October 04, 2022

Linear Pair of AnglesDefinition, Axiom, Examples

The linear pair of angles is an essential concept in geometry. With multiple real-life functions, you'd be amazed to find how useful this figure can be. While you may think it has no relevant in your life, we all must learn the concept to ace those tests in school.

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

What Is a Linear Pair of Angles?

Linearity, angles, and intersections are theories that continue to be relevant as you move ahead with geometry and more complex theorems and proofs. We will answer this question with a straightforward definition in this unique point.


A linear pair of angles is the term provided to two angles that are positioned 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 are adjacent on the same line and pair up to form a straight line. The sum of the angles in a linear pair will at all times produce a straight angle equivalent
times to 180 degrees.

It is important to bear in mind that linear pairs are always 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 always supplementary angles.

It is essential to make clear that, although the linear pair are constantly adjacent angles, adjacent angles never constantly linear pairs.

The Linear Pair Axiom

Through the precise explanation, we will explore the two axioms earnestly to completely understand every example given 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 considered obvious and self-explanatory. A linear pair of angles has two axioms associated with them.

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

The second axiom establishes that if two angles makes a linear pair, then uncommon arms of both angles create a straight angle among them. This is commonly called a straight line.

Examples of Linear Pairs of Angles

To imagine these axioms better, here are some diagram examples with their corresponding answers.

Example One

As we can see in this example, we have two angles that are adjacent to each other. As you can see in the diagram, the adjacent angles form a linear pair due to the fact that the sum 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 every angles creates 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 example, the linear pairs are:

∠A and ∠B

∠B and ∠C

∠C and ∠D

∠D and ∠A

Example Three

This case represents an intersection of three lines. Let's take note of the axiom and properties of linear pairs.

∠A 150 degrees

∠B: 50 degrees

∠C: 160 degrees

None of the angle totals add up to 180 degrees. As a effect, we can conclude that this example has no linear pair unless we stretch a straight line.

Uses of Linear Pair of Angles

Now that we have learned what linear pairs are and have looked at some cases, let's see how this theorem can be applied in geometry and the real-life scenario.

In Real-Life Scenarios

There are several utilizations of linear pairs of angles in real life. One familiar example is architects, who utilize these axioms in their day-to-day job to determine if two lines are perpendicular and makes a straight angle.

Builders and construction professionals also use expertise in this field to make their job easier. They employ linear pairs of angles to make sure that two close walls form a 90-degree angle with the floor.

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

In Geometry

Linear pairs of angles also perform a role in geometry proofs. A common proof that utilizes linear pairs is the alternate interior angles concept. 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 as well depends on linear pairs of angles. Even though 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 previously mentioned two rules, you are only required to know the measurement of one angle to figure out the measure of the rest.

The theorem of linear pairs is also employed for more complex applications, such as determining the angles in polygons. It’s critical to grasp the basics of linear pairs, so you are ready for more complex geometry.

As you can see, linear pairs of angles are a comparatively simple theorem with some engaging applications. Later when you're out and about, observe if you can see some linear pairs! And, if you're taking a geometry class, take notes on how linear pairs might be helpful in proofs.

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