October 04, 2022

Linear Pair of AnglesDefinition, Axiom, Examples

The linear pair of angles is an essential concept in geometry. With multiple real-life applications, you'd be astonished to figure out how relevant this figure can be. Although you might think it has no use in your life, we all should understand the concept to nail those examinations in school.

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

What Is a Linear Pair of Angles?

Linearity, angles, and intersections are theories that continue to be relevant as you go forward in geometry and more complex theorems and proofs. We will answer this question with a simple explanation in this single point.


A linear pair of angles is the name designated 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 aligned on the same line and together create a straight line. The sum of the angles in a linear pair will at all times make a straight angle equivalent
times to 180 degrees.

It is essential to note that linear pairs are at all times at adjacent angles. They share a common vertex and a common arm. This implies that at all times form on a straight line and are at all times supplementary angles.

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

The Linear Pair Axiom

Over the definition simplified, we will examine the two axioms critical to fully understand any example given to you.

Initiate with definition of what an axiom is. It is a mathematical postulate or hypothesis that is accepted without proof; it is deemed obvious and self-evident. A linear pair of angles has two axioms connected 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 establishes 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 envision these axioms better, here are some diagram examples with their individual explanations.

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 image, the adjacent angles form a linear pair because the total 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 have two lines intersect, producing four angles. Not all angles creates a linear pair, but respective 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 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 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 diagram has no linear pair until we extend a straight line.

Implementations of Linear Pair of Angles

At the moment we have explored what linear pairs are and have looked at some examples, let’s check how this concept can be used in geometry and the real world.

In Real-Life Situations

There are many applications of linear pairs of angles in real-world. One common example is architects, who apply these axioms in their daily work to identify if two lines are perpendicular and creates a straight angle.

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

Engineers also utilizes linear pairs of angles frequently. They do so by working out the weight on the 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 concept. This concept explains that if two lines are parallel and intersected by a transversal line, the alternate interior angles made are congruent.

The proof of vertical angles additionally replies on linear pairs of angles. Although the adjacent angles are supplementary and sum up to 180 degrees, the opposite vertical angles are always equal to each other. Because of previously mentioned two rules, you only need to figure out the measurement of any one angle to determine the measure of the rest.

The theorem of linear pairs is further employed for more complicated implementation, such as measuring the angles in polygons. It’s critical to understand the basics of linear pairs, so you are ready for more progressive geometry.

As you can see, linear pairs of angles are a comparatively simple concept with some fascinating implementations. Next time you're out and about, take note if you can spot any linear pairs! And, if you're attending a geometry class, take notes on how linear pairs may be useful in proofs.

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