October 04, 2022

Linear Pair of AnglesDefinition, Axiom, Examples

The linear pair of angles is an important subject in geometry. With several real-life uses, you'd be amazed to figure out how applicable this figure can be. Even though you might think it has no application in your life, we all must grasp the concept to nail those tests in school.

To save you time and offer this information easy to access, here is an introductory insight into the characteristics of a linear pair of angles, with images and examples to guide with your personal study sessions. We will also discuss few real-life and geometric uses.

What Is a Linear Pair of Angles?

Linearity, angles, and intersections are theories that remain to be applicable as you go forward in geometry and more sophisticated theorems and proofs. We will answer this question with a easy explanation in this single point.


A linear pair of angles is the term designated to two angles that are located 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 adjacent on the same line and together form 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 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 means that they always make on a straight line and are at all times supplementary angles.

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

The Linear Pair Axiom

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

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

The first axiom implies that if a ray is located 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 makes a straight angle among them. This is commonly called a straight line.

Examples of Linear Pairs of Angles

To imagine these axioms better, here are some diagram examples with their respective explanations.

Example One

Here in this example, we have two angles that are adjacent to each other. As you can see in the figure, the adjacent angles form a linear pair since the sum 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 instance, we possess two lines intersect, making four angles. Not all angles makes a linear pair, but each angle and the one next 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 case presents a common point of three lines. Let's look at the axiom and characteristics of linear pairs.

∠A 150 degrees

∠B: 50 degrees

∠C: 160 degrees

None of the angle totals add up to 180 degrees. As a result, we can come to the conclusion that this example has no linear pair unless we expand a straight line.

Applications of Linear Pair of Angles

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

In Real-Life Situations

There are multiple implementations of linear pairs of angles in real life. One familiar example 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 employ experts in this matter to make their job simpler. They employ linear pairs of angles to ensure that two adjacent walls form a 90-degree angle with the ground.

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

In Geometry

Linear pairs of angles as well play a role in geometry proofs. A regular proof that uses 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 additionally 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 above-mentioned two rules, you only need to figure out the measurement of one angle to determine the measure of the rest.

The concept of linear pairs is also employed for more sophisticated applications, such as working out the angles in polygons. It’s important to understand the basics of linear pairs, so you are prepared 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, observe if you can see some linear pairs! And, if you're attending a geometry class, bear in mind how linear pairs might be useful in proofs.

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