October 04, 2022

Linear Pair of AnglesDefinition, Axiom, Examples

The linear pair of angles is a significant concept in geometry. With multiple real-world applications, you'd be astonished to figure out how useful this figure can be. While you might wonder if it has no use in your life, we all should grasp the ideas to nail those tests in school.

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

What Is a Linear Pair of Angles?

Linearity, angles, and intersections are ideas that continue to be relevant as you move ahead with geometry and more sophisticated theorems and proofs. We will answer this question with a easy definition in this single point.


A linear pair of angles is the name provided to two angles that are positioned on a straight line and the total of their measurement of angles is 180 degrees.

To put it easily, linear pairs of angles are two angles that are aligned on the same line and together create a straight line. The total of the angles in a linear pair will always produce a straight angle equal to 180 degrees.

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

It is important to clarify that, while the linear pair are always adjacent angles, adjacent angles aren't always linear pairs.

The Linear Pair Axiom

Over the precise explanation, we will explore the two axioms earnestly to fully grasp 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 having to prove; it is considered 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, also known as a linear pair.

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

Examples of Linear Pairs of Angles

To envision these axioms better, here are a few figure examples with their respective explanations.

Example One

Here in this example, we have two angles that are neighboring each other. As you can notice in the image, the adjacent angles form a linear pair due to the fact that 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 possess two lines intersect, producing four angles. Not all angles form a linear pair, but each angle and the one close to it form 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 take note of 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 effect, we can conclude that this diagram has no linear pair unless we stretch one straight line.

Applications of Linear Pair of Angles

Now that we have gone through what linear pairs are and have looked at some instances, let's see how this theorem can be used in geometry and the real-life scenario.

In Real-Life Situations

There are several implementations of linear pairs of angles in real-world. One common case is architects, who apply these axioms in their day-to-day job to identify if two lines are perpendicular and form a straight angle.

Construction and Building professionals also use experts in this subject to make their job less complex. They use linear pairs of angles to make sure that two close walls make a 90-degree angle with the ground.

Engineers also utilizes linear pairs of angles frequently. 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 utilizes linear pairs is the alternate interior angles theorem. This concept expresses that if two lines are parallel and intersected by a transversal line, the alternate interior angles made are congruent.

The proof of vertical angles as well replies on linear pairs of angles. While the adjacent angles are supplementary and sum up to 180 degrees, the opposite vertical angles are always equal to one another. Because of previously mentioned two rules, you are only required to know the measurement of any one angle to determine the measurement of the rest.

The concept of linear pairs is also utilized for more sophisticated applications, such as working out the angles in polygons. It’s essential to grasp the fundamentals of linear pairs, so you are ready for more progressive geometry.

As you can see, linear pairs of angles are a somewhat easy theorem with few engaging implementations. Next time you're out and about, see 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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