October 04, 2022

Linear Pair of AnglesDefinition, Axiom, Examples

The linear pair of angles is a significant concept in geometry. With several real-life uses, you'd be surprised to find how applicable this figure can be. Although you might think it has no use in your life, we all must grasp the ideas to ace those examinations in school.

To save you time and create this data readily accessible, here is an introductory insight into the characteristics of a linear pair of angles, with diagrams and examples to guide with your personal study sessions. We will also discuss few real-world and geometric applications.

What Is a Linear Pair of Angles?

Linearity, angles, and intersections are theories that remain to be applicable as you go forward in geometry and more sophisticated theorems and proofs. We will answer this query with a easy definition in this single point.


A linear pair of angles is the term given to two angles that are positioned 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 are adjacent on the same line and together form a straight line. The total of the angles in a linear pair will always make 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 vertex and a common arm. This means that they always form on a straight line and are always supplementary angles.

It is important to clarify that, while the linear pair are constantly adjacent angles, adjacent angles aren't always linear pairs.

The Linear Pair Axiom

Over the definition clarified, we will explore the two axioms critical to fully comprehend every example provided to you.

First, let's define what an axiom is. It is a mathematical postulate or hypothesis that is approved without having to prove; it is deemed obvious and self-explanatory. A linear pair of angles has two axioms connected with them.

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

The second axiom states that if two angles produces a linear pair, then uncommon arms of both angles produces a straight angle among them. In other words, they make a straight line.

Examples of Linear Pairs of Angles

To visualize these axioms better, here are a few drawn examples with their respective explanations.

Example One

Here in this example, we have two angles that are neighboring one another. As you can observe in the image, the adjacent angles form a linear pair because the total of their measures 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

Here, we possess two lines intersect, producing four angles. Not every angles form 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 case presents an intersection of three lines. Let's observe 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 consequence, we can conclude that this example has no linear pair unless we stretch one straight line.

Implementations of Linear Pair of Angles

At the moment we have explored what linear pairs are and have observed some cases, let's see how this concept 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 example is architects, who apply these axioms in their day-to-day work to determine if two lines are perpendicular and makes a straight angle.

Builders and construction professionals also employ masters in this subject to make their job simpler. They utilize linear pairs of angles to assure that two adjacent walls make a 90-degree angle with the floor.

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

In Geometry

Linear pairs of angles as well perform a role in geometry proofs. A ordinary 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 created are congruent.

The proof of vertical angles as well depends on linear pairs of angles. Even though the adjacent angles are supplementary and add up to 180 degrees, the opposite vertical angles are at all times equivalent to each other. Because of previously mentioned two rules, you only need to figure out the measure of one angle to determine the measurement of the rest.

The theorem of linear pairs is also used for more complicated implementation, such as figuring out the angles in polygons. It’s critical to understand the basics of linear pairs, so you are prepared for more advanced geometry.

As shown, linear pairs of angles are a relatively simple theorem with few fascinating uses. Next time you're out and about, see if you can see any linear pairs! And, if you're attending a geometry class, be on the lookout for how linear pairs may be useful in proofs.

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