October 04, 2022

Linear Pair of AnglesDefinition, Axiom, Examples

The linear pair of angles is an essential concept in geometry. With several real-world uses, you'd be astonished to figure out how relevant this figure can be. Even though you may believe it has no application in your life, we all must understand the concept to ace those exams in school.

To save you time and create this information easy to access, here is an preliminary insight into the properties of a linear pair of angles, with diagrams and examples to guide with your personal study sessions. We will also discuss some real-world and geometric applications.

What Is a Linear Pair of Angles?

Linearity, angles, and intersections are theories that continue to be useful as you progress in geometry and more complicated theorems and proofs. We will answer this question with a easy definition in this single point.


A linear pair of angles is the name designated to two angles that are situated on a straight line and the total of their angles measure 180 degrees.

To put it easily, linear pairs of angles are two angles that sit 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 create a straight angle equivalent
times to 180 degrees.

It is crucial to note that linear pairs are at all times 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, 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 any example given to you.

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

The first axiom establishes that if a ray stands on a line, the adjacent angles will make a straight angle, also known as a linear pair.

The second axiom states 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 a few figure examples with their respective answers.

Example One

In this instance, we have two angles that are adjacent to each other. As you can observe in the figure, the adjacent angles form a linear pair because 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

In this instance, we possess two lines intersect, producing four angles. Not all angles form a linear pair, but each angle and the one next to it form 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 case shows 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 totals add up to 180 degrees. As a consequence, we can conclude that this example has no linear pair until we stretch one straight line.

Applications of Linear Pair of Angles

Now that we have explored what linear pairs are and have observed some examples, let’s check how this concept can be implemented in geometry and the real-life scenario.

In Real-Life Situations

There are many implementations of linear pairs of angles in real life. One familiar case is architects, who utilize these axioms in their day-to-day job to check if two lines are perpendicular and form a straight angle.

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

Engineers also apply linear pairs of angles frequently. They do so by calculating the weight on the beams and trusses.

In Geometry

Linear pairs of angles additionally play a function in geometry proofs. A common proof that employs linear pairs is the alternate interior angles theorem. This theorem explains 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 replies on linear pairs of angles. While the adjacent angles are supplementary and add up to 180 degrees, the opposite vertical angles are always equal to one another. Because of previously mentioned two rules, you only need to figure out the measure of any one angle to work out the measurement of the rest.

The theorem of linear pairs is further utilized for more sophisticated uses, such as determining the angles in polygons. It’s important to understand the basics of linear pairs, so you are prepared for more complex geometry.

As demonstrated, linear pairs of angles are a somewhat simple theorem with some interesting implementations. Later when you're out and about, observe if you can spot 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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