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 uses, you'd be astonished to discover how useful this figure can be. While you may believe it has no relevant in your life, we all must grasp the ideas to ace those examinations in school.

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

What Is a Linear Pair of Angles?

Linearity, angles, and intersections are concepts that remain to be useful as you progress in geometry and more complex theorems and proofs. We will answer this question with a straightforward definition in this single point.


A linear pair of angles is the name provided to two angles that are situated on a straight line and the total 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 pair up to create a straight line. The sum of the angles in a linear pair will at all times make a straight angle equivalent
times to 180 degrees.

It is crucial to note 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 at all times supplementary angles.

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

The Linear Pair Axiom

Over the definition clarified, we will explore the two axioms seriously to completely comprehend every example given to you.

Let’s start by defining what an axiom is. It is a mathematical postulate or assumption that is accepted without proof; it is considered evident and self-explanatory. A linear pair of angles has two axioms connected with them.

The first axiom implies that if a ray stands on a line, the adjacent angles will form a straight angle, namely called 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 between them. In other words, they make a straight line.

Examples of Linear Pairs of Angles

To imagine these axioms better, here are some figure examples with their individual explanations.

Example One

As we can see in this instance, we have two angles that are next to each other. As you can see in the image, the adjacent angles form a linear pair due to the fact that the total of their measures is equivalent to 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, creating four angles. Not every angles creates a linear pair, but each 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 case shows a common point of three lines. Let's observe the axiom and characteristics of linear pairs.

∠A 150 degrees

∠B: 50 degrees

∠C: 160 degrees

None of the angle combinations add up to 180 degrees. As a result, we can conclude that this example has no linear pair until we expand one straight line.

Implementations of Linear Pair of Angles

Now that we have learned what linear pairs are and have looked at some examples, let’s check how this concept can be implemented in geometry and the real world.

In Real-World Scenarios

There are multiple implementations of linear pairs of angles in real life. One common case is architects, who use these axioms in their daily job to identify if two lines are perpendicular and creates a straight angle.

Construction and Building professionals also utilize expertise in this field to make their work easier. They employ linear pairs of angles to ensure that two close walls form a 90-degree angle with the floor.

Engineers also apply linear pairs of angles frequently. They do so by figuring out the tension in 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 expresses that if two lines are parallel and intersected by a transversal line, the alternate interior angles created are congruent.

The proof of vertical angles as well depends on linear pairs of angles. While the adjacent angles are supplementary and add up to 180 degrees, the opposite vertical angles are at all times equal to one another. Because of these two rules, you only need to know the measure of one angle to work out the measure of the rest.

The theorem of linear pairs is subsequently used for more complex uses, such as figuring out the angles in polygons. It’s essential to grasp the basics of linear pairs, so you are prepared for more progressive geometry.

As you can see, linear pairs of angles are a somewhat simple theorem with few fascinating implementations. Next time you're out and about, observe if you can notice some linear pairs! And, if you're attending a geometry class, take notes on how linear pairs might be useful in proofs.

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