# Linear Pair of AnglesDefinition, Axiom, Examples

The linear pair of angles is a significant concept in geometry. With multiple real-world functions, you'd be amazed to discover how applicable this figure can be. Although you may believe it has no relevant in your life, we all should learn the ideas to ace those examinations in school.

To save you time and offer this information easily accessible, here is an preliminary insight into the properties of a linear pair of angles, with visualizations and examples to help with your personal study sessions. We will also discuss some real-life and geometric applications.

## What Is a Linear Pair of Angles?

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

### Explanation

A linear pair of angles is the term given to two angles that are situated 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 form a straight line. The sum of the angles in a linear pair will at all times create 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 apex and a common arm. This implies that they always form on a straight line and are at all times supplementary angles.

It is essential to explain that, although the linear pair are at all times adjacent angles, adjacent angles aren't always linear pairs.

### The Linear Pair Axiom

Through the precise explanation, we will study the two axioms seriously to completely understand every example provided to you.

Let’s start by defining what an axiom is. It is a mathematical postulate or assumption that is acknowledged without having to prove; it is deemed obvious and self-explanatory. 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 implies that if two angles create a linear pair, then uncommon arms of both angles create a straight angle between them. This is also known as a straight line.

## Examples of Linear Pairs of Angles

To envision these axioms better, here are a few diagram examples with their individual answers.

### Example One

As we can see in this example, we have two angles that are neighboring each other. As you can notice in the diagram, the adjacent angles form a linear pair due to the fact that the total of their measurement 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 possess two lines intersect, creating four angles. Not every angles creates a linear pair, but each angle and the one close 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 example represents 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 sum up to 180 degrees. As a consequence, we can come to the conclusion that this diagram has no linear pair until we expand one straight line.

## Implementations of Linear Pair of Angles

Now that we have learned what linear pairs are and have looked at some instances, let’s understand how this theorem can be used in geometry and the real world.

### In Real-World Scenarios

There are multiple implementations of linear pairs of angles in real-world. One such case is architects, who apply these axioms in their day-to-day job to identify if two lines are perpendicular and creates a straight angle.

Construction and Building professionals also use experts in this field to make their job easier. They utilize linear pairs of angles to make sure that two adjacent walls form a 90-degree angle with the ground.

Engineers also uses 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 common proof that utilizes 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 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 equivalent to each other. Because of these two rules, you are only required to figure out the measure of one angle to figure out the measure of the rest.

The concept of linear pairs is subsequently used for more sophisticated uses, such as determining the angles in polygons. It’s important to understand the fundamentals of linear pairs, so you are ready for more progressive geometry.

As demonstrated, linear pairs of angles are a relatively simple theorem with several engaging implementations. Later when you're out and about, see if you can notice some linear pairs! And, if you're participating in a geometry class, bear in mind how linear pairs may be useful in proofs.

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