October 04, 2022

Linear Pair of AnglesDefinition, Axiom, Examples

The linear pair of angles is a significant subject in geometry. With so many real-world applications, you'd be amazed to discover how applicable this figure can be. Even though you might wonder if it has no relevant in your life, we all should understand the ideas to nail those examinations in school.

To save you time and create this info readily accessible, here is an introductory insight into the characteristics of a linear pair of angles, with visualizations and examples to assist with your private study sessions. We will also talk about few real-life and geometric applications.

What Is a Linear Pair of Angles?

Linearity, angles, and intersections are ideas 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 explanation 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 pair up to form a straight line. The total of the angles in a linear pair will at all times 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 suggests that they always create on a straight line and are at all times supplementary angles.

It is essential to clarify that, even though the linear pair are constantly adjacent angles, adjacent angles aren't always linear pairs.

The Linear Pair Axiom

Over the precise explanation, we will examine the two axioms earnestly to completely comprehend any example given to you.

Initiate with definition of what an axiom is. It is a mathematical postulate or assumption that is approved without proof; it is believed clear and self-evident. 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 create 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. This is commonly called a straight line.

Examples of Linear Pairs of Angles

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

Example One

Here in this instance, we have two angles that are adjacent to each other. As you can observe in the figure, the adjacent angles form a linear pair since the total of their measurement equals 180 degrees. They are also supplementary angles, as 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 all angles form 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 case, the linear pairs are:

∠A and ∠B

∠B and ∠C

∠C and ∠D

∠D and ∠A

Example Three

This case shows convergence 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 consequence, we can come to the conclusion that this image has no linear pair unless 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 cases, let's see how this concept can be applied in geometry and the real world.

In Real-Life Situations

There are multiple implementations of linear pairs of angles in real life. One common example is architects, who apply these axioms in their day-to-day job to identify if two lines are perpendicular and form a straight angle.

Builders and construction professionals also utilize expertise in this subject to make their job easier. They use linear pairs of angles to make sure that two adjacent walls form a 90-degree angle with the floor.

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

In Geometry

Linear pairs of angles as well play a role in geometry proofs. A regular proof that employs linear pairs is the alternate interior angles concept. 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 as well replies on linear pairs of angles. Even though the adjacent angles are supplementary and sum up to 180 degrees, the opposite vertical angles are at all times equal to each other. Because of these two rules, you only need to know the measurement of one angle to work out the measure of the rest.

The concept of linear pairs is also utilized for more sophisticated uses, such as measuring the angles in polygons. It’s essential to grasp the fundamentals of linear pairs, so you are prepared for more progressive geometry.

As shown, linear pairs of angles are a comparatively simple concept with few interesting implementations. Next time you're out and about, observe if you can spot any linear pairs! And, if you're participating in a geometry class, be on the lookout for how linear pairs might be helpful in proofs.

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