October 04, 2022

Linear Pair of AnglesDefinition, Axiom, Examples

The linear pair of angles is an essential subject in geometry. With so many real-world functions, you'd be astonished to discover how applicable this figure can be. Even though you may wonder if it has no use in your life, we all must grasp the concept to ace those exams in school.

To save you time and create this info readily accessible, here is an introductory insight into the properties of a linear pair of angles, with visualizations and examples to help with your private study sessions. We will also discuss few real-world and geometric uses.

What Is a Linear Pair of Angles?

Linearity, angles, and intersections are theories that exist to be relevant as you progress in geometry and more sophisticated theorems and proofs. We will answer this query with a simple definition in this single point.


A linear pair of angles is the term designated to two angles that are positioned on a straight line and the total of their angles measure 180 degrees.

To put it easily, linear pairs of angles are two angles that are aligned on the same line and pair up to form 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 important to keep 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 essential to make clear that, even though the linear pair are at all times adjacent angles, adjacent angles not at all times linear pairs.

The Linear Pair Axiom

With the definition simplified, we will study the two axioms critical to fully comprehend 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 considered 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 implies that if two angles produces a linear pair, then uncommon arms of both angles produces 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 a few drawn examples with their respective answers.

Example One

In this instance, we have two angles that are adjacent to one another. As you can notice in the figure, the adjacent angles form a linear pair since the total of their measures is equivalent to 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 form a linear pair, but respective 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 example, the linear pairs are:

∠A and ∠B

∠B and ∠C

∠C and ∠D

∠D and ∠A

Example Three

This case represents an intersection of three lines. Let's take note of the axiom and characteristics 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 figure has no linear pair until 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 understand how this theorem can be utilized in geometry and the real world.

In Real-World Situations

There are multiple uses of linear pairs of angles in real-world. One such case is architects, who use these axioms in their daily work to check if two lines are perpendicular and form a straight angle.

Construction and Building professionals also employ experts in this subject to make their work less complex. They employ linear pairs of angles to assure that two adjacent walls make a 90-degree angle with the ground.

Engineers also utilizes linear pairs of angles regularly. They do so by working out the pressure on the beams and trusses.

In Geometry

Linear pairs of angles as well perform a function in geometry proofs. A ordinary proof that utilizes linear pairs is the alternate interior angles theorem. This concept states that if two lines are parallel and intersected by a transversal line, the alternate interior angles formed are congruent.

The proof of vertical angles also depends on linear pairs of angles. Even though 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 previously mentioned two rules, you are only required to determine the measure of any one angle to determine the measurement of the rest.

The concept of linear pairs is further used for more sophisticated uses, such as figuring out the angles in polygons. It’s essential to grasp the basics of linear pairs, so you are ready for more advanced geometry.

As shown, linear pairs of angles are a somewhat simple theorem with several interesting implementations. Later when you're out and about, take note if you can notice some linear pairs! And, if you're participating in a geometry class, be on the lookout for how linear pairs may be helpful in proofs.

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