October 04, 2022

Linear Pair of AnglesDefinition, Axiom, Examples

The linear pair of angles is an important subject in geometry. With several real-life functions, you'd be amazed to discover how useful this figure can be. While you may think it has no relevant in your life, we all should learn the ideas to nail those tests in school.

To save you time and create this information easily accessible, here is an preliminary insight into the properties of a linear pair of angles, with diagrams and examples to assist 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 theories that remain to be useful as you progress in geometry and more complex theorems and proofs. We will answer this query with a simple explanation in this unique 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 easily, linear pairs of angles are two angles that are adjacent on the same line and together create a straight line. The sum of the angles in a linear pair will at all times make a straight angle equivalent
times to 180 degrees.

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

It is crucial to clarify that, even though the linear pair are constantly adjacent angles, adjacent angles aren't always linear pairs.

The Linear Pair Axiom

Through the definition simplified, we will study the two axioms earnestly to fully understand any example thrown at you.

First, let's define what an axiom is. It is a mathematical postulate or assumption that is accepted without having to prove; it is deemed obvious and self-explanatory. A linear pair of angles has two axioms connected with them.

The first axiom implies that if a ray stands on a line, the adjacent angles will create a straight angle, making them a linear pair.

The second axiom implies that if two angles produces a linear pair, then uncommon arms of both angles create a straight angle between them. This is also known as a straight line.

Examples of Linear Pairs of Angles

To imagine these axioms better, here are a few figure examples with their individual explanations.

Example One

Here in this example, we have two angles that are adjacent to each other. As you can see 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

In this instance, we possess two lines intersect, producing four angles. Not every angles form a linear pair, but each angle and the one adjacent to it makes a linear pair.

∠A 30 degrees

∠B: 150 degrees

∠C: 30 degrees

∠D: 150 degrees

In this example, 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 observe the axiom and properties of linear pairs.

∠A 150 degrees

∠B: 50 degrees

∠C: 160 degrees

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

Applications of Linear Pair of Angles

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

In Real-Life Situations

There are several applications of linear pairs of angles in real life. One familiar example is architects, who apply these axioms in their day-to-day job to check if two lines are perpendicular and creates a straight angle.

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

Engineers also uses linear pairs of angles regularly. They do so by working out the tension in beams and trusses.

In Geometry

Linear pairs of angles as well perform a function in geometry proofs. A common proof that employs linear pairs is the alternate interior angles concept. This theorem explains that if two lines are parallel and intersected by a transversal line, the alternate interior angles created 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 each other. Because of these two rules, you are only required to know the measurement of one angle to work out the measure of the rest.

The theorem of linear pairs is also employed for more sophisticated applications, such as figuring out the angles in polygons. It’s critical to understand the basics of linear pairs, so you are ready for more complex geometry.

As demonstrated, linear pairs of angles are a relatively simple concept with several fascinating implementations. Next time you're out and about, take note if you can spot some linear pairs! And, if you're taking a geometry class, be on the lookout for how linear pairs might be useful in proofs.

Enhance Your Geometry Skills using Grade Potential

Geometry is fun and useful, majorly if you are looking into the domain of construction or architecture.

Despite that, if you're having problem understanding linear pairs of angles (or any other concept in geometry), think about signing up for a tutoring session with Grade Potential. One of our expert tutors will help you comprehend the material and nail your next test.