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 functions, you'd be amazed to discover how useful this figure can be. Even though you might believe it has no application in your life, we all should learn the ideas to ace those exams in school.

To save your time and make this info easily accessible, here is an introductory insight into the properties 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 applications.

What Is a Linear Pair of Angles?

Linearity, angles, and intersections are theories that continue to be useful as you progress in geometry and more complex theorems and proofs. We will answer this question with a easy explanation in this unique point.


A linear pair of angles is the name designated 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 create a straight line. The sum of the angles in a linear pair will always make a straight angle equal to 180 degrees.

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

It is crucial to clarify that, although the linear pair are at all times adjacent angles, adjacent angles never constantly linear pairs.

The Linear Pair Axiom

With the definition clarified, we will examine the two axioms earnestly to fully grasp every example provided to you.

First, let's define what an axiom is. It is a mathematical postulate or hypothesis that is approved without proof; it is deemed evident and self-explanatory. A linear pair of angles has two axioms linked with them.

The first axiom implies that if a ray stands on a line, the adjacent angles will form a straight angle, namely called a linear pair.

The second axiom establishes that if two angles create a linear pair, then uncommon arms of both angles makes a straight angle between them. In other words, they make a straight line.

Examples of Linear Pairs of Angles

To visualize these axioms better, here are a few figure examples with their corresponding explanations.

Example One

Here in this instance, we have two angles that are neighboring each other. As you can notice in the image, 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, creating four angles. Not all angles makes 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 instance presents an intersection 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 conclude that this example has no linear pair unless we expand a straight line.

Uses of Linear Pair of Angles

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

In Real-World Situations

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

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

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

In Geometry

Linear pairs of angles as well play a role in geometry proofs. A ordinary 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 made are congruent.

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

The theorem of linear pairs is also used for more complicated applications, such as determining the angles in polygons. It’s essential to grasp the fundamentals of linear pairs, so you are ready for more complex geometry.

As shown, linear pairs of angles are a relatively simple theorem with some fascinating applications. Next time you're out and about, see if you can spot some linear pairs! And, if you're taking a geometry class, take notes on how linear pairs might be helpful in proofs.

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