October 04, 2022

Linear Pair of AnglesDefinition, Axiom, Examples

The linear pair of angles is an essential subject in geometry. With several real-world applications, you'd be surprised to figure out how useful this figure can be. Although you might think it has no relevant in your life, we all should learn the ideas to nail those tests in school.

To save you time and make this information easily accessible, here is an introductory insight into the properties of a linear pair of angles, with images and examples to guide with your private study sessions. We will also discuss few real-life and geometric applications.

What Is a Linear Pair of Angles?

Linearity, angles, and intersections are theories that remain to be useful as you move ahead with geometry and more complex theorems and proofs. We will answer this query 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 angles measure 180 degrees.

To put it simply, linear pairs of angles are two angles that are adjacent on the same line and together create 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 essential to keep in mind that linear pairs are always at adjacent angles. They share a common apex and a common arm. This implies that they always create on a straight line and are always supplementary angles.

It is important 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 simplified, we will examine the two axioms critical to completely grasp every example thrown at you.

Initiate with definition of what an axiom is. It is a mathematical postulate or assumption that is acknowledged without having to prove; it is considered evident and self-evident. 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 form a straight angle, also known as a linear pair.

The second axiom establishes that if two angles makes a linear pair, then uncommon arms of both angles create a straight angle among them. This is also known as a straight line.

Examples of Linear Pairs of Angles

To visualize these axioms better, here are a few drawn examples with their corresponding answers.

Example One

In this instance, we have two angles that are adjacent to one another. As you can observe in the image, 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 have two lines intersect, producing four angles. Not every angles form a linear pair, but respective angle and the one close to it form 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 instance presents 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 sum up to 180 degrees. As a result, we can come to the conclusion that this image has no linear pair unless we stretch one straight line.

Applications of Linear Pair of Angles

Now that we have gone through what linear pairs are and have observed some examples, let’s check how this concept can be applied in geometry and the real world.

In Real-Life Scenarios

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

Builders and construction professionals also utilize experts in this matter to make their work easier. They use linear pairs of angles to assure that two adjacent walls make a 90-degree angle with the ground.

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

In Geometry

Linear pairs of angles as well play a role in geometry proofs. A ordinary proof that utilizes linear pairs is the alternate interior angles concept. This concept expresses 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 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 each other. Because of above-mentioned two rules, you are only required to determine the measure of one angle to determine the measurement of the rest.

The concept of linear pairs is also utilized for more complex uses, such as figuring out the angles in polygons. It’s critical to grasp the fundamentals of linear pairs, so you are ready for more advanced geometry.

As demonstrated, linear pairs of angles are a relatively easy theorem with some engaging implementations. Later when you're out and about, observe if you can spot some linear pairs! And, if you're attending a geometry class, bear in mind how linear pairs may be helpful in proofs.

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