October 04, 2022

Linear Pair of AnglesDefinition, Axiom, Examples

The linear pair of angles is an essential subject in geometry. With several real-life applications, you'd be surprised to discover how applicable this figure can be. Although you might believe it has no application in your life, we all should grasp the concept to nail those examinations in school.

To save your time and create this data easily accessible, here is an introductory insight into the properties of a linear pair of angles, with visualizations and examples to help with your private study sessions. We will also talk about some real-world and geometric uses.

What Is a Linear Pair of Angles?

Linearity, angles, and intersections are ideas that exist to be useful as you progress in geometry and more sophisticated theorems and proofs. We will answer this query with a straightforward explanation in this unique point.


A linear pair of angles is the term designated to two angles that are situated on a straight line and have the sum of their measurement of angles is 180 degrees.

To put it simply, linear pairs of angles are two angles that sit on the same line and together create a straight line. The total of the angles in a linear pair will at all times make a straight angle equal to 180 degrees.

It is essential to note that linear pairs are always at adjacent angles. They share a common vertex and a common arm. This means that they always create on a straight line and are always supplementary angles.

It is essential to explain that, although the linear pair are constantly adjacent angles, adjacent angles not at all times linear pairs.

The Linear Pair Axiom

Through the definition simplified, we will examine the two axioms earnestly to completely comprehend any example provided to you.

Initiate with definition of what an axiom is. It is a mathematical postulate or assumption that is approved without having to prove; it is considered clear and self-explanatory. A linear pair of angles has two axioms linked with them.

The first axiom establishes that if a ray is located on a line, the adjacent angles will make a straight angle, also known as 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 between them. This is commonly called a straight line.

Examples of Linear Pairs of Angles

To imagine these axioms better, here are some drawn examples with their individual explanations.

Example One

As we can see in this example, we have two angles that are adjacent to one another. As you can observe in the diagram, the adjacent angles form a linear pair because the sum of their measurement equals 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

In this instance, we possess two lines intersect, producing four angles. Not all angles creates a linear pair, but each angle and the one adjacent 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 instance shows convergence 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 combinations sum up to 180 degrees. As a effect, we can come to the conclusion that this diagram has no linear pair unless we stretch a straight line.

Uses of Linear Pair of Angles

At the moment we have gone through what linear pairs are and have looked at some examples, let’s check how this concept can be implemented in geometry and the real-life scenario.

In Real-World Situations

There are many utilizations of linear pairs of angles in real-world. One such case is architects, who use these axioms in their daily job to determine if two lines are perpendicular and makes a straight angle.

Construction and Building professionals also utilize expertise in this field to make their work simpler. 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 working out the pressure on the beams and trusses.

In Geometry

Linear pairs of angles also perform a role in geometry proofs. A regular proof that uses linear pairs is the alternate interior angles theorem. This theorem expresses that if two lines are parallel and intersected by a transversal line, the alternate interior angles formed are congruent.

The proof of vertical angles as well depends 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 one another. Because of these two rules, you are only required to know the measure of one angle to determine the measurement of the rest.

The concept of linear pairs is further employed for more complex applications, such as measuring the angles in polygons. It’s important to understand the basics of linear pairs, so you are prepared for more advanced geometry.

As you can see, linear pairs of angles are a somewhat easy theorem with several engaging implementations. Next time you're out and about, see if you can spot any linear pairs! And, if you're attending a geometry class, be on the lookout for how linear pairs may be helpful in proofs.

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