# Linear Pair of AnglesDefinition, Axiom, Examples

The linear pair of angles is an essential concept in geometry. With several real-life applications, you'd be surprised to find how relevant this figure can be. Even though you may think it has no use in your life, we all must understand the concept to nail those examinations in school.

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

## What Is a Linear Pair of Angles?

Linearity, angles, and intersections are concepts that continue to be applicable as you progress in geometry and more complex theorems and proofs. We will answer this query with a easy explanation in this unique point.

### Explanation

A linear pair of angles is the term designated to two angles that are located 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 are adjacent on the same line and together create a straight line. The sum of the angles in a linear pair will always create a straight angle equal to 180 degrees.

It is essential to keep in mind that linear pairs are always at adjacent angles. They share a common apex and a common arm. This means that they always make on a straight line and are at all times supplementary angles.

It is crucial to clarify that, while the linear pair are always adjacent angles, adjacent angles not at all times linear pairs.

### The Linear Pair Axiom

Over the definition clarified, we will explore the two axioms earnestly to completely grasp any example provided to you.

Initiate with definition of what an axiom is. It is a mathematical postulate or assumption that is acknowledged without proof; it is believed obvious and self-evident. A linear pair of angles has two axioms linked with them.

The first axiom establishes that if a ray stands on a line, the adjacent angles will create a straight angle, namely called a linear pair.

The second axiom implies that if two angles makes 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 corresponding explanations.

### Example One

As we can see in this example, we have two angles that are adjacent to each other. As you can observe in the image, the adjacent angles form a linear pair since the total of their measurement is equivalent to 180 degrees. They are also supplementary angles, since 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, making four angles. Not every angles creates 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 case, the linear pairs are:

∠A and ∠B

∠B and ∠C

∠C and ∠D

∠D and ∠A

### Example Three

This instance presents a common point of three lines. Let's observe the axiom and characteristics of linear pairs.

∠A 150 degrees

∠B: 50 degrees

∠C: 160 degrees

None of the angle totals add up to 180 degrees. As a result, we can conclude that this figure has no linear pair unless we expand a straight line.

## Uses of Linear Pair of Angles

At the moment we have learned what linear pairs are and have observed some cases, let's see how this theorem can be utilized in geometry and the real world.

### In Real-World Scenarios

There are several implementations of linear pairs of angles in real life. One common example is architects, who use these axioms in their day-to-day work to determine if two lines are perpendicular and makes a straight angle.

Builders and construction professionals also use expertise in this matter to make their job less complex. They utilize linear pairs of angles to assure that two close walls form a 90-degree angle with the floor.

Engineers also utilizes linear pairs of angles regularly. They do so by figuring out the tension in beams and trusses.

### In Geometry

Linear pairs of angles as well perform a role in geometry proofs. A ordinary proof that utilizes linear pairs is the alternate interior angles theorem. This concept explains that if two lines are parallel and intersected by a transversal line, the alternate interior angles formed are congruent.

The proof of vertical angles additionally replies on linear pairs of angles. While the adjacent angles are supplementary and add up to 180 degrees, the opposite vertical angles are at all times equal to one another. Because of above-mentioned two rules, you only need to know the measurement of one angle to figure out the measurement of the rest.

The theorem of linear pairs is subsequently utilized for more complicated implementation, such as figuring out 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 relatively easy theorem with several engaging applications. Next time you're out and about, observe if you can spot some linear pairs! And, if you're participating in a geometry class, be on the lookout for how linear pairs may be useful in proofs.

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