1 and 2 share a common vertex and a common side. For this reason, they are adjacent. The pair of angles 2 and 3 also share a common vertex and common side, making them adjacent as well. But what about angles 1 and 3? While these angles still share a common vertex, they have no side in common, so they are not adjacent.

Let’s now consider angle 4. Is it adjacent to any of the other angles? While it does share a common side with angle 3, they do not have the same vertex. For this reason, angles 3 and 4 are not adjacent. Angle 4 is also not adjacent to angles 1 and 2 because it has neither a common vertex nor a common side between them.