October 04, 2022

Linear Pair of AnglesDefinition, Axiom, Examples

The linear pair of angles is an important concept in geometry. With so many real-world uses, you'd be surprised to figure out how relevant this figure can be. While you might think it has no use in your life, we all should understand the ideas to nail those examinations in school.

To save your time and make this information easily accessible, here is an introductory insight into the properties of a linear pair of angles, with images and examples to guide with your personal study sessions. We will also talk about some real-world and geometric applications.

What Is a Linear Pair of Angles?

Linearity, angles, and intersections are concepts that remain to be relevant as you go forward in geometry and more complex theorems and proofs. We will answer this question with a simple explanation in this unique point.


A linear pair of angles is the name given to two angles that are located on a straight line and have the sum of their angles measure 180 degrees.

To put it easily, linear pairs of angles are two angles that are adjacent on the same line and pair up to form a straight line. The total of the angles in a linear pair will always produce a straight angle equivalent
times to 180 degrees.

It is essential to keep in mind that linear pairs are at all times at adjacent angles. They share a common apex and a common arm. This suggests that they always form on a straight line and are at all times supplementary angles.

It is important to clarify that, although the linear pair are constantly adjacent angles, adjacent angles never constantly linear pairs.

The Linear Pair Axiom

Through the definition clarified, we will study the two axioms critical to fully comprehend any example provided to you.

Let’s start by defining what an axiom is. It is a mathematical postulate or hypothesis that is accepted without proof; it is deemed clear 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 form a straight angle, making them a linear pair.

The second axiom establishes that if two angles makes a linear pair, then uncommon arms of both angles produces a straight angle between them. This is commonly called a straight line.

Examples of Linear Pairs of Angles

To visualize these axioms better, here are a few diagram examples with their respective answers.

Example One

As we can see in this instance, we have two angles that are next to each other. As you can see in the diagram, the adjacent angles form a linear pair due to the fact that the total of their measurement equals 180 degrees. They are also supplementary angles, because 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

In this example, we possess two lines intersect, making four angles. Not all angles creates a linear pair, but each 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 example represents a common point of three lines. Let's look at the axiom and properties of linear pairs.

∠A 150 degrees

∠B: 50 degrees

∠C: 160 degrees

None of the angle combinations add up to 180 degrees. As a consequence, we can come to the conclusion that this image has no linear pair until we expand one straight line.

Applications of Linear Pair of Angles

Now that we have gone through 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 several implementations of linear pairs of angles in real-world. One familiar case is architects, who utilize these axioms in their daily work to check if two lines are perpendicular and form a straight angle.

Construction and Building professionals also employ masters in this matter to make their work less complex. They use linear pairs of angles to make sure that two adjacent walls form a 90-degree angle with the floor.

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

In Geometry

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

The proof of vertical angles also depends on linear pairs of angles. Even though the adjacent angles are supplementary and sum up to 180 degrees, the opposite vertical angles are at all times equal to one another. Because of these two rules, you are only required to figure out the measure of one angle to determine the measure of the rest.

The concept of linear pairs is also utilized for more sophisticated implementation, such as working out the angles in polygons. It’s critical to grasp the basics of linear pairs, so you are ready for more complex geometry.

As you can see, linear pairs of angles are a somewhat easy concept with few interesting applications. Later when you're out and about, see if you can spot some linear pairs! And, if you're taking a geometry class, bear in mind how linear pairs may be useful in proofs.

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