October 04, 2022

Linear Pair of AnglesDefinition, Axiom, Examples

The linear pair of angles is an essential concept in geometry. With so many real-life applications, you'd be surprised to discover how applicable this figure can be. Even though you may believe it has no relevant in your life, we all must learn the ideas to nail those exams in school.

To save your time and offer this data easily accessible, here is an preliminary insight into the properties of a linear pair of angles, with diagrams and examples to assist 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 theories that continue to be applicable as you go forward in geometry and more complex theorems and proofs. We will answer this query with a straightforward definition in this unique point.


A linear pair of angles is the term designated to two angles that are situated on a straight line and have the sum of their measurement of angles is 180 degrees.

To put it simply, linear pairs of angles are two angles that are aligned on the same line and together form a straight line. The sum of the angles in a linear pair will always create a straight angle equivalent
times to 180 degrees.

It is important to note that linear pairs are at all times at adjacent angles. They share a common vertex and a common arm. This means that at all times make on a straight line and are at all times supplementary angles.

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

The Linear Pair Axiom

Through the precise explanation, we will study the two axioms critical to completely comprehend any example provided to you.

Initiate with definition of what an axiom is. It is a mathematical postulate or assumption that is accepted without proof; it is deemed obvious and self-explanatory. A linear pair of angles has two axioms associated with them.

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

The second axiom establishes that if two angles produces a linear pair, then uncommon arms of both angles makes a straight angle between them. This is commonly called a straight line.

Examples of Linear Pairs of Angles

To envision these axioms better, here are a few drawn examples with their respective explanations.

Example One

Here in this example, we have two angles that are next to each other. As you can notice in the figure, the adjacent angles form a linear pair because 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 possess two lines intersect, producing four angles. Not every angles creates 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 instance, the linear pairs are:

∠A and ∠B

∠B and ∠C

∠C and ∠D

∠D and ∠A

Example Three

This example shows an intersection 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 sum up to 180 degrees. As a effect, we can come to the conclusion that this example has no linear pair until we stretch a straight line.

Uses of Linear Pair of Angles

At the moment we have explored what linear pairs are and have observed some examples, let’s check how this concept can be utilized in geometry and the real world.

In Real-Life Situations

There are several uses of linear pairs of angles in real life. One such case is architects, who utilize these axioms in their day-to-day job to identify if two lines are perpendicular and creates a straight angle.

Builders and construction professionals also employ expertise in this matter to make their work simpler. They employ linear pairs of angles to make sure that two close walls create a 90-degree angle with the floor.

Engineers also uses linear pairs of angles regularly. They do so by figuring out the tension in beams and trusses.

In Geometry

Linear pairs of angles as well perform a role in geometry proofs. A ordinary proof that utilizes linear pairs is the alternate interior angles theorem. 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 replies on linear pairs of angles. Even though the adjacent angles are supplementary and add up to 180 degrees, the opposite vertical angles are always equal to one another. Because of previously mentioned two rules, you are only required to know the measure of one angle to determine the measurement of the rest.

The concept of linear pairs is subsequently used for more sophisticated implementation, such as working out the angles in polygons. It’s critical to grasp the fundamentals of linear pairs, so you are ready for more progressive geometry.

As you can see, linear pairs of angles are a somewhat easy theorem with some interesting implementations. Next time you're out and about, take note if you can notice 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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