October 04, 2022

Linear Pair of AnglesDefinition, Axiom, Examples

The linear pair of angles is an important subject in geometry. With multiple real-world functions, you'd be surprised to discover how useful this figure can be. While you may believe it has no use in your life, we all should grasp the ideas to nail those examinations in school.

To save you time and make this information readily accessible, here is an introductory insight into the characteristics of a linear pair of angles, with diagrams and examples to guide with your private study sessions. We will also discuss few real-world and geometric applications.

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 simple definition in this unique point.


A linear pair of angles is the term 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 at all times create a straight angle equivalent
times to 180 degrees.

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

It is essential to make clear that, although the linear pair are always adjacent angles, adjacent angles not at all times linear pairs.

The Linear Pair Axiom

With the precise explanation, we will study the two axioms critical to completely understand every example given to you.

Initiate with definition of what an axiom is. It is a mathematical postulate or assumption that is approved without having to prove; it is believed evident 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 produces a linear pair, then uncommon arms of both angles produces a straight angle between them. In other words, they make a straight line.

Examples of Linear Pairs of Angles

To imagine these axioms better, here are a few diagram examples with their individual answers.

Example One

Here in this instance, we have two angles that are neighboring one another. 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

Here, we have two lines intersect, producing four angles. Not all angles form a linear pair, but each angle and the one next to it makes 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 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 diagram has no linear pair unless we extend a straight line.

Uses of Linear Pair of Angles

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

In Real-Life Scenarios

There are several utilizations of linear pairs of angles in real life. One such case is architects, who use these axioms in their daily job to identify if two lines are perpendicular and form a straight angle.

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

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

In Geometry

Linear pairs of angles also perform a role in geometry proofs. A regular proof that utilizes linear pairs is the alternate interior angles concept. This theorem expresses that if two lines are parallel and intersected by a transversal line, the alternate interior angles formed are congruent.

The proof of vertical angles additionally depends on linear pairs of angles. Although the adjacent angles are supplementary and add up to 180 degrees, the opposite vertical angles are at all times equivalent to one another. Because of above-mentioned two rules, you only need to determine the measurement of any one angle to determine the measure of the rest.

The theorem of linear pairs is further used for more complicated applications, such as determining the angles in polygons. It’s critical to understand the fundamentals of linear pairs, so you are prepared for more advanced geometry.

As you can see, linear pairs of angles are a comparatively simple concept with several interesting applications. Next time you're out and about, take note if you can see some linear pairs! And, if you're attending a geometry class, take notes on how linear pairs may be useful in proofs.

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