October 04, 2022

Linear Pair of AnglesDefinition, Axiom, Examples

The linear pair of angles is a significant concept in geometry. With so many real-life functions, you'd be astonished to figure out how applicable this figure can be. While you might wonder if it has no relevant in your life, we all should understand the concept to nail those exams in school.

To save you time and offer this data easily accessible, here is an introductory insight into the characteristics of a linear pair of angles, with diagrams and examples to help with your personal study sessions. We will also discuss few 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 move ahead with geometry and more complex theorems and proofs. We will answer this query with a straightforward definition in this unique point.


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

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

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

The Linear Pair Axiom

With 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 acknowledged without proof; it is considered obvious and self-evident. A linear pair of angles has two axioms connected with them.

The first axiom implies 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 states that if two angles create 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 envision these axioms better, here are some figure examples with their respective explanations.

Example One

In this instance, we have two angles that are next to one another. As you can see in the diagram, the adjacent angles form a linear pair since the total of their measurement 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 possess two lines intersect, producing four angles. Not every angles form a linear pair, but each 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 case, the linear pairs are:

∠A and ∠B

∠B and ∠C

∠C and ∠D

∠D and ∠A

Example Three

This example shows an intersection of three lines. Let's observe 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 diagram has no linear pair until we extend one straight line.

Applications of Linear Pair of Angles

At the moment we have learned what linear pairs are and have observed some instances, let's see how this theorem can be applied in geometry and the real-life scenario.

In Real-Life Scenarios

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

Construction and Building professionals also employ expertise in this subject to make their job easier. They utilize linear pairs of angles to assure that two adjacent walls form a 90-degree angle with the floor.

Engineers also uses linear pairs of angles regularly. They do so by working out the weight on the beams and trusses.

In Geometry

Linear pairs of angles additionally perform a function in geometry proofs. A ordinary proof that utilizes linear pairs is the alternate interior angles theorem. This theorem states 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 replies on linear pairs of angles. Even though the adjacent angles are supplementary and sum up to 180 degrees, the opposite vertical angles are always equal to each other. Because of above-mentioned two rules, you only need to determine the measure of any one angle to figure out the measure of the rest.

The theorem of linear pairs is also utilized for more complex uses, such as figuring out the angles in polygons. It’s critical to understand the basics of linear pairs, so you are ready for more progressive geometry.

As shown, linear pairs of angles are a comparatively easy theorem with several fascinating implementations. Later when you're out and about, observe if you can spot any linear pairs! And, if you're taking a geometry class, bear in mind how linear pairs might be useful in proofs.

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