# Linear Pair of AnglesDefinition, Axiom, Examples

The linear pair of angles is a significant subject in geometry. With multiple real-life applications, you'd be surprised to figure out how applicable this figure can be. Although you might think it has no use in your life, we all should grasp the ideas to nail those examinations in school.

To save your time and create this information easy to access, here is an preliminary insight into the properties of a linear pair of angles, with diagrams and examples to assist with your personal study sessions. We will also talk about few real-world and geometric uses.

## What Is a Linear Pair of Angles?

Linearity, angles, and intersections are theories that exist to be relevant as you go forward in geometry and more complicated theorems and proofs. We will answer this question with a easy definition in this unique point.

### Definition

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 easily, linear pairs of angles are two angles that are aligned on the same line and pair up to create a straight line. The sum of the angles in a linear pair will always produce a straight angle equal to 180 degrees.

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

It is crucial to make clear 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 study the two axioms seriously to fully grasp any example provided to you.

First, let's define what an axiom is. It is a mathematical postulate or assumption that is accepted without proof; it is believed clear and self-explanatory. A linear pair of angles has two axioms associated with them.

The first axiom implies that if a ray stands on a line, the adjacent angles will create 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 makes a straight angle between them. This is also known as a straight line.

## Examples of Linear Pairs of Angles

To imagine these axioms better, here are a few drawn examples with their respective explanations.

### Example One

Here in this instance, we have two angles that are neighboring each other. As you can observe in the diagram, the adjacent angles form a linear pair since the sum of their measurement is equivalent to 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, creating four angles. Not all angles form a linear pair, but respective 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 example, the linear pairs are:

∠A and ∠B

∠B and ∠C

∠C and ∠D

∠D and ∠A

### Example Three

This example presents 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 add up to 180 degrees. As a effect, we can come to the conclusion that this example has no linear pair unless we extend one straight line.

## Applications of Linear Pair of Angles

At the moment we have gone through what linear pairs are and have looked at some examples, let's see how this theorem can be utilized in geometry and the real world.

### In Real-Life Scenarios

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

Construction and Building professionals also employ masters in this matter to make their work simpler. They utilize linear pairs of angles to assure that two adjacent walls create a 90-degree angle with the ground.

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

### In Geometry

Linear pairs of angles also play a function in geometry proofs. A ordinary proof that uses 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 formed are congruent.

The proof of vertical angles also depends on linear pairs of angles. Even though 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 only need to determine the measure of one angle to figure out the measure of the rest.

The concept of linear pairs is further employed for more complicated applications, such as figuring out the angles in polygons. It’s critical 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 relatively easy concept with some fascinating implementations. Later when you're out and about, see if you can see some linear pairs! And, if you're taking a geometry class, be on the lookout for how linear pairs might be useful in proofs.

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