October 04, 2022

Linear Pair of AnglesDefinition, Axiom, Examples

The linear pair of angles is a significant subject in geometry. With several real-world uses, you'd be surprised to discover how relevant this figure can be. Even though you may believe it has no application in your life, we all should understand the ideas to ace those exams in school.

To save you time and offer this info readily accessible, here is an preliminary insight into the characteristics of a linear pair of angles, with visualizations and examples to help 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 sophisticated theorems and proofs. We will answer this question with a straightforward explanation 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 angles measure 180 degrees.

To put it simply, linear pairs of angles are two angles that sit on the same line and together form a straight line. The total of the angles in a linear pair will at all times make a straight angle equivalent
times to 180 degrees.

It is important to bear in mind that linear pairs are at all times at adjacent angles. They share a common apex and a common arm. This means that they always create on a straight line and are always supplementary angles.

It is important to explain that, even though the linear pair are always adjacent angles, adjacent angles never constantly linear pairs.

The Linear Pair Axiom

Through the definition simplified, we will explore the two axioms critical to fully comprehend every example given to you.

Initiate with definition of what an axiom is. It is a mathematical postulate or assumption that is accepted without having to prove; it is believed obvious and self-evident. A linear pair of angles has two axioms associated with them.

The first axiom states that if a ray is located on a line, the adjacent angles will make 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 produces 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 diagram examples with their corresponding explanations.

Example One

In this instance, we have two angles that are adjacent to one another. As you can see in the figure, the adjacent angles form a linear pair since the sum of their measurement is equivalent to 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

In this example, we possess two lines intersect, making four angles. Not all angles makes a linear pair, but respective angle and the one adjacent 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 observe the axiom and properties of linear pairs.

∠A 150 degrees

∠B: 50 degrees

∠C: 160 degrees

None of the angle combinations sum up to 180 degrees. As a effect, we can come to the conclusion that this example has no linear pair unless we expand a straight line.

Uses of Linear Pair of Angles

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

In Real-World Situations

There are several uses of linear pairs of angles in real life. One common case is architects, who use these axioms in their day-to-day work to check if two lines are perpendicular and creates a straight angle.

Construction and Building professionals also use masters in this subject to make their job less complex. They use linear pairs of angles to ensure that two close walls create a 90-degree angle with the floor.

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

In Geometry

Linear pairs of angles as well perform a function in geometry proofs. A common proof that utilizes linear pairs is the alternate interior angles concept. This theorem states that if two lines are parallel and intersected by a transversal line, the alternate interior angles made are congruent.

The proof of vertical angles also depends on linear pairs of angles. Although 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 these two rules, you only need to know the measurement of one angle to determine the measure of the rest.

The theorem of linear pairs is further utilized for more sophisticated applications, such as measuring the angles in polygons. It’s critical to understand the basics of linear pairs, so you are prepared for more advanced geometry.

As you can see, linear pairs of angles are a relatively simple theorem with some engaging implementations. Next time you're out and about, see if you can spot any linear pairs! And, if you're taking a geometry class, be on the lookout for how linear pairs might be helpful in proofs.

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