October 04, 2022

Linear Pair of AnglesDefinition, Axiom, Examples

The linear pair of angles is a significant subject in geometry. With multiple real-life uses, you'd be amazed to find how applicable this figure can be. Even though you may wonder if it has no application in your life, we all should understand the concept to nail those exams in school.

To save you time and make this information easily accessible, here is an preliminary insight into the properties of a linear pair of angles, with images and examples to assist with your personal study sessions. We will also discuss few real-life and geometric applications.

What Is a Linear Pair of Angles?

Linearity, angles, and intersections are ideas that continue to be useful as you progress in geometry and more complex theorems and proofs. We will answer this question with a straightforward definition in this single point.


A linear pair of angles is the term designated to two angles that are positioned on a straight line and the total of their angles measure 180 degrees.

To put it simply, linear pairs of angles are two angles that sit on the same line and together create a straight line. The total of the angles in a linear pair will always create a straight angle equivalent
times to 180 degrees.

It is crucial to keep in mind that linear pairs are at all times at adjacent angles. They share a common vertex and a common arm. This suggests that at all times create on a straight line and are at all times supplementary angles.

It is important to make clear that, although the linear pair are always adjacent angles, adjacent angles aren't always linear pairs.

The Linear Pair Axiom

Over the definition clarified, we will study the two axioms earnestly to fully comprehend every example thrown at you.

Let’s start by defining what an axiom is. It is a mathematical postulate or assumption that is acknowledged without proof; it is considered evident and self-evident. A linear pair of angles has two axioms linked with them.

The first axiom establishes that if a ray is located on a line, the adjacent angles will make a straight angle, also known as a linear pair.

The second axiom implies that if two angles create a linear pair, then uncommon arms of both angles makes a straight angle between them. In other words, they make a straight line.

Examples of Linear Pairs of Angles

To envision these axioms better, here are some diagram examples with their respective answers.

Example One

In this example, we have two angles that are next to one another. As you can observe in the diagram, the adjacent angles form a linear pair since the sum of their measures equals 180 degrees. They are also supplementary angles, since they share a side and a common vertex.

Angle A: 75 degrees

Angle B: 105 degrees

Sum of Angles A and B: 75 + 105 = 180

Example Two

Here, we have two lines intersect, creating four angles. Not every angles creates a linear pair, but respective angle and the one close to it makes a linear pair.

∠A 30 degrees

∠B: 150 degrees

∠C: 30 degrees

∠D: 150 degrees

In this case, the linear pairs are:

∠A and ∠B

∠B and ∠C

∠C and ∠D

∠D and ∠A

Example Three

This case shows a common point of three lines. Let's observe the axiom and characteristics of linear pairs.

∠A 150 degrees

∠B: 50 degrees

∠C: 160 degrees

None of the angle totals sum up to 180 degrees. As a effect, we can come to the conclusion that this image has no linear pair unless we stretch one straight line.

Implementations of Linear Pair of Angles

Now that we have explored what linear pairs are and have observed some cases, let’s understand how this theorem can be implemented in geometry and the real world.

In Real-World Situations

There are multiple utilizations of linear pairs of angles in real-world. One common case is architects, who utilize these axioms in their day-to-day job to establish if two lines are perpendicular and creates a straight angle.

Construction and Building professionals also employ masters in this field to make their job simpler. They use linear pairs of angles to assure that two adjacent walls make a 90-degree angle with the floor.

Engineers also uses linear pairs of angles regularly. They do so by calculating the pressure on the beams and trusses.

In Geometry

Linear pairs of angles additionally play a role in geometry proofs. A regular proof that uses linear pairs is the alternate interior angles concept. This concept states that if two lines are parallel and intersected by a transversal line, the alternate interior angles created are congruent.

The proof of vertical angles additionally replies on linear pairs of angles. While the adjacent angles are supplementary and add up to 180 degrees, the opposite vertical angles are at all times equal to each other. Because of these two rules, you are only required to figure out the measurement of any one angle to work out the measurement of the rest.

The concept of linear pairs is also employed for more complicated implementation, such as determining the angles in polygons. It’s critical to grasp the basics of linear pairs, so you are ready for more progressive geometry.

As you can see, linear pairs of angles are a comparatively easy theorem with some fascinating uses. Later when you're out and about, observe if you can notice some linear pairs! And, if you're attending a geometry class, be on the lookout for how linear pairs might be useful in proofs.

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