October 04, 2022

Linear Pair of AnglesDefinition, Axiom, Examples

The linear pair of angles is an essential concept in geometry. With several real-world functions, you'd be surprised to find how applicable this figure can be. While you may believe it has no use in your life, we all must grasp the concept to ace those exams in school.

To save you time and create this information easily accessible, here is an introductory insight into the characteristics of a linear pair of angles, with diagrams and examples to help with your private 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 go forward in geometry and more complicated theorems and proofs. We will answer this query with a straightforward definition in this unique point.


A linear pair of angles is the term given to two angles that are situated 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 aligned on the same line and together form a straight line. The total 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 vertex and a common arm. This means that at all times create on a straight line and are at all times supplementary angles.

It is essential to clarify that, even though the linear pair are constantly adjacent angles, adjacent angles never constantly linear pairs.

The Linear Pair Axiom

With the definition simplified, we will examine the two axioms seriously to fully understand any example thrown at you.

First, let's define what an axiom is. It is a mathematical postulate or hypothesis that is acknowledged without proof; 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 stands on a line, the adjacent angles will create a straight angle, namely called a linear pair.

The second axiom states that if two angles produces a linear pair, then uncommon arms of both angles makes 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 next to each other. As you can see in the diagram, the adjacent angles form a linear pair because the sum of their measures equals 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 example, we have two lines intersect, creating four angles. Not every angles creates a linear pair, but each angle and the one close 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 instance presents 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 consequence, we can come to the conclusion that this figure has no linear pair unless we expand a straight line.

Uses of Linear Pair of Angles

Now that we have explored what linear pairs are and have looked at some examples, let's see how this theorem can be used in geometry and the real world.

In Real-Life Scenarios

There are multiple utilizations of linear pairs of angles in real-world. One familiar example is architects, who use these axioms in their day-to-day work to establish if two lines are perpendicular and makes a straight angle.

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

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

In Geometry

Linear pairs of angles as well play a function in geometry proofs. A regular 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 replies on linear pairs of angles. While the adjacent angles are supplementary and sum up to 180 degrees, the opposite vertical angles are always equivalent to each other. Because of above-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 further utilized for more sophisticated uses, such as measuring the angles in polygons. It’s essential to understand the fundamentals of linear pairs, so you are ready for more advanced geometry.

As you can see, linear pairs of angles are a comparatively easy concept with some engaging uses. Later when you're out and about, observe if you can spot some linear pairs! And, if you're participating in a geometry class, take notes on how linear pairs may be helpful in proofs.

Level Up Your Geometry Skills through Grade Potential

Geometry is entertaining and beneficial, majorly if you are curious about the domain of architecture or construction.

Still, if you're struggling to understand linear pairs of angles (or any other theorem in geometry), think about signing up for a tutoring session with Grade Potential. One of our expert instructors can guide you comprehend the material and nail your next exam.