October 04, 2022

Linear Pair of AnglesDefinition, Axiom, Examples

The linear pair of angles is an essential concept in geometry. With so many real-world applications, you'd be astonished to find how relevant this figure can be. While you may think it has no relevant in your life, we all should learn the ideas to ace those exams in school.

To save you time and offer this data readily accessible, here is an introductory insight into the properties of a linear pair of angles, with diagrams and examples to help with your private study sessions. We will also discuss some 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 move ahead with geometry and more sophisticated theorems and proofs. We will answer this question with a straightforward definition in this unique point.


A linear pair of angles is the name given to two angles that are situated on a straight line and have the sum of their measurement of angles is 180 degrees.

To put it easily, linear pairs of angles are two angles that sit on the same line and pair up to form a straight line. The total of the angles in a linear pair will always create 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 means that at all times form on a straight line and are at all times supplementary angles.

It is essential to make clear that, while the linear pair are always adjacent angles, adjacent angles aren't always linear pairs.

The Linear Pair Axiom

Through the definition simplified, we will examine the two axioms earnestly to completely understand every example given to you.

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

The first axiom establishes that if a ray stands on a line, the adjacent angles will create a straight angle, making them a linear pair.

The second axiom establishes that if two angles produces a linear pair, then uncommon arms of both angles produces 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 drawn examples with their individual explanations.

Example One

Here in this example, we have two angles that are next to one another. As you can observe in the diagram, the adjacent angles form a linear pair due to the fact that the sum of their measurement equals 180 degrees. They are also supplementary angles, since 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 have two lines intersect, producing four angles. Not every angles creates a linear pair, but respective angle and the one next 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 a common point 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 combinations sum up to 180 degrees. As a consequence, we can come to the conclusion that this diagram has no linear pair until we expand a straight line.

Implementations of Linear Pair of Angles

At the moment we have gone through what linear pairs are and have observed some instances, let's see how this theorem can be applied in geometry and the real-life scenario.

In Real-World Situations

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

Construction and Building professionals also employ masters in this matter to make their work less complex. They employ linear pairs of angles to assure that two adjacent walls form a 90-degree angle with the ground.

Engineers also utilizes 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 perform a role in geometry proofs. A common proof that employs linear pairs is the alternate interior angles theorem. This concept 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. Although the adjacent angles are supplementary and sum up to 180 degrees, the opposite vertical angles are always equal to one another. Because of previously mentioned two rules, you are only required to figure out the measurement of one angle to figure out the measure of the rest.

The theorem of linear pairs is also used for more sophisticated implementation, such as measuring the angles in polygons. It’s critical to grasp the fundamentals of linear pairs, so you are prepared for more progressive geometry.

As you can see, linear pairs of angles are a relatively easy theorem with few engaging implementations. Later when you're out and about, observe if you can see any linear pairs! And, if you're participating in a geometry class, be on the lookout for how linear pairs may be helpful in proofs.

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