October 04, 2022

Linear Pair of AnglesDefinition, Axiom, Examples

The linear pair of angles is an important concept in geometry. With so many real-life applications, you'd be astonished to figure out how relevant this figure can be. While you may believe it has no relevant in your life, we all should learn the concept to nail those examinations in school.

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

What Is a Linear Pair of Angles?

Linearity, angles, and intersections are theories that remain to be relevant as you progress in geometry and more sophisticated theorems and proofs. We will answer this query with a straightforward definition in this single point.


A linear pair of angles is the term provided to two angles that are located on a straight line and the total of their measurement of angles is 180 degrees.

To put it simply, linear pairs of angles are two angles that sit on the same line and pair up to create a straight line. The total of the angles in a linear pair will always produce a straight angle equal to 180 degrees.

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

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

The Linear Pair Axiom

Over the definition simplified, we will explore the two axioms critical to completely comprehend any 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 believed obvious and self-evident. A linear pair of angles has two axioms connected with them.

The first axiom implies that if a ray is located on a line, the adjacent angles will make a straight angle, namely called a linear pair.

The second axiom implies 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 visualize these axioms better, here are some figure examples with their corresponding explanations.

Example One

As we can see in this instance, we have two angles that are adjacent to each other. As you can notice in the image, the adjacent angles form a linear pair due to the fact that the total of their measures 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, making four angles. Not all angles creates a linear pair, but each 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 case, the linear pairs are:

∠A and ∠B

∠B and ∠C

∠C and ∠D

∠D and ∠A

Example Three

This instance represents an intersection 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 effect, we can come to the conclusion that this example has no linear pair until we stretch one straight line.

Implementations of Linear Pair of Angles

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

In Real-World Situations

There are several utilizations of linear pairs of angles in real-world. One common case is architects, who utilize these axioms in their day-to-day job to determine if two lines are perpendicular and form a straight angle.

Construction and Building professionals also employ expertise in this field to make their job simpler. They employ linear pairs of angles to make sure that two adjacent walls form a 90-degree angle with the floor.

Engineers also utilizes linear pairs of angles frequently. They do so by figuring out the tension in beams and trusses.

In Geometry

Linear pairs of angles additionally perform a function in geometry proofs. A ordinary proof that uses linear pairs is the alternate interior angles concept. This theorem expresses that if two lines are parallel and intersected by a transversal line, the alternate interior angles made are congruent.

The proof of vertical angles as well depends on linear pairs of angles. While 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 previously mentioned two rules, you are only required to figure out the measure of one angle to figure out the measure of the rest.

The concept of linear pairs is subsequently employed for more complex implementation, such as working out the angles in polygons. It’s important to grasp the basics of linear pairs, so you are ready for more progressive geometry.

As shown, linear pairs of angles are a comparatively easy theorem with several interesting implementations. Next time you're out and about, take note if you can notice some linear pairs! And, if you're attending a geometry class, take notes on how linear pairs may be useful in proofs.

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