October 04, 2022

Linear Pair of AnglesDefinition, Axiom, Examples

The linear pair of angles is an essential subject in geometry. With several real-life functions, you'd be amazed to discover how useful this figure can be. Even though you may believe it has no application in your life, we all must learn the concept to nail those exams in school.

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

What Is a Linear Pair of Angles?

Linearity, angles, and intersections are concepts that remain to be applicable as you go forward in geometry and more complicated theorems and proofs. We will answer this question with a simple explanation in this single 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 easily, linear pairs of angles are two angles that are aligned on the same line and pair up to create a straight line. The sum of the angles in a linear pair will at all times create a straight angle equal to 180 degrees.

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

It is important to clarify that, although the linear pair are constantly adjacent angles, adjacent angles not at all times linear pairs.

The Linear Pair Axiom

Through the definition clarified, we will explore the two axioms critical to completely comprehend every example provided to you.

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

The first axiom states that if a ray is located on a line, the adjacent angles will form a straight angle, making them a linear pair.

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

Examples of Linear Pairs of Angles

To visualize these axioms better, here are some diagram examples with their corresponding answers.

Example One

Here in this example, we have two angles that are neighboring each other. As you can notice in the diagram, the adjacent angles form a linear pair since the total 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 example, we have two lines intersect, making four angles. Not all angles form a linear pair, but respective angle and the one adjacent 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 example 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 totals add up to 180 degrees. As a result, we can conclude that this example has no linear pair until we extend a straight line.

Implementations of Linear Pair of Angles

At the moment we have gone through what linear pairs are and have observed some instances, let’s check how this theorem can be utilized in geometry and the real-life scenario.

In Real-Life Situations

There are multiple utilizations of linear pairs of angles in real life. One such example is architects, who use these axioms in their day-to-day work to determine if two lines are perpendicular and creates a straight angle.

Construction and Building professionals also employ masters in this subject to make their job simpler. They utilize linear pairs of angles to assure that two adjacent walls form a 90-degree angle with the ground.

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

In Geometry

Linear pairs of angles additionally play a function in geometry proofs. A ordinary proof that employs linear pairs is the alternate interior angles concept. This theorem 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 replies on linear pairs of angles. While the adjacent angles are supplementary and sum up to 180 degrees, the opposite vertical angles are always equivalent to one another. Because of previously mentioned two rules, you only need to know the measure of any one angle to determine the measure of the rest.

The concept of linear pairs is subsequently used for more complex applications, such as working out the angles in polygons. It’s essential to understand the fundamentals of linear pairs, so you are ready for more progressive geometry.

As demonstrated, linear pairs of angles are a somewhat simple concept with several interesting applications. Later when you're out and about, observe if you can notice any linear pairs! And, if you're attending a geometry class, be on the lookout for how linear pairs may be useful in proofs.

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