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 uses, you'd be astonished to find how relevant this figure can be. While you may believe it has no relevant in your life, we all must learn the ideas to nail those examinations 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 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 concepts that exist to be applicable as you progress in geometry and more complex 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 positioned on a straight line and the total of their measurement of angles is 180 degrees.

To put it easily, linear pairs of angles are two angles that are aligned on the same line and pair up to create a straight line. The total of the angles in a linear pair will at all times create a straight angle equivalent
times to 180 degrees.

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

It is important to make clear that, even though the linear pair are at all times adjacent angles, adjacent angles aren't always linear pairs.

The Linear Pair Axiom

With the definition clarified, we will study the two axioms seriously to completely grasp every example thrown at you.

Initiate with definition of what an axiom is. It is a mathematical postulate or hypothesis that is accepted without proof; it is considered clear and self-evident. A linear pair of angles has two axioms connected with them.

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

The second axiom states that if two angles create a linear pair, then uncommon arms of both angles makes a straight angle between them. This is also known as a straight line.

Examples of Linear Pairs of Angles

To imagine these axioms better, here are some figure examples with their respective 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 figure, the adjacent angles form a linear pair because the total 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

Here, we possess two lines intersect, producing four angles. Not all angles form 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 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 take note of 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 conclude that this example has no linear pair unless we expand one straight line.

Applications of Linear Pair of Angles

At the moment we have learned what linear pairs are and have observed some examples, let's see how this concept can be used in geometry and the real world.

In Real-World Scenarios

There are several applications of linear pairs of angles in real-world. One such example is architects, who apply these axioms in their daily job to establish if two lines are perpendicular and creates a straight angle.

Construction and Building professionals also utilize experts in this subject to make their work easier. They employ linear pairs of angles to assure that two close walls create a 90-degree angle with the ground.

Engineers also uses linear pairs of angles regularly. 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 regular proof that employs linear pairs is the alternate interior angles theorem. This concept 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 additionally 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 each other. Because of previously mentioned two rules, you are only required to determine the measurement of one angle to determine the measurement of the rest.

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

As demonstrated, linear pairs of angles are a relatively easy concept with few engaging uses. Next time you're out and about, take note if you can spot some linear pairs! And, if you're attending a geometry class, take notes on how linear pairs may be helpful in proofs.

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