October 04, 2022

Linear Pair of AnglesDefinition, Axiom, Examples

The linear pair of angles is a significant concept in geometry. With several real-life uses, you'd be surprised to find how relevant this figure can be. While you might believe it has no relevant in your life, we all must grasp the ideas to nail those examinations in school.

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

What Is a Linear Pair of Angles?

Linearity, angles, and intersections are theories that exist to be relevant as you go forward in geometry and more complicated theorems and proofs. We will answer this query with a simple definition in this unique 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 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 total 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 suggests that they always create on a straight line and are always supplementary angles.

It is crucial to explain that, although the linear pair are always adjacent angles, adjacent angles not at all times linear pairs.

The Linear Pair Axiom

Through the definition clarified, we will explore the two axioms seriously to fully grasp any example given to you.

First, let's define what an axiom is. It is a mathematical postulate or assumption that is accepted without proof; it is deemed evident and self-explanatory. A linear pair of angles has two axioms associated 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 create 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 some diagram examples with their corresponding answers.

Example One

In this instance, we have two angles that are neighboring one another. As you can observe in the diagram, the adjacent angles form a linear pair because the sum 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, making four angles. Not all angles form a linear pair, but respective angle and the one close to it makes 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 instance presents convergence of three lines. Let's look at the axiom and properties of linear pairs.

∠A 150 degrees

∠B: 50 degrees

∠C: 160 degrees

None of the angle combinations add up to 180 degrees. As a effect, we can conclude that this image has no linear pair unless we extend a straight line.

Applications 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 theorem can be utilized in geometry and the real world.

In Real-World Scenarios

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

Builders and construction professionals also utilize experts in this subject to make their work simpler. They employ linear pairs of angles to ensure that two close walls make a 90-degree angle with the floor.

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

In Geometry

Linear pairs of angles additionally play a role in geometry proofs. A regular proof that uses linear pairs is the alternate interior angles concept. This concept explains that if two lines are parallel and intersected by a transversal line, the alternate interior angles created are congruent.

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

The theorem of linear pairs is also utilized for more complex uses, such as measuring the angles in polygons. It’s essential to understand the basics of linear pairs, so you are ready for more advanced geometry.

As demonstrated, linear pairs of angles are a comparatively simple theorem with some interesting uses. Next time you're out and about, take note if you can notice any linear pairs! And, if you're participating in a geometry class, take notes on how linear pairs might be helpful in proofs.

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