October 04, 2022

Linear Pair of AnglesDefinition, Axiom, Examples

The linear pair of angles is an important subject in geometry. With so many real-life functions, you'd be astonished to discover how relevant this figure can be. While you may think it has no use in your life, we all must learn the concept to ace those tests in school.

To save your time and offer this information easily 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 applications.

What Is a Linear Pair of Angles?

Linearity, angles, and intersections are ideas that remain to be relevant as you go forward in geometry and more complex theorems and proofs. We will answer this question with a easy explanation in this single point.


A linear pair of angles is the name given to two angles that are situated 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 form a straight line. The total of the angles in a linear pair will always make a straight angle equal to 180 degrees.

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

It is crucial to make clear that, even though the linear pair are always adjacent angles, adjacent angles never constantly linear pairs.

The Linear Pair Axiom

Through the precise explanation, we will examine the two axioms seriously to completely 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 proof; it is believed obvious and self-explanatory. A linear pair of angles has two axioms linked with them.

The first axiom establishes that if a ray is located on a line, the adjacent angles will make 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 visualize these axioms better, here are a few figure examples with their respective explanations.

Example One

As we can see in this example, we have two angles that are adjacent to each other. As you can notice in the figure, the adjacent angles form a linear pair due to the fact that the sum of their measurement 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 have two lines intersect, producing four angles. Not every angles form a linear pair, but respective 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 example, the linear pairs are:

∠A and ∠B

∠B and ∠C

∠C and ∠D

∠D and ∠A

Example Three

This instance presents an intersection 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 add up to 180 degrees. As a result, we can conclude that this image has no linear pair unless we expand one straight line.

Uses of Linear Pair of Angles

At the moment we have explored what linear pairs are and have observed some instances, let’s check how this theorem can be implemented in geometry and the real world.

In Real-Life Scenarios

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

Builders and construction professionals also utilize experts in this subject to make their job easier. They use linear pairs of angles to make sure that two adjacent walls create a 90-degree angle with the floor.

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

In Geometry

Linear pairs of angles also play a function in geometry proofs. A ordinary 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 made are congruent.

The proof of vertical angles also replies on linear pairs of angles. Even though the adjacent angles are supplementary and add up to 180 degrees, the opposite vertical angles are always equivalent to each other. Because of above-mentioned two rules, you are only required to figure out the measure of any one angle to work out the measure of the rest.

The theorem of linear pairs is also used for more sophisticated applications, such as determining the angles in polygons. It’s important to grasp the fundamentals of linear pairs, so you are prepared for more complex geometry.

As shown, linear pairs of angles are a relatively simple theorem with few fascinating applications. Next time you're out and about, take note if you can spot some 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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