October 04, 2022

Linear Pair of AnglesDefinition, Axiom, Examples

The linear pair of angles is a significant subject in geometry. With so many real-world functions, you'd be surprised to find how useful this figure can be. While you may wonder if it has no relevant in your life, we all should grasp the ideas to nail those tests 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 diagrams and examples to help 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 exist to be useful as you go forward in geometry and more complicated theorems and proofs. We will answer this question with a simple definition in this single 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 sit on the same line and together form a straight line. The sum of the angles in a linear pair will always produce a straight angle equal to 180 degrees.

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

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

The Linear Pair Axiom

Through the definition clarified, we will explore the two axioms earnestly to completely grasp 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 clear and self-evident. A linear pair of angles has two axioms connected with them.

The first axiom states that if a ray stands on a line, the adjacent angles will create a straight angle, namely called 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 among them. This is commonly called a straight line.

Examples of Linear Pairs of Angles

To envision these axioms better, here are some drawn examples with their corresponding answers.

Example One

Here in this example, we have two angles that are neighboring each other. As you can observe in the image, the adjacent angles form a linear pair due to the fact that the total of their measures 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 instance, we possess two lines intersect, making four angles. Not all angles form a linear pair, but each angle and the one adjacent to it form 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 an intersection 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 totals sum up to 180 degrees. As a result, we can come to the conclusion that this example has no linear pair unless we extend one straight line.

Applications of Linear Pair of Angles

At the moment we have gone through what linear pairs are and have observed some instances, let’s understand how this theorem can be applied in geometry and the real-life scenario.

In Real-Life Scenarios

There are multiple implementations of linear pairs of angles in real life. One common case is architects, who utilize these axioms in their day-to-day work to determine if two lines are perpendicular and form a straight angle.

Builders and construction professionals also utilize experts in this field to make their work less complex. They use linear pairs of angles to assure that two close walls create a 90-degree angle with the ground.

Engineers also apply linear pairs of angles frequently. They do so by figuring out the pressure on the beams and trusses.

In Geometry

Linear pairs of angles also perform a role in geometry proofs. A ordinary 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 made are congruent.

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

The theorem of linear pairs is further utilized for more complicated uses, such as measuring the angles in polygons. It’s important to grasp the fundamentals of linear pairs, so you are ready for more complex geometry.

As demonstrated, linear pairs of angles are a relatively easy theorem with several engaging implementations. Later when you're out and about, take note if you can see some linear pairs! And, if you're taking a geometry class, be on the lookout for how linear pairs might be useful in proofs.

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